{"text":"\\section{Introduction}\n\nArbitrary-spin massless particles are expected to play a crucial role in the understanding of Quantum Gravity. Lower-spin theories may be realized as low-energy limits of\nspontaneously-broken higher-spin gauge theories since lower-spin symmetries are subgroups of higher-spin ones. It is believed that the tensionless limit of string theory\nis a theory of higher-spin gauge fields. The study of fermionic fields is interesting in this regard because they are required by supersymmetry.\n\nHigher-spin gauge fields can be described in the framework of two different formulations: frame-like and metric-like. The frame-like formulation generalizes the Cartan formulation\nof gravity where the gauge fields are described in terms of differential forms carrying irreducible representations of the fiber Lorentz group. This is available\nin Minkowski~\\cite{V-flat,hygra,AD} as well as in Anti-de Sitter (AdS)~\\cite{Vasiliev:1986td,Lopatin:1987hz,V-AdS,Vasiliev:2001wa} spaces.\nThe metric-like formulation, on the other hand, is a generalization of the metric formulation of linearized gravity~\\cite{deWit}. Originally developed by\nFronsdal~\\cite{Fronsdal:1978rb,Fronsdal:1978vb} and Fang--Fronsdal~\\cite{FF,Fang:1979hq}, it encodes the degrees of freedom of higher-spin particles in symmetric tensors and tensor-spinors.\nIn this approach, the construction of a gauge-invariant action for a higher-spin field requires that the field and the gauge parameter obey some off-shell algebraic constraints\n(see~\\cite{Rahman:2015pzl,Campoleoni:2017vds} for a recent review). Note that the latter requirement can be avoided by recourse to other formulations~\\cite{Francia:2002aa,Bekaert:2003az,Buchbinder:2004gp,Francia:2005bu,Bekaert:2006ix,Francia:2007qt,Buchbinder:2007ak,Buchbinder:2007vq,Francia:2007ee,Campoleoni:2009gs} (see\nAppendix~\\ref{sec:A}).\n\nBoth these approaches are geometric, albeit in different manners, in that the frame-like formulation extends Cartan geometry whereas the metric-like formulation extends Riemannian geometry.\nThe latter is however a particular gauge of the former just like in the case of gravity. The construction of interacting theories for higher-spin fields, fermions in particular,\nappears to be in dire need of the frame-like formulation. The metric-like formulation, in contrast, seems rather clumsy in managing the non-linearities required by gauge-theoretic consistency.\nYet it has the advantage of having a simplified field content that may make some features of the interactions more transparent. Understanding the connections\nbetween the two \nmay therefore provide valuable information~\\cite{Campoleoni:2012hp,Fredenhagen:2014oua,Campoleoni:2014tfa,Boulanger:2015ova}.\n\nIn this article, we will focus exclusively on higher-spin gauge fermions. These fields appear naturally in the supersymmetric versions of Vasiliev\ntheory~\\cite{Konstein:1989ij,Vasiliev:1990vu,Vasiliev:1992av,Sezgin:1998gg,Sezgin:1998eh,Sezgin:2002ru,Engquist:2002vr} (see~\\cite{Sezgin:2012ag} for a recent review) and also\nin the tensionless limit of superstring theory compactified on AdS$_5\\times S^5$. The frame-like formulation of gauge fermions~\\cite{V-flat,AD,hygra,V-AdS} has been discussed more\nrecently by various authors~\\cite{Alkalaev:2001mx,Alkalaev:2006hq,Sorokin:2008tf,Z,Z2,Skvortsov:2010nh}.\nThe Fang-Fronsdal metric-like approach for higher-spin fermions, on the other hand, has been studied in\narbitrary dimensions in Ref.~\\cite{Hallowell:2005np,Metsaev:2006zy,Metsaev:2013wza}. We will consider the free theory of a spin $s=n+\\tfrac{1}{2}$ massless fermionic field\nin flat and AdS spaces. Although we consider Majorana fermions for simplicity, our main results are valid almost verbatim for Dirac fermions in arbitrary spacetime dimensions.\nA crucial property of frame-like fermions in flat space is their shift symmetry w.r.t.~a gauge parameter which is an irreducible tensor-spinor in the fiber space with the symmetry\nproperty of the Young diagram $\\mathbb{Y}(n-1,1)$. This symmetry makes it almost manifest that the free Lagrangian is equivalent to that of the metric-like formulation~\\cite{V-flat}.\nIn AdS space, however, the constraints on this parameter may receive nontrivial corrections which vanish in the flat limit~\\cite{Sorokin:2008tf,Z}. This is tantamount to having no such\ncorrections provided that some appropriate mass-like terms appear in the gauge transformation. In other words, one can have a gauge-invariant Lagrangian description for frame-like fermions\nin AdS space that does not deform of the flat-space constraints on the field and the gauge parameters.\n\nThe organization of this article is as follows. In the remaining of this section we spell out our notations and conventions. A review of frame-like higher-spin massless fermions in flat space\nappears in Section~\\ref{sec:FFF}, where we write down the free Lagrangian~\\cite{Z,Skvortsov:2010nh} and discuss its gauge symmetries along with the constraints on the field and the gauge parameters.\nWe also show how this theory simplifies in $D=3, 4$. Section~\\ref{sec:FFA} formulates the free theory in AdS space with a trivial but convenient modification of the well-known mass-like\nterm~\\cite{Sorokin:2008tf,Z}. By virtue of judiciously-chosen terms in the gauge transformation, we ensure that the constraints on the field and the gauge parameters mimic their flat-space counterparts.\nThe value of the mass parameter, determined uniquely by gauge invariance, is in complete agreement with the known results~\\cite{Metsaev:2013wza,Metsaev:2003cu}.\nIn Section~\\ref{sec:Eqv}, we demonstrate explicitly the equivalence of the frame-like Lagrangian to the metric-like one at the free level. We conclude in Section~\\ref{sec:remarks} with some remarks,\nespecially on the subtleties\nthat may arise in an interacting theory. An appendix summarizes the essentials of the metric-like formulation of higher-spin gauge fermions.\n\n\\subsubsection*{Conventions \\& Notations}\\label{subsec:convnot}\nWe adopt the conventions of Ref.~\\cite{Freedman:2012zz}, with mostly positive metric signature $(-+\\cdots+)$.\nThe expression $(i_1\\cdots i_n)$  denotes a totally symmetric one in all the indices $i_1,\\cdots,i_n$ with no normalization factor, e.g., $(i_1i_2)=i_1i_2+i_2i_1$ etc.\nThe totally antisymmetric expression $[i_1\\cdots i_n]$ has the same normalization.\nThe number of terms appearing in the (anti-)symmetrization is assumed to be the possible minimum.\nA prime will denote a trace w.r.t. the background metric, e.g., $A^{\\prime}=\\bar{g}^{\\mu\\nu}A_{\\mu\\nu}=A^{\\mu}_{~\\mu}$.\nThe Levi-Civita symbol is normalized as $\\varepsilon_{01\\ldots D-1}= +1$, where $D$ is the spacetime dimension.\n\nFiber indices and world indices will respectively be denoted with lower case Roman letters and Greek letters.\nRepeated indices with the same name (appearing all as either covariant or contravariant ones)\nare (anti-)symmetrized with the minimum number of terms. This results in the following rules: $a(k)a=aa(k)=(k+1)a(k+1)$,~\n$a(k)a(2)=a(2)a(k)=\\binom{k+2}{2}\\,a(k+2)$,~ $a(k)a(k')=a(k')a(k)=\\binom{k+k'}{k}\\,a(k+k')$ etc,\nwhere $a(k)$ has a unit weight by convention, and so the proportionality coefficient gives the weight of the right hand side.\n\nThe $\\gamma$-matrices satisfy the Clifford algebra: $\\{\\gamma^a,\\gamma^b\\}=+2\\eta^{ab}$, and $\\gamma^{a\\,\\dagger}=\\eta^{aa}\\gamma^a$.\nTotally antisymmetric products of $\\gamma$-matrices, $\\gamma^{a_1\\ldots a_r}=\\tfrac{1}{r!}\\gamma^{[a_1}\\gamma^{a_2}\\cdots\\gamma^{a_r]}$, have unit weight.\nA ``slash'' will denote a contraction with $\\gamma$-matrix, e.g., $\\displaystyle{\\not{\\!\\!A\\!\\,}}=\\gamma^a A_a$.\n\nA Majorana spinor $\\chi$ obeys the reality condition: $\\chi^C=\\chi$. Two Majorana spinors $\\chi_{1,\\,2}$ follow the bilinear identity:\n$\\bar\\chi_1\\gamma^{a_1\\ldots a_r}\\chi_2=t_r\\,\\bar\\chi_2\\gamma^{a_1\\ldots a_r}\\chi_1$, where a ``bar'' denotes Majorana conjugation, and $t_r=\\pm 1$,\ndepending on the value of $r$ and spacetime dimensionality~\\cite{Freedman:2012zz}.\n\n\\section{Frame-like Fermions in Flat Space}\\label{sec:FFF}\n\nIn the frame-like formulation, a fermion of spin $s=n+\\tfrac{1}{2}$ is described by a vielbein-like 1-form $\\Psi^{a(n-1)}$, which is\na symmetric rank-($n-1$) irreducible tensor-spinor in the fiber space:\n\\begin{equation}\\label{fr1}\n\\Psi^{a(n-1)}=\\Psi_\\mu{}^{a(n-1)}dx^\\mu,\\qquad \\gamma_a\\Psi^{ab(n-2)}=0.\n\\end{equation}\nThe Minkowski background is described by the vielbein $\\bar{e}^{\\,a}=\\bar{e}_\\mu^{\\,a}dx^\\mu$ that satisfies $\\eta_{ab}\\bar{e}_\\mu^{\\,a}\\bar{e}_\\nu^{\\,b}=\\eta_{\\mu\\nu}$, and the spin-connection\n$\\bar{\\omega}^{ab}=\\bar{\\omega}_\\mu{}^{ab}dx^\\mu=-\\bar{\\omega}_\\mu{}^{ba}dx^\\mu$, which fulfill the following equations:\n\\begin{equation}\\label{fr2}\nT^a\\equiv d\\bar{e}^{\\,a}+\\bar{\\omega}^a{}_b\\bar{e}^{\\,b}=0,\\qquad \\rho^{ab}\\equiv d\\bar{\\omega}^{ab}+\\bar{\\omega}^a{}_c\\bar{\\omega}^{cb}=0.\n\\end{equation}\nIn the Cartesian coordinates, in particular, the solution of Eqs.~(\\ref{fr2}) is given by $\\bar{e}_\\mu^{\\,a}=\\delta_\\mu^a$ and $\\bar{\\omega}_\\mu{}^{ab}=0$. We will however work\nwith a generic coordinate system in order to facilitate the transition to AdS space. The following quantities will be useful in the subsequent discussion:\n\\begin{eqnarray}\n  {}^*\\bar{e}_{a_1}\\ldots\\bar{e}_{a_p}&\\equiv&\\tfrac{1}{(D-p)!}\\,\\epsilon_{a_1\\ldots a_pa_{p+1}\\ldots a_D}\\bar{e}^{\\,a_{p+1}}\\ldots\\bar{e}^{\\,a_D},\\label{fr2a}\\\\\n  \\eta^{a_1a_2|b_1b_2}&\\equiv&\\tfrac{1}{2}\\left(\\eta^{a_1b_1}\\eta^{a_2b_2}-\\eta^{a_1b_2}\\eta^{a_2b_1}\\right).\\label{fr2b}\n\\end{eqnarray}\nThe frame-like free action for a Majorana gauge fermion, in arbitrary dimensions\\footnote{Majorana fermions exist in $D=3,4,8,9,10$ and $11$. In dealing with such objects it is\nimportant to assume the anti-commuting nature of fermions already at the classical level (before quantization).}, reads~\\cite{Z,Skvortsov:2010nh}:\n\\begin{equation}\\label{fr3}\nS=-\\tfrac{1}{2}\\int\\left[\\bar{\\Psi}_{b_1c(n-2)}\\mathcal{A}^{a_1a_2a_3,\\,b_1b_2}\\hat{D}\\Psi_{b_2}{}^{c(n-2)}\\right]{}^*\\bar{e}_{a_1}\\bar{e}_{a_2}\\bar{e}_{a_3},\n\\end{equation}\nwhere $\\hat{D}$ denotes the Lorentz covariant derivative, and\n\\begin{equation}\\label{fr4}\n\\mathcal{A}^{a_1a_2a_3,\\,b_1b_2}\\equiv\\tfrac{1}{6n}\\left(\\gamma^{a_1a_2a_3}\\eta^{b_1b_2}+2(n-1)\\eta^{b_1b_2|[a_1a_2}\\gamma^{a_3]}\\right).\n\\end{equation}\nThe action~(\\ref{fr3}) enjoys the following gauge invariance:\n\\begin{equation}\\label{gauge-fr0}\n\\delta \\Psi^{a(n-1)}=\\hat{D}\\zeta^{a(n-1)}+\\bar{e}_b\\lambda^{b,\\,a(n-1)},\n\\end{equation}\nwhere the 0-form gauge parameters $\\zeta^{a(n-1)}$ and $\\lambda^{b,\\,a(n-1)}$ are irreducible tensor-spinors of rank ($n-1$) and rank $n$ respectively with the symmetry of the Young diagrams\n$\\mathbb{Y}(n-1)$ and $\\mathbb{Y}(n-1,1)$, i.e.,\n\\begin{equation}\\label{YD-1}\n\\zeta^{a(n-1)}\\sim\\begin{aligned}\n&\\underbrace{\\begin{tabular}{|c|c|c|c|}\\hline\n  $\\phantom{a}$&\\multicolumn{2}{|c|}{$~\\cdots~$}&\\phantom{a}\\\\\\hline\n\\end{tabular}}_{n-1}\n\\end{aligned}~,\n\\qquad\\qquad\n\\lambda^{b,\\,a(n-1)}\\sim\\overbrace{\\begin{aligned}\n&\\begin{tabular}{|c|c|c|c|}\\hline\n   $\\phantom{a}$&\\multicolumn{2}{|c|}{$~\\cdots~$}&\\phantom{a}\\\\\\hline\n\\end{tabular}\\\\[-4pt]\n&\\begin{tabular}{|c|}\n   $\\phantom{a}$\\\\\\hline\n\\end{tabular}\n\\end{aligned}}^{n-1}~.\n\\end{equation}\nThese irreducible tensor-spinors are subject to the following constraints:\n\\begin{equation}\\label{identity-fr1}\n\\gamma_b\\zeta^{ba(n-2)}=0,\\qquad \\gamma_b\\lambda^{b,\\,a(n-1)}=0,\\qquad \\gamma_c\\lambda^{b,\\,ca(n-2)}=0,\\qquad \\lambda^{a,\\,a(n-1)}=0.\n\\end{equation}\n\nIt is obvious that the action~(\\ref{fr3}) is invariant, up to a total derivative term, under the gauge transformation of the parameter $\\zeta^{a(n-1)}$, since $\\hat{D}^2=0$ in flat space.\nTo prove the shift symmetry w.r.t. the parameter $\\lambda^{b,\\,a(n-1)}$, let us make use of the identity: $\\bar{e}^c{}^*\\bar{e}_{a_1}\\bar{e}_{a_2}\\bar{e}_{a_3}={}^*\\bar{e}_{[a_1}\\bar{e}_{a_2}\\delta_{a_3]}^c$,\nso that the variation of the action can be written as\n\\begin{equation}\\label{gauge-fr2}\n\\delta_\\lambda S=-3\\int\\left[\\bar{\\Psi}_{b_1}{}^{c(n-2)}\\mathcal{A}^{a_1a_2a_3,\\,b_1b_2}\\hat{D}\\lambda_{a_3,\\,b_2c(n-2)}\\right]{}^*\\bar{e}_{a_1}\\bar{e}_{a_2}.\n\\end{equation}\nNow, let us take a careful look at the identity:\n\\begin{eqnarray}\n  6n\\mathcal{A}^{a_1a_2a_3,\\,b_1b_2}&=&\\left(\\gamma^{a_1a_2}\\eta^{b_1b_2}+2(n-1)\\eta^{a_1a_2|b_1b_2}\\right)\\gamma^{a_3}+(n-1)\\gamma^{[a_1}\\eta^{a_2]b_1}\\eta^{a_3b_2}\\nonumber\\\\\n  &&~~~~~~~~~~~~~~~~~~~~~~~~-\\gamma^{[a_1}\\eta^{a_2]a_3}\\eta^{b_1b_2}-(n-1)\\gamma^{[a_1}\\eta^{a_2]b_2}\\eta^{a_3b_1}.\\label{fr5c}\n\\end{eqnarray}\nWhen plugged into the gauge variation~(\\ref{gauge-fr2}), the first line on the right hand side of this identity gives vanishing contribution on account of the\n$\\gamma$-trace constraints~(\\ref{identity-fr1}) on the gauge parameter $\\lambda^{b,\\,a(n-1)}$. The two terms in the second line, on the other hand, cancel each other, thanks\nto the property $\\lambda^{a,\\,a(n-1)}=0$. This proves the shift symmetry since $\\delta_\\lambda S=0$.\n\nLet us count the number independent of components of the parameters $\\zeta^{a(n-1)}$ and $\\lambda^{b,\\,a(n-1)}$. Because the frame indices are $\\gamma$-traceless,\nthe number of possible values each index can take is essentially ($D-1$). Then it is easy to compute the number of components of the corresponding Young diagrams~(\\ref{YD-1});\nthey respectively turn out to be $\\binom{D+n-3}{n-1}f_D$ and $(n-1)\\binom{D+n-3}{n}f_D$, where\n\\begin{equation}\\label{fD-defined} f_D\\equiv2^{D\/2+((-)^D-5)\/4},\\end{equation}\nfor a Majorana fermion in $D$ dimensions. On the other hand, one needs to take into account the vanishing of the trace when one contracts two indices from different rows of\n$\\lambda^{b,\\,a(n-1)}$, which removes $\\binom{D+n-4}{n-2}f_D$ components. Therefore, the total numbers are given by\n\\begin{equation}\\label{z-dof} \\Delta_\\zeta=\\binom{D+n-3}{n-1}f_D,\\qquad \\Delta_\\lambda=(n-1)\\binom{D+n-3}{n}f_D-\\binom{D+n-4}{n-2}f_D.\\end{equation}\nThis counting will be useful later on.\n\n\\subsubsection*{\\underline{Special Case: $D=3$}}\n\nThe case of $D=3$ is important in the context of hypergravity theories~\\cite{hygra} (see also~\\cite{Troncoso} for a recent discussion). In this case,\nnote that the quantity ${}^*\\bar{e}_{a_1}\\bar{e}_{a_2}\\bar{e}_{a_3}$ reduces to the Levi-Civita tensor $\\epsilon_{a_1a_2a_3}$. Furthermore, one has at one's disposal the useful\n$D$-dimensional identity:\n\\begin{equation}\\label{fr5b}\n  \\mathcal{A}^{a_1a_2a_3,\\,b_1b_2}=\\tfrac{1}{6}\\gamma^{a_1a_2a_3}\\eta^{b_1b_2}+\\left(\\tfrac{n-1}{6n}\\right)\\gamma^{a_1a_2a_3b_1b_2}-\\left(\\tfrac{n-1}{12n}\\right)\\left(\\gamma^{b_1}\\gamma^{b_2}\\gamma^{a_1a_2a_3}+\\gamma^{a_1a_2a_3}\\gamma^{b_1}\\gamma^{b_2}\\right).\n\\end{equation}\nThe second term on the right hand side in the above identity is zero in $D=3$, whereas the last term gives vanishing contribution because of the $\\gamma$-trace condition on the field.\nOn account of the relation: $\\gamma^{a_1a_2a_3}\\epsilon_{a_1a_2a_3}=(3!)\\mathbb{I}$, therefore, the action~(\\ref{fr3}) reduces to the well-known Aragone-Deser form~\\cite{hygra}:\n\\begin{equation}\\label{fr6}\nS_{D=3}=-\\tfrac{1}{2}\\int\\bar{\\Psi}_{a(n-1)}\\hat{D}\\Psi^{a(n-1)}.\n\\end{equation}\nOn the other hand, the gauge symmetry~(\\ref{gauge-fr0})--(\\ref{identity-fr1}) reduces to\n\\begin{equation}\\label{gauge3D}\n\\delta \\Psi^{a(n-1)}=\\hat{D}\\zeta^{a(n-1)},\\qquad \\gamma_b\\zeta^{ba(n-2)}=0.\n\\end{equation}\nThis is because in $D=3$ the shift parameter $\\lambda^{b,\\,a(n-1)}$ is trivial but $\\zeta^{a(n-1)}$ is not,\n\\begin{equation}\\label{bagh} \\Delta_\\lambda=0,\\qquad \\Delta_\\zeta=n,\\end{equation}\nas one can easily see from Eq.~(\\ref{z-dof}).\n\n\\subsubsection*{\\underline{Special Case: $D=4$}}\n\nIn this case, the quantity ${}^*\\bar{e}_{a_1}\\bar{e}_{a_2}\\bar{e}_{a_3}$ reduces to the 1-form $\\epsilon_{a_1a_2a_3b}\\bar{e}^{\\,b}$, while only the first piece on the right hand side of the identity~(\\ref{fr5b})\ncontributes. Then the dimension-dependent identity: $\\gamma^{a_1a_2a_3}=-i\\epsilon^{a_1a_2a_3b}\\gamma_5\\gamma_{b}$, reduces the action~(\\ref{fr3}) to\n\\begin{equation}\\label{fr7}\nS_{D=4}=-\\tfrac{i}{2}\\int\\bar{\\Psi}_{a(n-1)}\\gamma_5\\gamma_b\\bar{e}^{\\,b}\\hat{D}\\Psi^{a(n-1)}.\n\\end{equation}\nBecause $\\Delta_\\zeta=n(n+1)\\neq0,~\\Delta_\\lambda=(n-1)(n+2)\\neq0$, both the parameters $\\zeta^{a(n-1)}$ and $\\lambda^{b,\\,a(n-1)}$ are nontrivial, and so the gauge symmetry has the full general form of~(\\ref{gauge-fr0}).\nThe Lagrangian~(\\ref{fr7}) appeared in both Ref.~\\cite{V-flat} and~\\cite{AD}, but only the former reference could correctly identify the gauge symmetries.\n\n\\section{Frame-like Fermions in AdS Space}\\label{sec:FFA}\n\nThe AdS background is described by the vielbein $\\bar{e}^{\\,a}=\\bar{e}_\\mu^{\\,a}dx^\\mu$ that satisfies $\\eta_{ab}\\bar{e}_\\mu^a\\bar{e}_\\nu^b=\\bar{g}_{\\mu\\nu}$, and the spin-connection\n$\\bar{\\omega}^{ab}=\\bar{\\omega}_\\mu{}^{ab}dx^\\mu=-\\bar{\\omega}_\\mu{}^{ba}dx^\\mu$, which fulfill the following equations:\n\\begin{equation}\\label{fr2AdS}\nT^a\\equiv d\\bar{e}^{\\,a}+\\bar{\\omega}^a{}_b\\bar{e}^{\\,b}=0,\\qquad \\rho^{ab}\\equiv d\\bar{\\omega}^{ab}+\\bar{\\omega}^a{}_c\\bar{\\omega}^{cb}=-\\frac{1}{l^2}\\bar{e}^{\\,a}\\bar{e}^{\\,b},\n\\end{equation}\nwhere $l$ is the AdS radius.\nLet us write the free action for a Majorana gauge fermion in AdS space by augmenting the kinetic term, already studied in the context of flat space, by a mass term:\n\\begin{eqnarray}\nS&=&-\\tfrac{1}{2}\\int\\left[\\bar{\\Psi}_{b_1c(n-2)}\\mathcal{A}^{a_1a_2a_3,\\,b_1b_2}\\hat{D}\\Psi_{b_2}{}^{c(n-2)}\\right]{}^*\\bar{e}_{a_1}\\bar{e}_{a_2}\\bar{e}_{a_3}\\nonumber\\\\\n&&-\\tfrac{1}{2}\\mu\\int\\left[\\bar{\\Psi}_{b_1c(n-2)}\\mathcal{B}^{a_1a_2,\\,b_1b_2}\\Psi_{b_2}{}^{c(n-2)}\\right]{}^*\\bar{e}_{a_1}\\bar{e}_{a_2},\\label{fr3Ad}\n\\end{eqnarray}\nwhere $\\mu$ is some parameter with the dimensions of mass, to be specified later, and\n\\begin{equation}\\label{fr4Ad}\n\\mathcal{B}^{a_1a_2,\\,b_1b_2}\\equiv\\tfrac{1}{2n}\\left[\\gamma^{a_1a_2}\\eta^{b_1b_2}+2(n-1)\\eta^{a_1a_2|b_1b_2}-\\tfrac{1}{2}\\left(\\tfrac{n-1}{D+2n-4}\\right)\\left(\\gamma^{b_1}\\gamma^{b_2}\\gamma^{a_1a_2}+\\gamma^{a_1a_2}\\gamma^{b_1}\\gamma^{b_2}\\right)\\right].\n\\end{equation}\nNote that our choice of $\\mathcal{B}^{a_1a_2,\\,b_1b_2}$ differs from that of Ref.~\\cite{Sorokin:2008tf,Z} by a trivial term which vanishes upon implementing\nthe constraint on the field. Yet this term will be useful for our purpose.\n\nIt suffices to consider, invoking another mass parameter $\\tilde{\\mu}$, the gauge transformation:\n\\begin{equation}\\label{gauge-fr0Ad}\n\\delta \\Psi^{a(n-1)}=\\hat{D}\\zeta^{a(n-1)}+\\tilde{\\mu}\\bar{e}_b\\left[\\gamma^b\\zeta^{a(n-1)}-\\left(\\tfrac{2}{D+2n-4}\\right)\\gamma^a\\zeta^{a(n-2)b}\\right]+\\bar{e}_b\\lambda^{b,\\,a(n-1)},\n\\end{equation}\nwhich is compatible with the $\\gamma$-trace constraint, $\\gamma_a\\Psi^{ab(n-2)}=0$, on the field without requiring any modification of the\nproperties~(\\ref{YD-1}) and~(\\ref{identity-fr1}) of the gauge parameters. In other words, the choice of this gauge transformation~(\\ref{gauge-fr0Ad}) is such that the field and the gauge\nparameters mimic their flat-space properties. This point is implicit in the choice made in Ref.~\\cite{Sorokin:2008tf,Z}.\n\nTo see that the shift transformation w.r.t.~the parameter $\\lambda^{b,\\,a(n-1)}$ is a symmetry of the Lagrangian~(\\ref{fr3Ad}), let us first note that the invariance of the kinetic term\nfollows exactly the flat-space logic. Then, from the variation of the mass term, we have\n\\begin{equation}\\label{mass-var}\n\\delta_\\lambda S=-2\\mu\\int\\left[\\bar{\\Psi}_{b_1c(n-2)}\\mathcal{B}^{a_1a_2,\\,b_1b_2}\\lambda_{a_2,\\,b_2}{}^{c(n-2)}\\right]{}^*\\bar{e}_{a_1}.\n\\end{equation}\nOn account of the identity:\n\\begin{eqnarray}\n  2n\\mathcal{B}^{a_1a_2,\\,b_1b_2}&=&\\eta^{b_1b_2}\\gamma^{a_1}\\gamma^{a_2}+(n-1)\\eta^{a_1b_1}\\eta^{a_2b_2}-\\tfrac{1}{2}\\left(\\tfrac{n-1}{D+2n-4}\\right)\\left(\\gamma^{a_1a_2b_1}\\gamma^{b_2}+\\eta^{b_2[a_1}\\gamma^{a_2]}\\right)\\nonumber\\\\\n  &&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-\\eta^{a_1a_2}\\eta^{b_1b_2}-(n-1)\\eta^{a_1b_2}\\eta^{a_2b_1},\\label{fr4Ad1}\n\\end{eqnarray}\nwe then see that $\\delta_\\lambda S=0$. The cancellations happen in much the same way as the identity~(\\ref{fr5c}) eliminates contributions from the kinetic term.\n\nThe symmetry requirement of the Lagrangian~(\\ref{fr3Ad}) w.r.t.~the $\\zeta$-transformation in~(\\ref{gauge-fr0Ad}) would relate the mass parameters $\\mu$ and $\\tilde{\\mu}$ to each other\nand with the inverse AdS radius. There are a priori three kinds on contributions resulting from the $\\zeta$-transformation: 2-derivative, 1-derivative and 0-derivative ones. Not surprisingly,\nby virtue of the commutator formula:\n\\begin{equation}\\label{dsq-Ad}\n\\hat{D}^2\\zeta_{a(n-1)}=-\\frac{1}{l^2}\\bar{e}^{\\,b}\\bar{e}^{\\,c}\\left[\\eta_{ab}\\zeta_{ca(n-2)}+\\tfrac{1}{4}\\gamma_{bc}\\zeta_{a(n-1)}\\right],\n\\end{equation}\nthe 2-derivative piece actually reduces to a 0-derivative piece. The explicit computation makes use of the identities:\n$\\bar{e}^{\\,b}\\bar{e}^{\\,c}{}^*\\bar{e}_{a_1}\\bar{e}_{a_2}\\bar{e}_{a_3}={}^*\\bar{e}_{[a_1}\\delta^b_{a_2}\\delta^c_{a_3]}$ and\n$\\bar{e}^{\\,b}{}^*\\bar{e}_{a_1}\\bar{e}_{a_2}={}^*\\bar{e}_{[a_1}\\delta^b_{a_2]}$, and leads straightforwardly to\n\\begin{equation}\\label{AdS-last0} -\\tfrac{(D+2n-3)(D+2n-4)}{4\\,n}\\tfrac{1}{l^2}-\\tfrac{(D-2)(D+2n-3)}{n(D+2n-4)}\\mu\\tilde{\\mu}=0,\\end{equation}\nin order that the even-derivative terms cancel each other. Cancellation of the 1-derivative terms,\non the other hand, requires that the following condition be met:\n\\begin{equation}\\label{mm-rel}-(D-2)\\tilde{\\mu}-\\mu=0.\\end{equation}\nConditions~(\\ref{AdS-last0}) and~(\\ref{mm-rel}) can be combined into the relation:\n\\begin{equation}\\label{AdS-last} \\mu^2 l^2=\\left(n+\\tfrac{D-4}{2}\\right)^2>0,\\end{equation}\nwhich gives, up to a sign, the real mass parameter $\\mu$ in terms of the inverse AdS radius. The parameter $\\tilde{\\mu}$ is then also determined from Eq.~(\\ref{mm-rel}).\nThis uniquely fixes the Lagrangian~(\\ref{fr3Ad}) as well as the gauge transformation~(\\ref{gauge-fr0Ad}) while the field and gauge parameters mimic their respective\nflat-space properties.\n\nThe physical significance of the mass parameter $\\mu$ will be made clear in the next section as we work out the gauge fixed equations of motion. To proceed,\nlet us forgo the language of differential forms and rewrite the action~(\\ref{fr3Ad}) as:\n\\begin{equation}\\label{fr3Ad2}\nS=-\\tfrac{1}{2}\\int d^Dx\\,\\bar{e}\\,\\bar{\\Psi}_{\\mu,\\,ac(n-2)}\\left(6\\mathcal{A}^{\\mu\\rho\\nu,\\,ab}\\hat{D}_\\rho+2\\mu \\mathcal{B}^{\\mu\\nu,\\,ab}\\right)\\Psi_{\\nu,\\,b}{}^{c(n-2)},\n\\end{equation}\nwhere $\\bar{e}\\equiv\\det\\bar{e}_\\mu^{\\,a}$ is the determinant of the background AdS vielbein. The resulting Lagrangian equations of motion for the frame-like fermion field\n$\\Psi_{\\mu,\\,a(n-1)}$ take the form:\n\\begin{equation}\\label{eom1}\n\\mathcal{R}^{\\mu,\\,a(n-1)}\\equiv\\left(\\tfrac{6}{n-1}\\right)\\left(\\mathcal{A}^{\\mu\\rho\\nu,\\,ab}\\hat{D}_\\rho+\\tfrac{1}{3}\\mu\\,\\mathcal{B}^{\\mu\\nu,\\,ab}\\right)\\Psi_{\\nu,\\,b}{}^{a(n-2)}=0.\n\\end{equation}\nHere, the normalization factor keeps the equations of motion well defined also for $n=1$, as we will see.\nWe emphasize that the equations of motion~(\\ref{eom1}) are $\\gamma$-traceless in the fiber indices, i.e.,\n\\begin{equation}\\label{xxx} \\gamma_b\\mathcal{R}^{\\mu,\\,ba(n-2)}=0,\\end{equation}\nas they should be. Actually, the very choices of $\\mathcal{A}^{\\mu\\rho\\nu,\\,ab}$ and $\\mathcal{B}^{\\mu\\nu,\\,ab}$ made respectively in Eqs~(\\ref{fr4}) and~(\\ref{fr4Ad})\nwere such that the action~(\\ref{fr3Ad2}) manifestly has the following form:\n\\begin{equation}\\label{fr3Ad2.5}\nS=-\\tfrac{1}{2}\\int d^Dx\\,\\bar{e}\\,\\bar{\\Psi}_{\\mu,\\,a(n-1)}\\mathcal{R}^{\\mu,\\,a(n-1)}.\n\\end{equation}\nClearly, the equations of motion~(\\ref{eom1}) share the gauge symmetries~(\\ref{gauge-fr0Ad}) of the action:\n\\begin{equation}\\label{gauge-xxx}\n\\delta \\Psi_{\\mu,\\,a(n-1)}=\\hat{D}_\\mu\\zeta_{a(n-1)}+\\tilde{\\mu}\\bar{e}_\\mu^{\\,b}\\left[\\gamma_b\\zeta_{a(n-1)}-\\left(\\tfrac{2}{D+2n-4}\\right)\\gamma_a\\zeta_{a(n-2)b}\\right]+\\bar{e}_\\mu^{\\,b}\\lambda_{b,\\,a(n-1)}\\,.\n\\end{equation}\nIn the next section we will fix these gauge symmetries to find among other things the number of physical degrees of freedom, which should match with that of a Majorana fermion\nof spin $s=n+\\tfrac{1}{2}$\\,.\n\n\\section{Equivalence of Frame- \\& Metric-like Formulations}\\label{sec:Eqv}\n\nThe first step to establish the equivalence of the frame- and metric-like descriptions of a gauge fermion is to find a match in the respective number of local degrees of freedom.\nTo count this for a frame-like fermion~\\cite{V-AdS}, we rewrite the equations of motion~(\\ref{eom1}) exclusively in terms of world indices:\n\\begin{equation}\\label{eom2}\n\\mathcal{R}^{\\mu,\\,\\alpha(n-1)}\\equiv\\left(\\gamma^{\\mu\\rho\\nu}\\nabla_\\rho+\\mu\\gamma^{\\mu\\nu}\\right)\\Psi_{\\nu,}{}^{\\alpha(n-1)}+\\tfrac{1}{2n}\\mathcal{C}^{\\mu\\nu\\beta,\\,\\alpha}\\Psi_{\\nu,\\,\\beta}{}^{\\alpha(n-2)}=0,\n\\end{equation}\nwhere $\\mathcal{C}^{\\mu\\nu\\beta,\\,\\alpha}$ is an operator antisymmetric in the $\\mu,\\nu,\\beta$ indices, given by\n\\begin{equation}\\label{eom3}\n\\mathcal{C}^{\\mu\\nu\\beta,\\,\\alpha}\\equiv\\left[\\gamma^\\alpha,\\gamma^{\\mu\\rho\\nu\\beta}\\right]\\nabla_\\rho-\\mu\\left\\{\\gamma^\\alpha,\\gamma^{\\mu\\nu\\beta}\\right\\}-\\left(\\tfrac{2}{D+2n-4}\\right)\\mu\\gamma^\\alpha\\gamma^{\\mu\\nu\\beta}.\n\\end{equation}\nSome of the dynamical modes however are not physical because of gauge invariance. In order to exclude the correct number of pure gauge modes, let us rewrite the gauge\ntransformations~(\\ref{gauge-xxx}) as:\n\\begin{equation}\\label{gauge-xw}\n\\delta \\Psi_{\\mu,\\,\\alpha(n-1)}=\\nabla_\\mu\\zeta_{\\alpha(n-1)}+\\tilde{\\mu}\\left[\\gamma_\\mu\\zeta_{\\alpha(n-1)}-\\left(\\tfrac{2}{D+2n-4}\\right)\\gamma_{\\alpha}\\zeta_{\\alpha(n-2)\\mu}\\right]+\\lambda_{\\mu,\\,\\alpha(n-1)}\\,.\n\\end{equation}\nNow one can use this freedom to choose the following covariant gauge:\n\\begin{equation}\\label{gf1}\n\\displaystyle{\\not{\\!\\Psi\\!\\,}}_{\\alpha(n-1)}\\equiv\\gamma^\\mu\\Psi_{\\mu,\\,\\alpha(n-1)}=0,\\qquad\\Longrightarrow\\qquad \\Psi'_{\\alpha(n-2)}\\equiv\\bar{g}^{\\mu\\nu}\\Psi_{\\mu,\\,\\nu\\alpha(n-2)}=0.\n\\end{equation}\nAs a consequence, the equations of motion~(\\ref{eom2}) reduce to the following form:\n\\begin{equation}\\label{eom-gf}\n\\left(\\displaystyle{\\not{\\!\\nabla\\!\\,}}-\\mu\\right)\\Psi^{\\mu,}{}_{\\alpha(n-1)}-\\gamma^\\mu\\nabla^\\nu\\Psi_{\\nu,\\,\\alpha(n-1)}+\\tfrac{1}{2n}\\mathcal{C}^{\\mu\\nu\\rho,}{}_\\alpha\\,\\chi_{\\nu,\\,\\rho\\alpha(n-2)}=0,\n\\end{equation}\nwhere $\\chi_{\\mu,\\,\\alpha(n-1)}$ is the irreducible part of the field $\\Psi_{\\mu,\\,\\alpha(n-1)}$ with the symmetry of the Young diagram $\\mathbb{Y}(n-1,1)$, i.e., it has exactly the same properties as\nthe gauge parameter $\\lambda_{\\mu,\\,\\alpha(n-1)}$. Its appearance in the last term of Eq.~(\\ref{eom-gf}) is easy to understand. The antisymmetry property of $\\mathcal{C}^{\\mu\\nu\\rho,\\,\\alpha}$\nremoves the completely symmetric part of $\\Psi_{\\mu,\\,\\alpha(n-1)}$, while the $\\gamma$-trace parts are trivial  by the gauge choice~(\\ref{gf1}).\n\nThe condition~(\\ref{gf1}) is however not a complete gauge fixing. This can be seen by taking its gauge variation, which results in the Dirac equation for $\\zeta_{\\alpha(n-1)}$:\n\\begin{equation}\\label{gf2}\n\\delta\\displaystyle{\\not{\\!\\Psi\\!\\,}}_{\\alpha(n-1)}\\equiv\\left[\\displaystyle{\\not{\\!\\nabla\\!\\,}}-\\left(\\tfrac{D+2n-2}{D+2n-4}\\right)\\mu\\right]\\zeta_{\\alpha(n-1)}=0.\n\\end{equation}\nNot only does this allow for nontrivial solutions for $\\zeta_{\\alpha(n-1)}$ but it also leaves $\\lambda_{\\mu,\\,\\alpha(n-1)}$ completely unaffected. Therefore, one can use to freedom of the shift parameter\n$\\lambda_{\\mu,\\,\\alpha(n-1)}$ to further gauge fix:\n\\begin{equation}\\label{gf3} \\chi_{\\mu,\\,\\alpha(n-1)}=0.\\end{equation}\nThis finally reduces the equations of motion~(\\ref{eom-gf}) to the Dirac form plus the divergence constraint:\n\\begin{equation}\\label{eom7}\n  \\left(\\displaystyle{\\not{\\!\\nabla\\!\\,}}-\\mu\\right)\\Psi_{\\mu,\\,\\alpha(n-1)}=0,\\qquad \\nabla^\\mu\\Psi_{\\mu,\\,\\alpha(n-1)}=0.\n\\end{equation}\nTo exhaust the residual freedom of $\\zeta_{\\alpha(n-1)}$ let us choose the gauge:\n\\begin{equation}\\label{gf9}\\Psi_{0,\\,\\alpha(n-1)}=0.\\end{equation}\nIts is easy to see that no residual freedom of $\\zeta_{\\alpha(n-1)}$ is left. A would-be residual parameter must obey some screened Poisson equation with no source term,\nwhich has no nontrivial solutions.\n\nThe count of local physical degrees of freedom is now immediate. The system~(\\ref{eom7}) describes $(D-1)\\Delta_\\zeta$ many dynamical variables, where $\\Delta_\\zeta$ is given in Eq.~(\\ref{z-dof}).\nBut the gauge choices~(\\ref{gf1}), (\\ref{gf3}) and~(\\ref{gf9}) respectively remove $\\Delta_\\zeta$, $\\Delta_\\lambda$ and $\\Delta_\\zeta$ degrees of freedom. Therefore, the total number of physical degrees\nof freedom is $(D-3)\\Delta_\\zeta-\\Delta_\\lambda$, which is the same as\n\\begin{equation}\\label{dof-adv} \\Delta_{\\text{Frame}}=\\binom{D+n-4}{n}f_D\\,.\\end{equation}\nThis confirms, in view of Eq.~(\\ref{dofF}), that the count matches in the two formulations: $\\Delta_\\text{Frame}=\\Delta_{\\text{Metric}}$\\,.\n\nThe physical significance of the mass parameter $\\mu$ is now clear from the Dirac equation in~(\\ref{eom7}). While Eq.~(\\ref{AdS-last}) says that $\\mu$ must be real, one may choose $\\mu>0$\nwithout any loss of generality. Then,\n\\begin{equation}\\label{AdS-last1} \\mu=\\frac{1}{l}\\left(n+\\tfrac{D-4}{2}\\right)>0.\\end{equation}\nOur $\\mu$ corresponds to the lowest value of the mass parameter $m$ for a fermion carrying a unitary irreducible representation of the AdS isometry algebra:\n\\begin{equation}\\label{kutta}  \\left(\\displaystyle{\\not{\\!\\nabla\\!\\,}}-m\\right)\\Psi_{\\mu,\\,\\alpha(n-1)}=0,\\qquad m\\geq\\mu>0.\\end{equation}\nThe bound saturates for the massless representation~\\cite{Metsaev:2006zy,Metsaev:2013wza,Metsaev:2003cu}, as we see.\n\nNext we will show that the two formulations are equivalent at the level of the free Lagrangian. With this end in view, let us decompose the fermion field\n$\\Psi_{\\mu,\\,\\alpha(n-1)}$ into totally symmetric, $\\gamma$-traceless mixed-symmetric and $\\gamma$-trace parts:\n\\begin{equation}\\label{decomp} \\Psi_{\\mu,\\,\\alpha(n-1)}=\\psi_{\\mu\\alpha(n-1)}+\\chi_{\\mu,\\,\\alpha(n-1)}+\\gamma_{[\\mu}\\theta_{\\alpha]\\alpha(n-2)},\\end{equation}\nwhere the fields appearing on the right hand side have the symmetry of the following Young diagrams:\n\\begin{equation}\\label{bilai1}\n\\psi_{\\alpha(n)}\\sim\\begin{aligned}\n&\\underbrace{\\begin{tabular}{|c|c|c|c|}\\hline\n  $\\phantom{a}$&\\multicolumn{2}{|c|}{$~\\cdots~$}&\\phantom{a}\\\\\\hline\n\\end{tabular}}_n\\,,\n\\end{aligned}\n\\quad\n\\chi_{\\mu,\\,\\alpha(n-1)}\\sim\\overbrace{\\begin{aligned}\n&\\begin{tabular}{|c|c|c|c|}\\hline\n   $\\phantom{a}$&\\multicolumn{2}{|c|}{$~\\cdots~$}&\\phantom{a}\\\\\\hline\n\\end{tabular}\\\\[-4pt]\n&\\begin{tabular}{|c|}\n   $\\phantom{a}$\\\\\\hline\n\\end{tabular}\n\\end{aligned}}^{n-1}\\,,\n\\quad\n\\theta_{\\alpha(n-1)}\\sim\\begin{aligned}\n&\\underbrace{\\begin{tabular}{|c|c|c|c|c|}\\hline\n  $\\phantom{a}$&\\multicolumn{2}{|c|}{$~\\cdots~$}&\\phantom{a}\\\\\\hline\n\\end{tabular}}_{n-1}\\,.\n\\end{aligned}\n\\end{equation}\nWe have imposed irreducibility conditions on $\\chi_{\\mu,\\,\\alpha(n-1)}$, so that it is subject to the following constraints:\n\\begin{equation}\\label{bilai2}\n\\gamma^\\mu\\chi_{\\mu,\\,\\alpha(n-1)}=0,\\qquad \\gamma^\\beta\\chi_{\\mu,\\,\\alpha(n-2)\\beta}=0,\\qquad \\chi_{\\alpha,\\,\\alpha(n-1)}=0.\n\\end{equation}\nOf course there will be additional constraints on the fields  $\\psi_{\\alpha(n)}$ and $\\theta_{\\alpha(n-1)}$ coming from the $\\gamma$-trace condition on the\nparent field $\\Psi_{\\mu,\\,\\alpha(n-1)}$ in the $\\alpha$-indices. To find them, let us first take a $\\gamma$-trace of Eq.~(\\ref{decomp}) in an $\\alpha$-index.\nThis results in\n\\begin{equation}\\label{bilai3} \\displaystyle{\\not{\\!\\psi\\!\\,}}_{\\mu\\alpha(n-2)}-(D-2)\\theta_{\\mu\\alpha(n-2)}-(n-1)\\gamma_\\mu\\displaystyle{\\not{\\!\\theta}}_{\\alpha(n-2)}+\\gamma_\\alpha\\displaystyle{\\not{\\!\\theta}}_{\\mu\\alpha(n-3)}=0.\\end{equation}\nAnother $\\gamma$-trace w.r.t. the $\\mu$-index gives\n\\begin{equation}\\label{bilai4}\n\\psi'_{\\alpha(n-2)}-(Dn-2n+2)\\displaystyle{\\not{\\!\\theta}}_{\\alpha(n-2)}-\\gamma_\\alpha\\theta'_{\\alpha(n-3)}=0.\n\\end{equation}\nNow a third $\\gamma$-trace in an $\\alpha$-index yields:\n\\begin{equation}\\label{bilai5} \\displaystyle{\\not{\\!\\psi\\!\\,}}^{\\,\\prime}_{\\alpha(n-3)}-(Dn+D-4)\\theta^{\\prime}_{\\alpha(n-3)}+\\gamma_\\alpha\\displaystyle{\\not{\\!\\theta}}^{\\,\\prime}_{\\alpha(n-4)}=0.\\end{equation}\nOn the other hand, one could also have obtained a triple $\\gamma$-trace by first contracting the $\\mu$ index with an $\\alpha$ index in Eq.~(\\ref{decomp}) and then taking a $\\gamma$ trace.\nThis however produces a different result:\n\\begin{equation}\\label{bilai6} \\displaystyle{\\not{\\!\\psi\\!\\,}}^{\\,\\prime}_{\\alpha(n-3)}-(D+n-4)\\theta^{\\prime}_{\\alpha(n-3)}+\\gamma_\\alpha\\displaystyle{\\not{\\!\\theta}}^{\\,\\prime}_{\\alpha(n-4)}=0.\\end{equation}\nEqs.~(\\ref{bilai5}) and~(\\ref{bilai6}) impose the following constraints:\n\\begin{equation}\\label{bilai7} \\displaystyle{\\not{\\!\\psi\\!\\,}}^{\\,\\prime}_{\\alpha(n-3)}=0,\\qquad \\theta^{\\prime}_{\\alpha(n-3)}=0,\\end{equation}\ni.e., the symmetric rank-$n$ field $\\psi_{\\alpha(n)}$ must be triply $\\gamma$-traceless, whereas the symmetric rank-$(n-1)$ field $\\theta_{\\alpha(n-1)}$ must be traceless. This in turn results,\nfrom Eqs.~(\\ref{bilai3}) and~(\\ref{bilai4}), in the following relation:\n\\begin{equation}\\label{bilai8}\n\\theta_{\\alpha(n-1)}=\\left(\\tfrac{1}{D-2}\\right)\\left[\\displaystyle{\\not{\\!\\psi\\!\\,}}_{\\alpha(n-1)}-\\left(\\tfrac{1}{nD-2n+2}\\right)\\gamma_\\alpha\\psi'_{\\alpha(n-2)}\\right].\n\\end{equation}\nFinally, plugging the above expression into the decomposition~(\\ref{decomp}), we obtain:\n\\begin{eqnarray}\n\\Psi_{\\mu,\\,\\alpha(n-1)}&=&\\psi_{\\mu\\alpha(n-1)}+\\chi_{\\mu,\\,\\alpha(n-1)}+\\left(\\tfrac{1}{D-2}\\right)\\left[\\gamma_{[\\mu}\\displaystyle{\\not{\\!\\psi\\!\\,}}_{\\alpha]\\alpha(n-2)}-\\left(\\tfrac{2}{Dn-2n+2}\\right)\\gamma_{\\mu\\alpha}\\psi'_{\\alpha(n-2)}\\right]\\nonumber\\\\\n&&~~~~~~~~~~~+\\tfrac{1}{(D-2)(Dn-2n+2)}\\left[(n-2)\\gamma_\\alpha\\gamma_\\mu\\psi'_{\\alpha(n-2)}-2\\bar{g}_{\\alpha(2)}\\psi'_{\\mu\\alpha(n-3)}\\right].\\label{bilai9}\n\\end{eqnarray}\nThis decomposition generalizes that of Ref.~\\cite{V-flat} to arbitrary dimensions.\n\nIt will be convenient to write the covariant equations of motion~(\\ref{eom2}) in the following form:\n\\begin{equation}\\label{ge1} \\mathcal{R}^{\\mu,\\,\\alpha(n-1)}\\equiv\\mathcal{O}^{\\mu\\nu,\\,\\alpha(n-1)\\beta(n-1)}\\Psi_{\\nu,\\,\\beta(n-1)}=0,\\end{equation}\nwhere we have defined the operator $\\mathcal O$ as:\n\\begin{equation}\\label{ge2}\n\\mathcal{O}^{\\mu\\nu,\\,\\alpha(n-1)\\beta(n-1)}\\equiv\\left(\\gamma^{\\mu\\rho\\nu}\\nabla_\\rho+\\mu\\gamma^{\\mu\\nu}\\right)\\bar{g}^{\\,\\alpha(n-1),\\,\\beta(n-1)}+\\tfrac{1}{2n(n-1)}\\mathcal{C}^{\\mu\\nu\\beta,\\,\\alpha}\\bar{g}^{\\,\\alpha(n-2),\\,\\beta(n-2)},\n\\end{equation}\nwith $\\bar{g}^{\\,\\alpha(k),\\,\\beta(k)}\\equiv\\tfrac{1}{k^2}\\bar{g}^{\\,\\alpha\\beta}\\bar{g}^{\\,\\alpha\\beta}\\ldots\\bar{g}^{\\,\\alpha\\beta}$ (multiplicity $k$) denoting the unit-strength symmetric tensor product of $k$ background\nmetric tensors. This enables us to present the corresponding Lagrangian as:\n\\begin{equation}\\label{ge3} \\tfrac{1}{\\sqrt{-\\bar{g}}}\\,\\mathcal{L}=-\\tfrac{1}{2}\\bar{\\Psi}_{\\mu,\\,\\alpha(n-1)}\\mathcal{O}^{\\mu\\nu,\\,\\alpha(n-1)\\beta(n-1)}\\Psi_{\\nu,\\,\\beta(n-1)}\\,.\\end{equation}\n\nWhen the decomposition~(\\ref{bilai9}) is plugged into the above Lagrangian, the irreducible mixed-symmetric part $\\chi_{\\mu,\\,\\alpha(n-1)}$ completely drops out, thanks to the shift symmetry.\nThe fact that the parameter $\\lambda_{\\mu,\\,\\alpha(n-1)}$ enjoys exactly the same properties as $\\chi_{\\mu,\\,\\alpha(n-1)}$ plays a crucial role in this regard. The resulting Lagrangian contains only\nthe completely symmetric part $\\psi_{\\alpha(n)}$ and can be viewed as a gauge-fixed version of the original Lagrangian~(\\ref{ge3}) with the gauge fixing: $\\chi_{\\mu,\\,\\alpha(n-1)}=0$. The explicit\nderivation of this Lagrangian is tedious but straightforward. The calculations can however be simplified by noting that, on account of the $\\gamma$-tracelessness of the equations of\nmotion~(\\ref{ge1}) in the $\\alpha$-indices, the Lagrangian splits into the sum of two pieces:\n\\begin{equation}\\label{ge4} \\tfrac{1}{\\sqrt{-\\bar{g}}}\\,\\mathcal{L}=-\\tfrac{1}{2}\\bar{\\Xi}_{\\mu,\\,\\alpha(n-1)}\\mathcal{O}^{\\mu\\nu,\\,\\alpha(n-1)\\beta(n-1)}\\Xi_{\\nu,\\,\\beta(n-1)}\n+\\tfrac{1}{2}\\bar{\\xi}_{\\mu,\\,\\alpha(n-2)}\\gamma_\\alpha\\mathcal{O}^{\\mu\\nu,\\,\\alpha(n-1)\\beta(n-1)}\\gamma_\\beta\\xi_{\\nu,\\,\\beta(n-2)}\\,,\\end{equation}\nwhere the tensor-spinors $\\Xi_{\\mu,\\,\\alpha(n-1)}$ and $\\xi_{\\mu,\\,\\alpha(n-2)}$ are given by:\n\\begin{eqnarray}\n  \\Xi_{\\mu,\\,\\alpha(n-1)} &=& \\psi_{\\mu\\alpha(n-1)}+\\left(\\tfrac{1}{D-2}\\right)\\left[(n-1)\\gamma_\\mu\\displaystyle{\\not{\\!\\psi\\!\\,}}_{\\alpha(n-1)}-\\left(\\tfrac{2}{Dn-2n+2}\\right)\\bar{g}_{\\mu\\alpha}\\psi'_{\\alpha(n-2)}\\right],\\nonumber\\\\\n  \\xi_{\\mu,\\,\\alpha(n-2)} &=& \\left(\\tfrac{1}{D-2}\\right)\\left[-\\displaystyle{\\not{\\!\\psi\\!\\,}}_{\\mu\\alpha(n-2)}+\\left(\\tfrac{1}{Dn-2n+2}\\right)\\left(n\\gamma_\\mu\\psi'_{\\alpha(n-2)}-\\gamma_\\alpha\\psi'_{\\mu\\alpha(n-3)}\\right)\\right].\\label{ge5}\n\\end{eqnarray}\nOne can explicitly carry out the calculations to get to the following result:\n\\begin{eqnarray}\n-\\tfrac{2}{\\sqrt{-\\bar{g}}}\\,\\mathcal{L}&=&\\bar{\\psi}_{\\alpha(n)}\\left(\\not{\\!\\nabla\\!}-\\mu\\right)\\psi^{\\alpha(n)}\n+n\\bar{\\displaystyle{\\not{\\!\\psi\\!\\,}}}_{\\alpha(n-1)}\\left(\\not{\\!\\nabla\\!}+\\mu\\right)\\displaystyle{\\not{\\!\\psi\\!\\,}}^{\\alpha(n-1)}\n-2n\\bar{\\displaystyle{\\not{\\!\\psi\\!\\,}}}_{\\alpha(n-1)}\\!\\nabla_\\mu\\psi^{\\mu\\alpha(n-1)}\\nonumber\\\\\n&&-\\tfrac{1}{4}n(n-1)\\bar{\\psi}'_{\\alpha(n-2)}\\left(\\not{\\!\\nabla\\!}-\\mu\\right)\\psi'^{\\,\\alpha(n-2)}\n-n(n-1)\\bar{\\psi}'_{\\alpha(n-2)}\\nabla_\\mu\\displaystyle{\\not{\\!\\psi\\!\\,}}^{\\,\\mu\\alpha(n-2)}.\\label{ge6}\n\\end{eqnarray}\nThis indeed coincides with the Lagrangian~(\\ref{f00}) for a metric-like gauge fermion in AdS space. Because only the symmetric part of the parent field $\\Psi_{\\mu,\\,\\alpha(n-1)}$\nappears in this Lagrangian, the corresponding gauge symmetry is obtained simply by a total symmetrization of the indices in Eq.~(\\ref{gauge-xw}). The result is:\n\\begin{equation}\\label{gauge-xwm}\n\\delta \\psi_{\\alpha(n)}=\\tfrac{1}{n}\\left(\\nabla_\\alpha\\zeta_{\\alpha(n-1)}-\\tfrac{1}{2l}\\gamma_{\\alpha}\\zeta_{\\alpha(n-1)}\\right),\n\\end{equation}\nwhich also matches perfectly with the metric-like gauge symmetry~(\\ref{tamm2}).\n\nThis hardly comes as a surprise. The symmetric part of $\\Psi_{\\mu,\\,\\alpha(n-1)}$ has all the characteristics of a metric-like gauge fermion; in particular it is\ntriple $\\gamma$-traceless as we have shown in Eq.~(\\ref{bilai7}). Moreover, it transforms w.r.t.~a symmetric $\\gamma$-traceless gauge parameter $\\zeta_\\alpha(n-1)$.\nThe gauge-invariant Lagrangian description for such a system is unique~\\cite{Hallowell:2005np,Metsaev:2006zy,Metsaev:2013wza}.\nSo, $\\psi_{\\alpha(n)}$ is a metric-like gauge fermion in every sense.\n\n\\section{Remarks}\\label{sec:remarks}\n\nIn this article, we have elaborated on some key features of higher-spin gauge fermions and the connections between their frame- and metric-like formulations\nat the free level. A gauge-invariant frame-like Lagrangian description in AdS space, with the constraints on the\nfields and the gauge parameters resembling their flat-space cousins, facilitates the explicit derivation of the corresponding metric-like Lagrangian as a gauge\nfixing. This derivation generalizes that of Ref.~\\cite{V-flat} to AdS space and arbitrary dimensions. Although the equivalence of the frame-\nand metric-like formulations at the free level may not come as a surprise, our work fills a gap in the literature.\n\nAs is well-known, the frame-like formulation packages the non-linearities in an interacting theory in a very efficient way. For higher-spin fermions this can be seen\nin a very simple setup: the Aragone-Deser hypergravity~\\cite{hygra}$-$a consistent gauge theory of a spin $s=n+\\tfrac{1}{2}$ massless Majorana fermion coupled to Einstein\ngravity in 3D flat space. While only fermion bilinears appear in the frame-like formulation~\\cite{hygra}, the metric-like formulation will also include four-fermion couplings\nthat originate from integrating out the spin-connection, just like in supergravity~\\cite{Freedman:2012zz}. Moreover, the fermion-bilinear terms will look more complicated\nin the metric-like variables. To see this, note that with frame-like fermions the cubic cross-coupling in the covariant language has the simple form~\\cite{ours}:\n\\begin{equation}\\label{last0}\n\\mathcal{L}_3\\sim\\bar\\Psi_{\\mu,\\,\\alpha(n-1)}\\gamma^{\\mu\\nu\\rho}\\gamma^{\\sigma\\lambda}\\Psi_{\\nu,}{}^{\\alpha(n-1)}\\partial_\\sigma h_{\\rho\\lambda},\n\\end{equation}\nwhere $h_{\\mu\\nu}$ is the metric perturbation. Because the irreducible hook part $\\chi_{\\mu,\\,\\alpha(n-1)}$ of the frame-like fermion is trivial in $D=3$, the\ndecomposition~(\\ref{bilai9}) amounts to a complicated field redefinition:\n\\begin{equation}\\label{last1}\n\\Psi_{\\mu,\\,\\alpha(n-1)}=\\psi_{\\mu\\alpha(n-1)}+\\gamma_{[\\mu}\\displaystyle{\\not{\\!\\psi\\!\\,}}_{\\alpha]\\alpha(n-2)}+\\left(\\tfrac{1}{n+2}\\right)\\left[n\\gamma_\\alpha\\gamma_\\mu\\psi'_{\\alpha(n-2)}-2\\eta_{\\mu\\alpha}\\psi'_{\\alpha(n-2)}+2\\eta_{\\alpha(2)}\\psi'_{\\mu\\alpha(n-3)}\\right],\n\\end{equation}\nwhere $\\psi_{\\alpha(n)}$ is the metric-like fermion. After this redefinition is performed, the cubic coupling~(\\ref{last0}) will look cumbersome in terms of the metric-like fermion.\nWithin the metric-like formulation, it would be more difficult to construct or to prove the consistency of this cubic coupling, say using the techniques of\nRef.~\\cite{Henneaux:2012wg,Henneaux:2013gba}. The fermion-bilinear cross-couplings do not stop at any finite order in the graviton fluctuations and the situation gets only worse\nat higher orders, while the frame-like formulation captures all the non-linearities in a very neat way~\\cite{hygra}.\n\nIn higher dimensions the difference between the two formulations becomes more drastic. The hook part of the frame-like fermion never shows up in the interacting Lagrangian because\nof the deformed shift symmetry. However, there appear the so-called ``extra'' fields: a set of additional fields that arises when one tries to construct a complete set of gauge-invariant\nobjects (curvatures)\\footnote{The extra fields are generalizations of the spin-connection. The number of extra fields depends on the spin; the higher the spin, the more are the\nextra fields needed for constructing curvatures. The extra fields however do not enter the free action, and so they are not expressed in terms of physical fields via equations of motion.\n}~\\cite{Fradkin:1986ka}.\nTo understand the role of these extra fields that are absent in the free Lagrangian, one may express them in terms of the physical fields by means of appropriate constraints\nimplemented via Lagrange multipliers~\\cite{Vasiliev:1986td,V-AdS,Fradkin:1986ka,Fradkin:1987ks,Fradkin:1986qy}. Then, up to pure gauge parts, the extra fields are given by\nderivatives of the physical fields. The extra fields therefore induce higher-derivative terms in the interactions, while their absence in the free Lagrangian merely reflects\nthe absence of higher-derivative kinetic terms.\nExplicit solution of the aforementioned constraints are difficult, and actually not needed. The main idea of the so-called Fradkin-Vasiliev\nformalism~\\cite{Fradkin:1986ka,Fradkin:1987ks,Fradkin:1986qy} is that one can treat the extra fields as independent variables since most of the gauge-invariant curvatures\nvanish on shell.\n\n\\subsection*{Acknowledgments}\n\nThe author is grateful to N.~Boulanger, A.~Campoleoni, G.~Lucena G\\'omez, M.~Henneaux, and especially to  E.~D.~Skvortsov for valuable inputs and useful comments.\nHe would like to thank the organizers of the 4th Mons Workshop on Higher Spin Gauge Theories (2017), during which this study was initiated.\n\n\\begin{appendix}\n\\numberwithin{equation}{section}\n\\section{Metric-like Formulation}\\label{sec:A}\n\nThe metric-like formulation of gauge fermions originated in the work of Fang and Fronsdal~\\cite{FF,Fang:1979hq}, who studied the massless limit of the Lagrangian for massive\nhigher-spin fermions. The Fang-Fronsdal Lagrangian can be derived uniquely by considering gauge invariance and supersymmetry transformations for a massless system involving\nthe pair of spins $\\left(s, s+\\tfrac{1}{2}\\right)$~\\cite{Curtright:1979uz}. The construction was later generalized for maximally symmetric spaces with arbitrary dimension in Ref.~\\cite{Hallowell:2005np,Metsaev:2006zy,Metsaev:2013wza}. In the metric-like formulation, a spin $s=n+\\tfrac{1}{2}$ gauge fermion is described by a completely symmetric rank-$n$\ntensor-spinor $\\psi_{\\mu(n)}$ in the world indices. It satisfies the triple $\\gamma$-trace condition:\n\\begin{equation}\\label{tg1}\\displaystyle{\\not{\\!\\psi\\!\\,}}'_{\\mu(n-3)}=0.\\end{equation}\nIt is convenient to describe metric-like theories in the operator formalism, where contraction and symmetrization of indices are realized through auxiliary variables\nand tensor operations are simplified in terms of operator calculus. Symmetric tensor-spinor fields are represented by:\n\\begin{equation}\\label{field}\\psi(x,u)=\\tfrac{1}{n!}\\,\\psi_{\\mu_1\\ldots\\mu_n}(x)\\,\\bar{e}^{\\,\\mu_1}_{a_1}(x)u^{a_1}\\,\\ldots\\,\\bar{e}^{\\,\\mu_n}_{a_n}(x)u^{a_n},\\end{equation}\nwhere $\\bar{e}^{\\,\\mu}_a(x)$ is the background vielbein and $u^a$ is an auxiliary tangent variable. The action of the covariant derivative is defined as\na differential operation involving both $x$ and $u$:\n\\begin{equation}\\label{covD}\\nabla_\\mu=\\bar{\\nabla}_\\mu+\\bar{\\omega}_\\mu{}^{ab}u_a\\tfrac{\\partial}{\\partial u^b},\\end{equation}\nwhere $\\bar{\\nabla}_\\mu$ is the standard covariant derivative acting on naked tensorial indices, and $\\bar{\\omega}_\\mu{}^{ab}$ the background spin connection.\nIn what follows we work only with the contracted auxiliary variable and the associated derivative:\n\\begin{equation}\\label{u-du} u^\\mu\\equiv \\bar{e}^{\\,\\mu}_{a}(x)u^{a},\\quad \\partial_u^\\mu\\equiv \\bar{e}^{\\,\\mu a}(x)\\tfrac{\\partial}{\\partial u^a}.\\end{equation}\nThe vielbein postulate then implies that $[\\nabla_\\mu,u^\\nu]=0$~as well as $[\\nabla_\\mu,\\partial_u^\\nu]=0$. The commutator of covariant derivatives on a spinor function of $u$ and $\\partial_u$\nwill be given by:\n\\begin{equation}\\label{commutator}[\\nabla_\\mu,\\nabla_\\nu]=R_{\\mu\\nu\\rho\\sigma}(x)u^\\rho\\partial_u^\\sigma+\\tfrac{1}{4}R_{\\mu\\nu\\rho\\sigma}(x)\\gamma^{\\rho\\sigma}.\\end{equation}\nOne would have to use the following set of operators~\\cite{Hallowell:2005np,Metsaev:2006zy,Metsaev:2013wza}:\n\\begin{equation}\\label{sett} \\mathbb{G}=\\left\\{\\displaystyle{\\not{\\!\\nabla\\!\\,}},\\,\\partial_u\\!\\cdot\\!\\nabla,\\,u\\!\\cdot\\!\\nabla,\\,\\displaystyle{\\not{\\!\\partial_u\\!\\,}},\\,\\displaystyle{\\not{\\!u\\!\\,}},\\,\\partial_u^2,\\,u^2,\\,u\\!\\cdot\\!\\partial_u\\right\\}.\\end{equation}\nThe set comprises eight operators: the Dirac operator $\\displaystyle{\\not{\\!\\nabla\\!\\,}}$, divergence $\\partial_u\\!\\cdot\\!\\nabla$, symmetrized-gradient $u\\!\\cdot\\!\\nabla$, $\\gamma$-trace $\\displaystyle{\\not{\\!\\partial_u\\!\\,}}$, symmetrized-$\\gamma$ $\\displaystyle{\\not{\\!u\\!\\,}}$, trace $\\partial_u^2$,\nsymmetrized-metric $u^2$ and rank $u\\!\\cdot\\!\\partial_u$.\nThese operators have nontrivial commutation relations because of $[\\partial_u^\\mu,u^\\nu]=\\bar{g}^{\\,\\mu\\nu}$ and the non-commutativity~(\\ref{commutator}) of the\ncovariant derivatives if the background is non-flat.\n\nThen, the Lagrangian for a massless fermionic field in AdS space can be written as (for a Majorana fermion, certain terms in the Lagrangian are equivalent up to total\nderivatives)~\\cite{Metsaev:2013wza}:\n\\begin{eqnarray}\n\\tfrac{1}{\\sqrt{-\\bar{g}}}\\,\\mathcal{L}&=&-\\tfrac{1}{2}\\bar{\\psi}(\\ast_n)\\left(\\displaystyle{\\not{\\!\\nabla\\!\\,}}-u\\!\\cdot\\!\\nabla\\displaystyle{\\not{\\!\\partial_u\\!\\,}}-\\displaystyle{\\not{\\!u\\!\\,}}\\,\\partial_u\\!\\cdot\\!\\nabla+\\displaystyle{\\not{\\!u\\!\\,}}\\,\\displaystyle{\\not{\\!\\nabla\\!\\,}}\\displaystyle{\\not{\\!\\partial_u\\!\\,}}+\\tfrac{1}{2}\\displaystyle{\\not{\\!u\\!\\,}}\\,u\\!\\cdot\\!\\nabla\\,\\partial_u^2+\\tfrac{1}{2}u^2\\,\\partial_u\\!\\cdot\\!\\nabla\\,\\displaystyle{\\not{\\!\\partial_u\\!\\,}}\\right)\n\\psi\\nonumber\\\\&&-\\tfrac{1}{2}\\bar{\\psi}(\\ast_n)\\left(-\\tfrac{1}{4}u^2\\displaystyle{\\not{\\!\\nabla\\!\\,}}\\,\\partial_u^2\\right)\\psi+\\tfrac{1}{2}\\mu\\,\\bar{\\psi}(\\ast_n)\\left(1-\\displaystyle{\\not{\\!u\\!\\,}}\\,\\displaystyle{\\not{\\!\\partial_u\\!\\,}}-\\tfrac{1}{4}u^2\\,\\partial_u^2\\right)\\psi,\\label{f00}\n\\end{eqnarray}\nwhere the operation: $(\\ast_k)\\equiv\\left(\\overleftarrow{\\partial_u}\\cdot\\overrightarrow{\\partial_u}\\right)^k$ enables contraction between two rank-$k$ tensor-spinors, and has the properties:\n$(\\ast_k)u^\\mu=k\\overleftarrow{\\partial_u}^\\mu(\\ast_{k-1})$ and $(\\ast_k)\\overrightarrow{\\partial_u}^\\mu=(k+1)^{-1}u^\\mu(\\ast_{k+1})$\\,. The mass parameter:\n\\begin{equation}\\label{tamm0}\\mu=\\frac{1}{l}\\left(n+\\tfrac{D-4}{2}\\right),\\end{equation}\nis uniquely fixed by gauge invariance~\\cite{Metsaev:2006zy,Metsaev:2013wza}, where $l$ is the AdS radius.\nThe gauge symmetry of the Lagrangian~(\\ref{f00}) is w.r.t.~a symmetric $\\gamma$-traceless rank-$(n-1)$ tensor-spinor parameter:\n\\begin{equation}\\label{tamm0.5}\n\\varepsilon=\\tfrac{1}{(n-1)!}\\,\\varepsilon_{\\mu_1\\ldots\\mu_{n-1}}u^{\\mu_1}\\ldots u^{\\mu_{n-1}}, \\qquad \\displaystyle{\\not{\\!\\partial_u\\!\\,}}\\varepsilon=0,\n\\end{equation}\nwhile the triple $\\gamma$-tracelessness condition~(\\ref{tg1}) on the field translates in the operator formalism to:\n\\begin{equation}\\label{tamm1} \\displaystyle{\\not{\\!\\partial_u\\!\\,}}\\partial_u^2\\psi=\\partial_u^2\\displaystyle{\\not{\\!\\partial_u\\!\\,}}\\psi=0.\\end{equation}\nExplicitly, the gauge transformations are given by:\n\\begin{equation}\\label{tamm2}\\delta\\psi=u\\!\\cdot\\!\\nabla\\varepsilon-\\frac{1}{2l}\\displaystyle{\\not{\\!u\\!\\,}}\\,\\varepsilon.\\end{equation}\nThis can be verified by using the commutator~(\\ref{commutator}), which reduces in AdS space to:\n\\begin{equation}\\label{commutator-AdS}[\\nabla_\\mu,\\nabla_\\nu]=-\\frac{1}{l^2}\\left(u_{[\\mu} d_{\\nu]}+\\tfrac{1}{2}\\gamma_{\\mu\\nu}\\right),\\end{equation}\nand the various commutators of the operators in $\\mathbb{G}$ given the properties~(\\ref{tamm0.5}) and~(\\ref{tamm1}).\n\nThe metric-like description of higher-spin gauge fermions in flat-space is easily obtained by taking the limit $l\\rightarrow\\infty$ of the\ngauge invariant system~(\\ref{f00})--(\\ref{commutator-AdS}). The degrees of freedom count in flat~\\cite{Rahman:2015pzl} and AdS~\\cite{Campoleoni:2017vds}\nspaces are of course the same, and is given by:\n\\begin{equation}\\label{dofF} \\Delta_{\\text{Metric}}=\\binom{D+n-4}{n}f_D,\\end{equation}\nwhere $f_D$ for a Majorana fermion is given in Eq.~(\\ref{fD-defined}), while for a Dirac fermion the value is twice as much.\nNote that Eq.~(\\ref{dofF}) counts the number of physical dynamical fields plus their conjugate momenta.\nIn AdS space, one of course gets the same number since the counting of dynamical equations, constraints and gauge freedom works in the same way.\n\nAs already mentioned in the Introduction, the $\\gamma$-trace constraints~(\\ref{tamm0.5})--(\\ref{tamm1}) on the gauge parameter and the higher-spin fermionic field can be avoided by\nrecourse to other formulations. These include the non-local formulation~\\cite{Francia:2002aa}, the BRST\nformulation~\\cite{Buchbinder:2004gp,Buchbinder:2007vq}, the higher-derivative compensator formulation~\\cite{Francia:2007qt}, the quartet formulation~\\cite{Buchbinder:2007ak}\nand the non-minimal formulation with no higher derivatives~\\cite{Campoleoni:2009gs}.\n\n\\end{appendix}\n\n\\bibliographystyle{ws-rv-van}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{INTRODUCTION}\n\nIn the theory of planet formation, planets are thought to have formed in\nprotoplanetary disks through mutual collisions and coalescence of\nplanetesimals. The formation process of planetesimals, on the other\nhand, still has a large uncertainty. Before the planetesimal formation,\ndust grains grow through their collisional coalescence in a\nprotoplanetary disk and settle to the disk mid-plane,  which forms a\ndense dust layer at the mid-plane \\citep[e.g.,][]{saf69,nak81,tan05,dul05}.\nPlanetesimals would be formed in the dust layer through gravitational\ninstability \\citep[e.g.,][]{gol73,sek98,you02},\nstreaming instability \\citep[e.g.,][]{you05,you07}\nor simple coalescence \\citep[e.g.,][]{wei93,bra08a,bra08b}.\nIn these models of planetesimal formation, motion of dust\ngrains is an important factor because it determines the spatial\ndistribution of dust grains and the collision speed between\nthem. Furthermore, their motion is governed by the drag forces from the\ndisk gas.\n\nGas drag forces on dust grains strongly depend \non their internal\nstructure (or their bulk densities). Most studies of dust growth in\nprotoplanetary disks have assumed compact structure of dust\ngrains. However, dust grains \ngrowing\nthrough mutual\ncollisions would actually be aggregates of (sub-micron) primitive grains\nand the aggregates have a fluffy structure with an extremely low bulk\ndensity, as reported by experimental and theoretical studies\n\\citep[e.g.,][]{blu04b,orm07,suy08,oku09,zsom11}.\nSuch fluffy aggregates\nhave large ratios of their \ngeometrical \ncross sections to masses, which\nsignificantly enhance \ngas drag forces\non them compared with compact\ndust grains. \nHence, in order to clarify dust growth and planetesimal formation \nin protoplanetary disks, we have to examine the internal structure and \nthe geometrical \ncross sections of dust aggregates \n(we use a term `cross section' in referring to `geometrical cross\nsection', hereafter). \n\n\\citet{suy08} (hereafter S08) performed $N$-body numerical simulations of\nsequential aggregate collisions to examine the compression process of\ngrowing aggregates. \nThe sequential collisions mean that we repeat collisions of\naggregates obtained at the previous collisions.\nWith such a simulation, we can observe a natural evolution of \nthe aggregate structure.\nTheir numerical results showed that large aggregates have an\nextremely low bulk density \nin spite of\ncompression at aggregate collisions. In the early stage of dust growth,\naggregates just stick without any restructuring because of their low \nimpact energy and they have a fluffy structure with an extremely \nlow bulk density as they grow. \nIn the later stage in which the impact energy exceeds a\ncritical energy, aggregates are gradually compressed. \nEven in this compression stage, their density remains very low.\nIt is found that the compressed aggregates have a low fractal\ndimension of 2.5. This structural feature causes the low density\nof the compressed aggregates.\nS08 also\nderived a formula describing the density evolution of growing\naggregates. To estimate their bulk densities, S08 \nused\nthe\nso-called gyration radii of the aggregates but did not examine their\ncross sections. \nHowever, cross sections are directly related to the gas drag \nforces\nrather than gyration radii. \nIt is necessary to clarify the evolution of\ncross sections of dust aggregates during their growth. \n\nCross sections of aggregates depend on their internal\nstructure. There are two simple aggregate models. \nOne is Ballistic Cluster-Cluster Aggregation (BCCA). \nA BCCA cluster is formed through\ncollisions between \ntwo equal-sized clusters. \nSecond is Ballistic Particle-Cluster Aggregation (BPCA). \nA BPCA cluster is formed through\ndeposition of small monomer particles on a large cluster. \nFor both BCCA and BPCA, restructuring is assumed to be\nnegligible at each collision.\nThe BCCA clusters have very fluffy and open structures and \nthe BPCA clusters have relatively\ncompact structures. \nFigure~\\ref{fig:spmintro} shows the ratios of cross\nsections to masses of aggregates. \nIt is shown that the cross section per mass strongly\ndepend on aggregate types. \nCross sections of dust aggregates are expected to be between those of\nBCCA and BPCA clusters. \nOne may consider that a cross section is approximately given by the\nsquare of a gyration radius.\nCross sections of aggregates are, however, generally\nindependent of their gyration radii, especially for highly fluffy\naggregates.\nThe non-dimensional ratio of the cross section to the square\nof the gyration radius gradually decreases with their growth for BCCA\nclusters (Minato et al.~2006; see also Fig.~6).\nOn the other hand, this ratio is almost constant in the growth\nof BPCA clusters. \nOkuzumi et al. (2009) proposed a useful relation between \nthe cross section and the gyration radius\nfor various aggregates formed through \nhit-and-stick growth (as well as BCCA and BPCA). \nPaszun and Dominik (2009) also derived another relation.\nNevertheless, it is not clear whether these relations are\nalso valid for aggregates compressed at collisions.\nWe check the validity of these relations,\nusing the resultant aggregates obtained by S08.\n\nOnce we find a valid relation between the cross sections and the \ngyration radii, it would be very helpful to describe the evolution \nof the cross sections because the compression model by S08\ncan describe gyration radii of growing aggregates.\nThe compression model by S08, however, has some limitations.\nThis model is not directly applicable to low-energy collisions\n(i.e., hit-and-stick collisions) or to non-equal-mass collisions.\nIn order to describe gyration radii and cross sections of aggregates\nfor all growth stages seamlessly, we further refine the compression model, \nby removing these limitations.\n\n\nIn laboratory experiments, \\citet{weidl09}\nexamined compression of aggregates consisting of \n1.5$\\mu$m-diameter SiO$_2$ spheres at their multiple \nrebounds and also developed an empirical compression model. \nInitial aggregates in their experiments\npossess a volume filling factor of $\\sim 0.1$, \nwhich is approximately equal to \nthat of BPCA clusters. On the other hand, S08 and the \npresent paper focus on the compression of fluffier \naggregates of which filling factor is between BCCA and \nBPCA clusters during their collisional growth. \nHence our compression model and theirs are\ncomplementary to each other. As mentioned above, dust \naggregates are expected to have much smaller bulk densities\nthan BPCA clusters at the early stage of their \ngrowth in protoplanetary disks. Our compression model is \nuseful as long as bulk densities of aggregates is lower \nthan that of BPCA clusters.\n\nIt should be noticed that S08 only considered head-on collisions of\naggregates in their numerical simulations of sequential\ncollisions. The oblique collisions are expected to hinder the\ncompression (Wada et al. 2007; Paszun and Dominik 2009).\nIn the present study, however, we use the results in S08 as the \nfirst step. We will examine the effects of oblique collisions \nin future work.\n\nIn the next section, we briefly summarize the results of S08. In Section\n3, we numerically \ncalculate\ncross sections for the aggregates obtained by S08. \nWe find that Okuzumi et al's relation between\nthe cross section and the gyration radius is valid\nfor compressed aggregates, too.\nIn Section 4, we refine the compression model by S08,\nby removing its limitations in a reasonable way.\nWe find that the refined compression model reproduces well \nboth of gyration radii and cross sections of aggregates\nobtained by the numerical simulation,\nwith the help of Okuzumi et al's relation.\nWe also check the validity of the refined model for non-equal-mass \ncollisions with additional numerical simulations of aggregate collisions.\nA summary is given in the last section.\n\\section{RESULTS OF AGGREGATE COMPRESSION IN $N$-BODY SIMULATIONS BY S08}\n\nSuyama et al.~(2008) performed $N$-body numerical simulations of head-on\naggregate collisions and examined the density evolution of aggregates\ngrowing through the collisions. We examine the cross section\nof the resultant aggregates obtained by S08. Before\nthat, we briefly describe the numerical results of S08.\n\nIn the simulations, aggregates consist of a large number of icy spherical\nparticles with a radius of $r_1=0.1\\mu$m.\nS08 adopted the particle-interaction\nmodel by Wada et al.~(2007). In the interaction model, \nrepulsive and adhesive forces in the normal direction between particles\nin contact\nare given by the JKR theory (Johnson et al. 1971).\nA\ntangential force and a torque also arise to\nresist the slide, roll, and twist motions between them. Aggregate\ncompression is regulated mainly by inelastic rolling motions of the constituent\nparticles (e.g., Dominik \\& Tielens 1997;\nWada et al. 2007, 2008 [hereafter W07,W08]~; G\\\"{u}ttler et al. 2010).\nThe rolling energy $E_\\mathrm{roll}$\nis the energy required for rolling \na particle on its contact neighbor by an angle of $\\pi\/2$.\nThe rolling\nenergy is given by (W07, S08)\n\\begin{equation}\n E_\\mathrm{roll}=6\\pi^2\\gamma r_1\\xi_\\mathrm{crit},\\label{eroll} \n\\end{equation}\nwhere $\\xi_\\mathrm{crit}$ is the critical displacement for inelastic\nrolling motion.\nThe parameter range of $\\xi_\\mathrm{crit}$ is set to be from 2 to 16\n$\\mathrm{\\mbox{\\AA}}$ in S08. \nA large\nrolling energy $E_\\mathrm{roll}$\nsuppresses the restructuring of\naggregates. \nTo examine the structure evolution of\ngrowing aggregates, S08 performed $N$-body simulations of sequential\ncollisions. Each\nsimulation starts from a collision of aggregates composed of two\nparticles (i.e., dimers) and ends with a collision of aggregates\ncomposed of 16,384 particles. \nThe resultant aggregate obtained in the previous\ncollision is used as initial aggregates at each collision in the\nsimulation of sequential collisions.\nThe impact velocity is constant in sequential collisions. \nFor various (constant) impact velocities and critical rolling displacements,\nthey performed a large number of runs of sequential collisions.\n\nAs an index of the \nsize of an aggregate,\nS08 adopted the\nradius of gyration, $r_g$, defined by\n\\begin{equation}\n r_g\\equiv\n  \\sqrt{\\sum_{i=1}^N\n  \\frac{|\\mbox{\\boldmath{$x$}}_{i}-\\mbox{\\boldmath{$x$}}_\\mathrm{M}|^2}\n  {N}\n  },\n\\end{equation}\nwhere $\\mbox{\\boldmath{$x$}}_i$ is the position of particle $i$,\n$\\mbox{\\boldmath{$x$}}_\\mathrm{M}$ is\nthe position of the center of mass of the aggregate, and $N$ is the\nnumber of particles composing the aggregate.\nUsing the radius of gyration, \nthe volume $V$ and \nthe bulk density \n$\\rho$ \nof the aggregate are \nevaluated to be (Mukai et al. 1992; W08)\n\\begin{equation}\n V(r_g)=\\frac{4\\pi}{3}\\left(\\sqrt{\\frac{5}{3}}r_g\\right)^3,\n\\label{vdef}\n\\end{equation}\n\\begin{equation}\n \\rho(r_g)=\\frac{m_1N}{V(r_g)},\n\\label{rhodef}\n\\end{equation}\nrespectively,\nwhere \n$\\sqrt{5\/3}r_{g}$ is the so-called characteristic radius of an aggregate\nand \n$m_1$ is the mass of a constituent particle.\n\nAggregates are expected to have a BCCA structure for collisions at\nsufficiently low velocity because of sticking together of equal-mass\naggregates without any restructuring. If the compression is effective at\ncollisions, the gyration radii of aggregates would become smaller than\nthose of BCCA clusters. It is meaningful to compare the\nobtained aggregates with BCCA clusters. \nSince BCCA clusters have a fractal dimension of $\\sim 2$, the radius of\ngyration of  the BCCA cluster is given for large $N$ by (e.g., Mukai et\nal. 1992; W08\n)\n\\footnote{Exactly speaking,\nequation (\\ref{bccarg}) is satisfied for BCCA clusters formed through\nhead-on (hit-and-stick) collisions, which have the fractal dimension\nof 2.0. When offset collisions are also included at the formation\nof BCCA, their fractal dimension is 1.9 and the gyration radii\nproportional to $N^{0.52}$ (Okuzumi et al. 2009). \nAlthough the later BCCA is more realistic,\nthe former BCCA is used in S08 and the present study \nsince S08 consider only \nhead-on collisions in their simulations.}\n\n\\begin{equation}\n r_{g,\\mathrm{BCCA}}\\simeq N^{0.50}r_1.\\label{bccarg}\n\\end{equation}\n\nFigure~\\ref{fig:rgrho}a shows the gyration radius of the aggregates in the\nsimulations of sequential collisions performed by S08 for various values of\nparameters, $\\xi_\\mathrm{crit}$ and the impact velocity\n$v_\\mathrm{imp}$.\nThe density of monomer particles is given by \n$\\rho_m (\\equiv 3m_1\/[4\\pi r_1^3])$. \nIn the simulation of sequential collisions, the size of growing\naggregates is dependent on \nthe direction of each collision.\nS08 did 30 runs of the simulation of sequential\ncollisions and obtained the averaged value of $r_g$ from 30 runs for each\n$\\xi_\\mathrm{crit}$ and $v_\\mathrm{imp}$.\nIn Figure~\\ref{fig:rgrho}a,  the horizontal axis is the number of the\nconstituent particles, $N$, in the growing aggregates and the vertical\naxis is the gyration radii divided by $N^{1\/2}r_1$ for comparison\nwith BCCA clusters.\nThe dashed line represents the radius of the BCCA cluster\nand it is \nalmost flat for\nlarge $N$\nas expected from equation (\\ref{bccarg}).\nThe size of small aggregates produced in our simulation is almost the same as\nthat of BCCA clusters. \nThis is because\nthe impact energy is small enough at the early stage of the aggregate\ngrowth and the compression is ineffective at\neach collision. As the aggregates grow,\nthe impact energy increases.\nWhen the impact energy attains to $E_\\mathrm{roll}$, the compression of\nthe aggregate starts: aggregates become smaller than BCCA clusters. \nThe critical number of\nparticles, $N_\\mathrm{crit}$, in the aggregate for compression is given\nby (W08, S08)\n\\begin{equation}\n N_\\mathrm{crit} = \\beta\\frac{8E_\\mathrm{roll}}{m_1v_\\mathrm{imp}^2},\n\\label{ncrit}\n\\end{equation}\nwhere $\\beta$ is a non-dimensional coefficient. In Figure~\\ref{fig:rgrho}a,\nwe also plot the critical number $N_\\mathrm{crit}$ with filled circles on\neach curve, by setting $\\beta=0.5$.\nFigure~\\ref{fig:rgrho}b shows bulk densities of growing aggregates in\nthese simulations, which evaluated with $r_g$ by\nequation~(\\ref{rhodef}). \nWe also plot the density of BCCA clusters.\nIt decreases as $N^{-0.50}$ for large $N$.\nAfter the onset of compression (i.e., $N > N_\\mathrm{crit}$), the\nbulk densities of the aggregates obtained by the numerical simulation \nare larger than that of the BCCA but still keep on decreasing gradually\nin all cases,\nindicative of the inefficiency of collisional compression.\nS08 also developed the compression model,\nwhich reproduces \nthe density evolution of growing aggregates in\nthe compression stage. We will describe the compression model \nin Section 4.\n\\section{GEOMETRICAL CROSS SECTIONS OF AGGREGATES PRODUCED IN THE\n SEQUENTIAL COLLISIONS}\n\nWe numerically \ncalculate\ncross sections of resultant aggregates \nobtained by\nS08. The cross section of a dust aggregate is given by\nthe area of the shadow of the aggregate projected onto a plane. The area\nof the shadow is calculated by counting the number of square meshes\nin the shadow (Fig.~\\ref{fig:ssmosikizu}). The width of the square\nmeshes is set to be\n$0.0055 r_1$. This width is much smaller than the radius of the\nmonomer particle, $r_1$, \nthough meshes with a much wider width are\ndrawn to emphasize them in Figure~\\ref{fig:ssmosikizu}.\nThe area of the shadow is dependent on the\nplane onto which the shadow is projected.\nWe calculate the areas of the shadows for 30 orientations randomly\nchosen and define the cross section of the aggregate\nby the mean values of the areas. Figure~\\ref{fig:ss1headon} shows the\ncross section\ncalculated in this way for  aggregates produced in the simulations of\nsequential collisions. \nThe vertical axis is the cross\nsection divided by $N \\pi r_1^2$, \nwhich corresponds to the (non-dimensional) cross section per mass. \nIf the overlapping of the monomer particles in the shadow is negligibly \nsmall, the value of the vertical axis \napproaches\nunity. A filled circle in Figure~\\ref{fig:ss1headon}a \nindicates the shadow area\nfor each orientation and the line shows the mean of them. \nThe cross section per mass\ndecreases as an aggregate grows in the simulation of sequential\ncollisions due to the overlapping of the constituent particles. \nSince S08 did 30 independent runs of sequential collisions, the mean cross\nsections are calculated and plotted as thin lines in\nFigure~\\ref{fig:ss1headon}b. \nThen we obtain the averaged value of the\n30 mean cross sections as shown by the thick line in\nFigure~\\ref{fig:ss1headon}b.\nThe dispersion of the mean geometrical cross section can be evaluated\nin Figure~\\ref{fig:ss1headon}b.\nThe standard deviation of the mean geometrical\ncross section is equal to or less than 11\\% of its averaged value\nduring the aggregate growth.\nIn this way, we did two kinds of averaging to calculate the cross section\nof growing aggregates in the simulation.\nWe show and discuss the cross sections of aggregates by using these\nfinally-obtained cross sections hereafter.\n\nFigure~\\ref{fig:ssheadon} shows the cross section of resultant\naggregates for various values\nof the parameter set ($\\xi_\\mathrm{crit}, v_\\mathrm{imp}$).\nThe averaged cross section of the BCCA cluster is also calculated and\nplotted. \nThe cross section of BCCA we obtained agrees \nwith the result of\n\\citet{min06b}.\nEven for BCCA, the ratio $S\/(N \\pi r_1^2)$\ngradually decreases with an increase in $N$ due to the overlapping of\nconstituent particles.\nIn the early stage of the aggregate growth (i.e. for small $N$), the cross\nsections of the resultant aggregates change almost along the line of\nBCCA. As the aggregates grow, however, their cross sections deviate from\nthe line of BCCA and become much smaller than that of BCCA, which\nis due to compression at collisions. This qualitative tendency \nis consistent\nwith the evolution of the radius of gyration (Fig.~\\ref{fig:rgrho}a).\nAlthough the change in the cross sections is gradual and the starting\npoints of compression in the cross sections are not clear compared with\nthose in the gyration radii, the starting points are also described by\nequation~(\\ref{ncrit}), by setting the parameter\n$\\beta$ to be 2.0.\nThe larger value of $\\beta$ than Figure~\\ref{fig:rgrho} indicates that\nthe onset of compression in the cross section is later than that in the\ngyration radius.\n\nIn Figure~\\ref{fig:ssrgheadon} we plot the ratio of the cross section $S$ \nto $\\pi r_g^2$ for all resultant aggregates with solid lines. \nThe ratio of  $S$ to $\\pi r_g^2$ decreases for small aggregates, \nwhich is consistent with the BCCA case (dotted lines). \nFor sufficiently large aggregates, the ratio increases as a result of\ntheir compression.\nOkuzumi et al. (2009) proposed a useful expression of \nthe cross section $S$ for aggregates formed through hit-and-sticks.\nThe expression is given by\n\\begin{equation}\nS(r_g, N) = \\left(\n\\frac{1}{S\\sub{BCCA}(N)} + \\frac{1}{ \\pi (5\/3) r_g^2}\n-\\frac{1}{ \\pi (5\/3) r\\sub{$g$, BCCA}(N)^2}\n\\right)^{-1},\n\\label{s-O09}\n\\end{equation}\nwhere the cross section of the BCCA cluster is given by \n(Minato et al. 2006)\n\\begin{eqnarray}\n\\frac{S_\\mathrm{BCCA}}{\\pi r_1^2}=\\left\\{\n \\begin{array}{cc}\n  12.5N^{0.685}\\exp(-2.53\/N^{0.0920}) &(N < 16),\\\\\n  0.352N+0.566N^{0.862} &(N \\geq 16).\\label{sbcca}\n \\end{array}\n  \\right.\n\\end{eqnarray}\nEquation~(\\ref{bccarg}) is used as the expression of\n$r\\sub{$g$,BCCA}(N)$.\nWe also plotted the cross sections obtained from equation (\\ref{s-O09})\nwith dashed lines in Figure~\\ref{fig:ssrgheadon}.\nIt is found that the expression by Okuzumi et al. (2009) \nreproduces the cross section surprisingly well for compressed \naggregates as well as hit-and-stick aggregates, by using the gyration \nradius $r_g$. In this expression, the information of compression\nis correctly included through the gyration radius.\nPaszun and Dominik (2009) also derived another relation\nbetween $S$ and the aggregate size (i.e., eq.[11] of their paper).\nFigure~7 is the same as the left-bottom panel of Figure~6\nbut the prediction by Paszun and Dominik is also plotted.\nAlthough the prediction by Paszun and Dominik is consistent with\nthe numerical results,\nit overestimates $S$ when the ratio $S\/(\\pi r_g^2)$ is larger than unity\n(i.e., for relatively compact aggregates)\nand underestimates for $S\/(\\pi r_g^2)<0.7$.\nThe underestimation was also reported by Okuzumi et al.\nThey found that the underestimation in Paszun and Dominik's model\nis severe especially for large and fluffy aggregates.\nFor other $\\xi_\\mathrm{c}$,\nwe also find the same trend as in the case of \n$\\xi_\\mathrm{c}=8\\mathrm{\\mbox{\\AA}}$ shown in \nFigure~\\ref{fig:ssrgheadon2}.\nHence it is concluded that the model of Okuzumi et al.~is \nmore accurate than that of Paszun and Dominik.\nOnce an accurate compression model describing $r_g$ is obtained,\nit enables us to calculate the evolution of the cross section with the\nhelp of Okuzumi et al's model.\n\n\n\\section{COMPRESSION MODEL}\nA compression model describing $r_g$ was developed by S08\nbut it has two limitations.\nThe model is not directly applicable to low-energy collisions\n(i.e., hit-and-stick collisions at the early growth stage) or \nto non-equal-mass collisions.\nIn order to describe gyration radii and cross sections of aggregates\nfor both the early hit-and-stick stage and the compression stage seamlessly,\nwe refine our compression model,\nby removing these limitations in a natural way.\nBefore that, we briefly describe the compression model by S08.\n\n\\subsection{Compression Models of W08 and S08}\nW08 developed a compression model by introducing the pressure\n(or the strength) of aggregates to explain their numerical results on\ncollisions between BCCA clusters.\nThe compression model of S08 is based on that of W08.\nAt a collision of two aggregates with the impact energy \n$E_\\mathrm{imp}$, the compression of the merged aggregate from the \ninitial volume, $V_\\mathrm{initial}$, to the final\nvolume, $V_\\mathrm{final}$, is described in the model of W08 by\n\\begin{equation}\n E_\\mathrm{imp} = -\\int_{V_\\mathrm{initial}}^{V_\\mathrm{final}} P dV.\n\\label{eimpp}\n\\end{equation}\nThe initial volume $V_\\mathrm{initial}$ is defined by the volume of the merged\naggregate at the moment that the two aggregates just stick. After the\nmoment of the sticking, the compression proceeds. The volumes before\nand after the compression are evaluated with the radius of gyration,\n$r_g$, as in equation~(\\ref{vdef}). The pressure $P$ of the aggregates\nis given by\n\\begin{equation}\n P = 2 \\left(\\frac{5}{3}\\right)^6  \n\\frac{b E_\\mathrm{roll}\\rho_m}{m_1}\n \\left(\\frac{\\rho}{\\rho_m }\\right)^{13\/3} N^{2\/3},\\label{pform}\n\\end{equation}\nwhere the fitting parameter $b$ is set to be 0.15. Note that the\npressure $P$ of the aggregates is dependent on the total number of\nconstituent particles (or the total mass) as well as the density.\nThat is, $P$ is not an intensive variable.\nThis strange property in the pressure comes from the fractal\nstructure of the aggregates. W08 showed with their simulation of\ncollisions between BCCA clusters that the compressed aggregates \nhave internal structures with a fractal dimension of\n2.5. The simulation of sequential collisions done by S08 showed that\ntheir resultant aggregates also have the same fractal dimension of 2.5.\n\nIn order to describe the compression of such fractal aggregates,\nW08 also introduced the fractal volume defined by\n\\begin{equation}\n V_f(r_g)\\equiv ar_g^{2.5},\n\\end{equation}\nwhere \nthe coefficient $a$ is given by \n$(9\\pi\/5)^{5\/4}\\Gamma(9\/4) \\simeq 7.7$.\nUsing the fractal volume, the fractal density is defined by\n\\begin{equation}\n \\rho_f(r_g)\\equiv\\frac{m_1N}{V_f(r_g)}=\\frac{m_1N}{a}r_g^{-2.5}\n  \\label{rhofdef}.\n\\end{equation}\nThe dimensions of the fractal volume and the fractal density differ\nfrom those of the ordinary volume and density. These fractal\nquantities are related with the ordinary quantities $V$ and $\\rho$ as\n\\begin{eqnarray}\n\\frac{V(r_g)}{v_m}&=&\\left(\\frac{5}{3}\\right)^{3\/2}\n \\left( \\frac{V_f(r_g)}{v_{f,1}}\\right)^{6\/5},\\label{ord_fra_v}\\\\\n \\frac{\\rho(r_g)}{\\rho_m}&=&\\left(\\frac{3}{5}\\right)^{3\/2} \n  \\left(\\frac{\\rho_f(r_g)}{\\rho_{f,1}}\\right)^{6\/5}\n N^{-1\/5},\\label{ord_fra}\n\\end{eqnarray}\nwhere $v_m$ ($=4\/3\\pi r_1^3$) is the volume of a monomer and $v_{f,1}$ is\ngiven by $ar_1^{2.5}$.\nUsing the fractal volume $V_f$ (and the fractal density $\\rho_f$),\nequation~(\\ref{eimpp}) is rewritten as\n\\begin{equation}\n E_\\mathrm{imp} = -\\int_{V_{f,\\mathrm{initial}}}^{V_{f,\\mathrm{final}}} P_f\n  dV_f,\n  \\label{eimppf}\n\\end{equation}\nwhere the fractal pressure $P_f$ is given by\n\\begin{equation}\n P_f \\equiv P \\frac{dV}{dV_f}\n     =  4 \\frac{bE_\\mathrm{roll}\\rho_{f,1}}{m_1}\n     \\left(\\frac{\\rho_f}{\\rho_{f,1}}\\right)^5.\\label{pf} \n\\end{equation}\nIt should be noticed that the fractal pressure is dependent on\n$\\rho_f$ but not on the total mass. \nThat is, $P_{f}$ is an intensive variable.\nEquation (\\ref{eimppf}) (or\n[\\ref{eimpp}]) reproduces the numerical results on the compression\nat collisions between BCCAs.\n\nS08 pointed out that W08's compression model\nneeds a minor modification to describe the compression of\npartially compressed aggregates at their collisions, which occurs\nin their simulation of sequential collisions. At the moment of\nsticking at each collision, large voids are produced in the merged\naggregate. The volume of the new voids is included in the initial\nvolume of the merged aggregate, $V_{f,\\mathrm{initial}}$ in\nequation~(\\ref{eimppf}).\nThe energy required for compression of the new voids is\n$\\sim E_\\mathrm{roll}$ and it is much smaller that that predicted by\nequation~(\\ref{eimppf})\nat collisions between partially compressed aggregates. To describe\nthe compression at such collisions, S08 modified W08's\nmodel. Since the energy required for the crush of the new voids is\nnegligible, the initial fractal volume of the merged aggregate in\nequation~(\\ref{eimppf}) is set to be the sum of the fractal volumes of two\ncolliding aggregates, by removing the volume of the new voids.\nThat is,\n\\begin{equation}\n V_{f,\\mathrm{initial}} = V_{f,1} + V_{f,2}\\label{vfinit}. \n\\end{equation}\nUsing equations~(\\ref{eimppf})-(\\ref{vfinit}), we have the (final)\nfractal density of the merged aggregate, $\\rho_{f,\\mathrm{final}}$,\nproduced at collisions of two equal-mass aggregates with the fractal\ndensity $\\rho_{f,0}~(=Nm_1\/[V_{f,1}+V_{f,2}])$\n\\begin{equation}\n \\left(\\frac{\\rho_{f,\\mathrm {final}}}{\\rho_{f,1}}\\right)^4=\n  \\left(\\frac{\\rho_{f,0}}{\\rho_{f,1}}\\right)^4\n   +\\frac{E_\\mathrm{imp}}{bNE_\\mathrm{roll}}.\\label{oldmodel}\n\\end{equation}\nwhere $N$ is the number of constituent particles in the merged one.\nEquation~(\\ref{oldmodel}) describes the density evolution of partially\ncompressed aggregates growing through mutual collisions.\nEquation~(\\ref{oldmodel}) with $b=0.15$ reproduces \nthe density evolution of growing aggregates in Figure~\\ref{fig:rgrho}\nfor $N>N_\\mathrm{crit}$, as seen in Figure~8 of S08.\n\n\n\\subsection{Refinement of the Compression Model}\n\nThe compression model by S08 is not applicable to the early growth stage \n($N<N_\\mathrm{crit}$) where the compression is ineffective. \nThis is because the energy\nrequired for the crush of the new voids is neglected in the model.\nThis limitation is removed, by taking into account the compression process \nof the new voids produced at sticking of two aggregates.\nFurthermore, the model of S08 assumes the collisions of equal-mass aggregates.\nWe also remove this limitation in a reasonable way.\nAt the extension of our compression model to non-equal-mass collisions, \nwe assume that the compressed aggregates have the fractal dimensions \nof 2.5 as well as in the case of equal-mass collisions. \nThe validity of the assumption will be discussed in Section 5.\n\nDensity evolution of aggregates at each collisions\nis divided into the following three steps: \n\\begin{enumerate}\n \\item Creation of new voids at the sticking of two aggregates.\n \\item Compression of the new voids in the merged aggregate. \n \\item Further compression of the merged aggregate after the crush of\n       the new voids.\n\\end{enumerate}\nThese steps are schematically explained by Figure~\\ref{fig:refinedmodel}.\nWe describe the change in the volume or the \ndensity of the aggregates at each step in detail below.\n\nIn Step~1, the density decreases because of the new voids in the merged\naggregate. The change in the density\nis described by Okuzumi et al. (2009).\nWhen the two aggregates with the masses $M_1$, $M_2$, and the\nvolumes $V_1$, $V_2$ ($\\le V_1$) collide with each other, the volumes\nof the merged aggregate $V'_{1+2}$ (before the compression)\nis given by\n\\begin{equation}\n V'_{1+2} = V_1 + V_2 + V_\\mathrm{void},\\label{vdash}\n\\end{equation}\nwhere the volume of the voids $V_\\mathrm{void}$ is obtained from\nthe empirical formula\n\\begin{equation}\n V_\\mathrm{void} =\\min\\left[\n                       \\chi\\sub{BCCA}-1.03\\ln\\left(\\frac{V_1+V_2}{2V_2}\\right),\n                       6.94\\right]V_2\n\\end{equation}\n(see Okuzumi et al. [2009] for the derivation).\nThe collision of equal-mass\naggregates is not assumed in Okuzumi et al. (2009).\nNote that $\\chi\\sub{BCCA} (\\equiv 2^{3\/d_f} -2)$ is 0.83 \nfor BCCA clusters formed through head-on collisions having the\nfractal dimension $d_f=2.0$.\nThe density of the merged aggregates before the compression\nis given by ($M_1+M_2)\/V'_{1+2}$. The fractal volume $V'_{f,1+2}$\nand the fractal density $\\rho'_{f,1+2}$ are obtained from\nequations~(\\ref{ord_fra_v}) and (\\ref{ord_fra}), respectively.\n\n\n\nIn Step~2 and 3, the merged aggregate is compressed\nand the density increases. Step~2 is the compression of the\nnew voids. We put the energy required for the crush of the\nnew voids to be $b'E_\\mathrm{roll}$, where $b'$ is the non-dimensional\nparameter. By fitting with results of the simulation,\nthis parameter is fixed to be $b'=3b$, as will be shown in the next\nsubsection. This relation would be reasonable because both parameters \nare related to the beginning of compression.\n\nWhen the impact energy $E_\\mathrm{imp}$ is smaller than\n$b'E_\\mathrm{roll}$,\nthe new voids are only partially compressed at Step~2.\nIn this case, the compression at Step 3 does not occur \nbecause of the small impact energy.\nThe impact energy $E_\\mathrm{imp}$ is \nevaluated using the reduced mass, $(1\/M_1+1\/M_2)^{-1}$.\nIn this case, the decrease in the fractal volume of the merged\naggregates, $\\Delta V_{f,\\mathrm{1+2}}$, is evaluated to be\n\\begin{equation}\n \\Delta V_{f,1+2} = \\frac{E_\\mathrm{imp}}{b'E_\\mathrm{roll}}\n  V_{f,\\mathrm{void}},\n\\end{equation}\nwhere\n\\begin{equation}\n V_{f,\\mathrm{void}} = V'_{f, 1+2} - V_{f,1} - V_{f,2}.\n\\label{vfvoid}\n\\end{equation}\nIt should be noticed that the fractal volume $V_{f,\\mathrm{void}}$\ndefined by equation~(\\ref{vfvoid}) is negative in collisions where\nthe volume ratio $V_1\/V_2$ is larger than $8 \\times 10^4$.\nA prescription for such high-volume-ratio collisions will be described \nat the end of this subsection.\nHere we consider the case where $V_{f,\\mathrm{void}}$ is positive.\nThen the final fractal volume $V_{f, 1+2}$ after the compression\nis given by\n\\begin{equation}\n V_{f, 1+2} = V'_{f, 1+2} - \\Delta V_{f,1+2}.\\label{modellow}\n\\end{equation}\nand the final fractal density $\\rho_{f,\\mathrm{final}}$ of the merged\naggregate is given by $(M_1+M_2)\/V_{f, 1+2}$.\n\nWhen the impact energy is larger than $b'E_\\mathrm{roll}$, \nthe new voids are crushed completely and the merged\naggregate is further compressed at Step~3. Since the fractal\nvolume of the merged aggregate is $V_{f,1}+V_{f,2}$ at the end\nof Step~2, the final fractal volume $V_{f, 1+2}$ after Step 3\nis obtained from the equation\n\\begin{equation}\nE_\\mathrm{imp} - b'E_\\mathrm{roll} =\n - \\int_{ V_{f,1}+V_{f,2} }^{ V_{f, 1+2} } P_f(\\rho_f) dV_f.\n\\label{modelhigh}\n\\end{equation}\nIntegrating the RHS of equation~(\\ref{modelhigh}), we obtain the final\nfractal density for $E_\\mathrm{imp} > b'E_\\mathrm{roll}$ as\n\\begin{equation}\n \\left(\\frac{\\rho_{f,\\mathrm {final}}}{\\rho_{f,1}}\\right)^4=\n  \\left(\\frac{\\rho_{f,0}}{\\rho_{f,1}}\\right)^4\n   +\\frac{E_\\mathrm{imp}-b'E_\\mathrm{roll}}{bNE_\\mathrm{roll}},\n   \\label{rhoevol_up1}\n\\end{equation}\nwhere we used \n\\begin{equation}\n\\rho_{f,0}= \\frac{M_1+M_2}{V_{f,1}+V_{f,2}}.\n\\label{rhof0}\n\\end{equation}\nIn the limit of\n$E_\\mathrm{imp}\\gg b'E_\\mathrm{roll}$,\nequation~(\\ref{rhoevol_up1}) is identical to\nequation~(\\ref{oldmodel}) (or the compression model by S08).\nThe final fractal volume $V_{f,1+2}$ is given by\n$(M_1+M_2)\/\\rho_{f,\\mathrm{final}}$.\n>From the fractal density, we obtain the gyration radius $r_g$,\nusing equation~(\\ref{rhofdef}). The cross section $S$\nis also obtained from Okuzumi et al's expression (eq.[\\ref{s-O09}]).\nIn this way, we can calculate the density evolution \n(i.e., the evolution of $r_g$ and $S$) at both low- and high-energy \ncollisions, using equation~(\\ref{modellow}) for \n$E_\\mathrm{imp} < b'E_\\mathrm{roll}$ and equation~(\\ref{rhoevol_up1}) for \n$E_\\mathrm{imp} > b'E_\\mathrm{roll}$.\n\nIn high-volume-ratio collisions where $V_1\/V_2 > 8\\times10^4$,\nas noticed above, $V_{f,\\mathrm{void}}$ is negative and\na special prescription is necessary. \nSince a negative $V_{f,\\mathrm{void}}$ means no voids,\nStep 2 should be omitted and \nthe merged aggregate is compressed only with Step 3.\nThat is, equations~(\\ref{modelhigh})-(\\ref{rhof0}) are used\nfor all impact energies in this case.\nIn equations~ (\\ref{modelhigh})-(\\ref{rhof0}), the terms of \n$b'E_\\mathrm{roll}$ is omitted and $V_{f,1}+V_{f,2}$ is replaced by \n$V'_{f,1+2}$ because Step 2 does not occur.\n\n\\subsection{Test of the Refined Compression Model}\n\nLet us test the refined compression model with the numerical results.\nUsing the refined compression model, we calculate the evolution of\ngyration radius for the same condition as the numerical simulations by\nS08 and also obtain the cross sections of the aggregates with \nequation~(\\ref{s-O09}). The results are shown in Figures~\\ref{fig:rgfit}\nand \\ref{fig:sfitoku}. In Figure~\\ref{fig:rgfit}, we plot the evolution\nof the gyration radius calculated with the refined compression model\nand compared it with the numerical results by S08. The parameters $b$ and \n$b'$ are set to be $b=0.15$ and $b'=3b~(=0.45)$, respectively.\nWith this setting of the parameters, the refined model reproduces well \nthe numerical results at both the early growth stage and the compression \nstage. Figure~\\ref{fig:sfitoku} shows the evolution of the cross sections\nand indicates that the refined model also succeeds in describing the \ncross sections with the help of equation~(\\ref{s-O09}). \n\nIn the above, the evolution of gyration radius of growing aggregates\nare calculated with the refined compression model\nand their cross sections are indirectly calculated, by using\n$r_g$ and equation~(\\ref{s-O09}).\nWe also propose another way to describe the cross sections of aggregates.\nWe define alternative characteristic sizes of aggregates $r_S$ by \n\\begin{equation}\nr_S=\\sqrt{S\/\\pi}.\n\\label{eq:rs}\n\\end{equation}\nIt would be possible to describe the evolution of $r_S$\ndirectly (instead of the gyration radius)\nwith the refined compression model in the following way.\nUsing this characteristic size $r_S$ instead of $\\sqrt{5\/3}r_g$,\nwe can define the volume and the bulk density of the aggregate\nby the similar equations to (\\ref{vdef}) and (\\ref{rhodef}).\nThe fractal volume and the fractal density are also defined in the same way.\nThen, applying the refined compression model to the fractal density\ndefined with $r_S$, we can describe the evolution of $r_S$\nas well as in the case of $r_g$. \nThe evolution of the cross section $S$ is calculated with \nequation~(\\ref{eq:rs}).\nThis is a direct way to describe the\ncross section rather than the above.\nIn this calculation of $S$,\nwe have to be cautious with the following two points.\nOne is the modification in equation~(\\ref{vdash}).\nAt a collision of sufficiently fluffy aggregates, \nequation~(\\ref{vdash}) can give the volume $V'_{1+2}$ larger\nthan that of the BCCA cluster with the same mass, $V_\\mathrm{BCCA}$\nwhen the size $r_S$ is used instead of $r_g$.\nSuch a large $V'_{1+2}$ is not realistic. In this case, we set\nthe volume $V'_{1+2} = V_\\mathrm{BCCA}$ instead of equation~(\\ref{vdash}).\nThe other point is the parameter $b$.\nAlthough $b$ is set to be 0.15 in the case of $r_g$,\nwe have to calibrate the parameter $b$ again in the case of $r_S$\nas a result of the fitting with numerical results.\nIn Figure~\\ref{fig:sfit}, we plot the evolution\nof $S$ calculated with this direct way.\nIn this calculation, the parameters are set to be $b=0.6$ and $b'=3b$.\nWe see that the refined model works well for the evolution of the\ncross sections with this direct way, too.\\footnote{\nIn the early growth stage of Figure~\\ref{fig:sfit},\nwe used $V'_{1+2} = V_\\mathrm{BCCA}$\ninstead of equation~(\\ref{vdash}) \nwhen the volume of equation~(\\ref{vdash})\nis larger than $V_\\mathrm{BCCA}$, as mentioned above.\n}\n\nIt is found that the refined compression model enables us to describe \nthe whole evolution of the radius of gyration $r_g$ and the cross \nsection $S$ of growing aggregates.\nNote that the refined model is applicable to non-equal-mass\ncollisions though the tests in Figure~\\ref{fig:rgfit}-\\ref{fig:sfit} are \ndone only in the equal-mass case. \nAt the extension of the refined model to non-equal-mass\ncollisions, the fractal dimension of compressed aggregates\nis assumed to be 2.5 even in the case of non-equal-mass collisions\nthough it is not verified with $N$-body simulations in non-equal-mass cases.\nIt is possible that a very large mass ratio increases the fractal dimension,\nas seen in BPCA clusters.\nOkuzumi et al. (2009) examined the effect of the mass ratio\non the fractal dimension for aggregates growing with hit-and-stick\ncollisions with $N$-body simulations. They showed that\nthe collisions with the mass ratio of 10 increases\nthe fractal dimension $d_f$ only by 0.1 (see their figure 6). Furthermore, \ncollisions with such a mass ratio have a major contribution at dust \ngrowth in protoplanetary disks, as shown by Okuzumi et al.~(2009,2012).\nHence the effect of non-equal-mass collisions would not \nchange largely the fractal dimension of compressed aggregates\nin the realistic growth process.\n\nTo confirm the validity of the refined model in non-equal-mass\ncollisions, we have further performed additional $N$-body simulations of \naggregate collisions. Similar to W08, we consider collisions of two BCCA \nclusters but their masses are not equal in the present case.\nThe projectile BCCA cluster consists of 1024 particles (or 4096 particles)\nwhile the number of constituent particles of the target BCCA cluster \nis 16384. Their mass ratio is $1\/16$ (or $1\/4$).\nThe constituent particles are icy ones with the radius with 0.1$\\mu$ m.\nThe impact velocity $v\\sub{imp}$ is a parameter.\nWe set $v\\sub{imp} \\le 4.4$ms$^{-1}$ since we focus on the compression \nprocess rather than fragmentation (Wada et al. 2007,2008).\nThe numerical results of compression at the non-equal-mass collisions\nare shown in Figure~\\ref{fig:additional}.\nThe predictions by the refined model with $b=0.15$ are plotted by solid \nlines and the dashed line indicates the formula by W08 (their equation~[45]).\nThe numerical results in the non-equal-mass collisions approximately \nagree with the predictions by the refined model though upper shifts by \n$\\sim$20\\% are observed in the case of $M_p\/M_t=1\/16$.\nAs well as in the case of equal-mass collisions,\nthe slope in $r_g$-$E\\sub{imp}$ relation is approximately given by -0.1,\nwhich indicates that compressed aggregates have the fractal dimension of \n2.5. The upper shifts in the numerical results indicate that the \ncompression requires larger impact energy in non-equal-mass collisions \nthan the equal-mass case. This effect in non-equal-mass collisions would \nbe included by adopting a larger parameter $b$ in the refined model.\nFurthermore, for high-mass-ratio collisions with $M_t\/M_p \\gg 10$,\nthe refined model is not verified through $N$-body simulations yet\nalthough such collisions have only a minor contribution in dust \ngrowth (Okuzumi et al.~2009,2012).\nIn the future work, the effect of non-equal-mass collisions\nshould be further examined in the numerical simulation of sequential \ncollisions as done by S08 in order to calibrate the parameter\n$b$ more accurately.\n\\section{SUMMARY}\nWe examined the evolution of the geometrical cross section of the\ngrowing (icy) aggregates obtained by $N$-body simulations of\nsequential head-on collisions (S08)\nand constructed to construct a refined compression model,\nwhich is applicable to the description of the evolution of both\ngeometrical cross sections and gyration radii of growing aggregates.\nThe results are summarized as follows:\n\\begin{enumerate}\n \\item We examined geometrical cross sections of the aggregates\n       produced in the simulation of sequential collisions done in our\n       previous paper. \n\tAs aggregates grow, compression becomes effective and makes their \n\tcross sections smaller than those of the BCCA clusters. The \n\tbeginning of the compression is given by equation~(\\ref{ncrit}), \n\tas seen in the evolution of the gyration radius.\n \\item The relation between the cross section and the gyration radius\n       seen in aggregates obtained by S08\n\tis well described by Okuzumi et al's expression.\n       This indicates that Okuzumi et al's expression is valid\n        for compressed aggregates as well as hit-and-stick aggregates.\n       If the evolution of the gyration radius is well described by\n       a compression model, Okuzumi et al's expression enables us to\n       calculate the cross section, too.\n \\item We further refined the compression model of S08, by including \n       the compression energy for the voids produced at the sticking \n       of two aggregates. The refined model is also extended to\n       non-equal-mass collisions in a reasonable way.\n\tWith the refined model, we can accurately reproduce the evolution \n        of both the gyration radius and the cross section of aggregates\n        obtained by S08 from their early growth stage.\n       The validity of the refined compression model\n       for non-equal-mass collisions is also checked by\n       additional numerical simulations of BCCA collisions.\n       Although S08 considered only icy aggregates in the numerical \n       simulation, our compression model would be also applicable to \n       silicate aggregates by using a suitable value of $E_\\mathrm{roll}$. \n\\end{enumerate}\n\nOur $N$-body simulations of aggregate collisions and the refined \ncompression model indicate that collisional compression is not so \neffective. As a result, dust aggregates (or initial planetesimal material) \nwould have extremely low bulk densities, as suggested by S08. Okuzumi \net al. (2012) showed that such extremely low bulk densities of aggregates \naccelerate their growth in protoplanetary disks and help their overcoming \nof the radial drift barrier against the planetesimal formation. However, \nsolar-system bodies do not have such low densities at present. Dust \naggregates (or planetesimals) should be compressed by other processes. \nA steady ram pressure due to the gas drag on aggregates and a \nself-gravity of sufficiently large aggregates would be candidates for \naggregate compression, as indicated by S08 and Okuzumi et al. (2012). \nFor relatively compact dust cakes \n($\\rho \\sim0.1$g$\/$cm$^3$) made of micron-sized silicate particles, \ncompression is observed at a pressure $> 100$Pa (Blum and \nSchr\\\"{a}pler 2004). However, for icy aggregates with very low bulk \ndensities ($\\rho \\ll 0.1$g$\/$cm$^3$), compressive strength \nhas not yet been measured. In future work, compression strength of \nvery fluffy aggregates should be measured in numerical simulations \nand laboratory experiments.\n\nIn the present study,\nwe focus on the aggregates obtained at head-on collisions. \nAt oblique collisions, the merged aggregates are elongated\n(W07; Paszun and Dominik 2009). Although our model\nindicates inefficient compression at aggregate collisions, the effect of \noblique collisions would further hinder compression.\nIn future work, we should clarify the validity of our compression model \nin the case where oblique collisions are included.\n\\acknowledgments\nThe authors would like to thank Hiroshi Kimura, \nTetsuo Yamamoto and Hiroshi Kobayashi for\ntheir valuable comments. We would also like to thank Takeshi Chigai for\ntechnical support with respect to the computer setup. This study was\nsupported by a Grant-in-Aid from JSPS(22540242, 22740299).\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\chapter[Aging rate: exploring the growth of the coherence length]{Aging rate: exploring the growth \\\\ of the coherence length} \\labch{aging_rate}\n\\setlength\\epigraphwidth{.5\\textwidth}\n\nExperiments in \\gls{SG}s are developed in out-of-equilibrium conditions most of the times\\footnote{As we have mentioned before, recent experiments in thin-film geometry~\\cite{guchhait:14,guchhait:15b} stands as honorable exceptions.}. Typically, the experimental setup consists of a system that it is rapidly cooled from $T_1>\\ensuremath{T_\\mathrm{c}}\\xspace$ to $T_2<\\ensuremath{T_\\mathrm{c}}\\xspace$, and its off-equilibrium evolution, which we have already termed as \\textit{aging\\index{aging}} (see~\\refsubsec{aging_memory_rejuvenation}), is studied.\n\nUnder these non-equilibrium conditions, it was originally predicted in the context of the droplet\\index{droplet!picture} theory (see~\\refsubsec{theoretical_pictures}) that domains\\index{magnetic domain} of correlated spins start to grow at the microscopic level~\\cite{fisher:88}. Although with some differences in the nature of the domains\\index{magnetic domain}, \\gls{RSB} also expects a similar behavior. The linear size of those domains\\index{magnetic domain} is known as \\textit{coherence length\\index{coherence length}}.\n\nThis coherence length\\index{coherence length} have been measured in numerical simulations~\\cite{huse:91,marinari:96,komori:99,komori:00} and also in experiments~\\cite{joh:99,bernardi:01} long time ago. The initial expectation~\\cite{fisher:88} for the growth of the coherence length\\index{coherence length} with the time was $\\xi \\sim \\left(\\log \\ensuremath{t_\\mathrm{w}}\\xspace\\right)^{1\/\\psi}$ and some numerical simulations found that ansatz to be compatible with their results (see, e.g.~\\cite{kisker:96}). However, the mainstream usually accepts the alternative growth functional form described by $\\xi \\sim \\ensuremath{t_\\mathrm{w}}\\xspace^{1\/z(T)}$ which better describes the results.\n\nThe aging rate\\index{aging!rate} $z(T) = {\\mathrm{d}} \\log \\ensuremath{t_\\mathrm{w}}\\xspace \/ {\\mathrm{d}} \\log \\xi$ was difficult to measure in traditional experiments based on the study of the shift of the peak in the relaxation\\index{relaxation} rate $S(\\ensuremath{t_\\mathrm{w}}\\xspace)$~\\cite{joh:99}. However, it can be now experimentally measured with excellent accuracy through the study of activation energies in \\gls{SG}s with thin-film geometry~\\cite{zhai:17}.\n\nA strong discrepancy has been found between numerical and experimental measurements of $z(T)$. We solve that discrepancy in Ref.~\\cite{janus:18} and this chapter is devoted to exposing the results of the cited reference that have been obtained in this thesis. In this chapter, we first motivate the work and describe the state of the art in~\\refsec{why_study_aging} and~\\refsec{how_can_study_aging}. Then, we describe the numerical simulation that we have performed in~\\refsec{numerical_simulation_aging}. We describe the problem in~\\refsec{controversy_aging} and finally we show the results in~\\refsec{large_xi_limit}.\n\n\\section{Why should we care about off-equilibrium dynamics?} \\labsec{why_study_aging}\nWe have stated several times that experiments and theory focus on different regimes, off-equilibrium, and equilibrium respectively. Moreover, we have also stated that, traditionally, simulations have been a powerful tool in theoretical research.\n\nThe increase of computational power has recently allowed us to promote the numerical simulations to a higher ``responsibility'' role in the development of the \\gls{SG} field. Now, our simulations are in the border of the experimental regime~\\cite{janus:08b,janus:09b,janus:17b,janus:18} and therefore, numerical work is in a privileged situation. On the one hand, it still has the classical advantages of the simulations: we are able to access microscopic configurations\\index{configuration} that are difficult to access from the experiments, and we have total control of our system which is desirable in many senses (for example, our protocols are totally reproducible with no source of error). On the other hand, our numerical data can be now confronted with the experimental one through mild extrapolations (see e.g. ~\\cite{janus:17b,janus:18,zhai-janus:20,zhai-janus:21}). This is extremely useful, not only because it allows us to test the numerical models but also because we can compute theoretical quantities, not accessible from experiments, in our experiment-compatible system at relevant time-scales.\n \nMoreover, in the last years, the development of a statics-dynamics dictionary~\\cite{barrat:01,janus:08b,janus:10,janus:10b,janus:17} has been a milestone in the development of the numerical research. According to the statics-dynamics equivalence\\index{statics-dynamics equivalence}, the off-equilibrium properties of an (effective) infinite system that ages for a finite-time $\\ensuremath{t_\\mathrm{w}}\\xspace$ with a coherence length\\index{coherence length} $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$, are tightly bounded with the equilibrium properties of a finite-size system with linear length $L \\sim \\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$. The study of out-equilibrium systems may be helpful in order to extend the statics-dynamics dictionary and establish new relations.\n\n\\section{Coherence length, a fundamental quantity to study off-equilibrium dynamics}\\labsec{how_can_study_aging}\nThe experimental study of out-equilibrium \\gls{SG}s is usually focused on the characterization of magnetic responses when an external magnetic field is applied. As we have already discussed in~\\refsec{experimental_spinglass}, both time scales, the aging\\index{aging} before turn on (off) the magnetic field and the subsequent magnetic evolution of the system, turned out to be essential to describe the off-equilibrium phenomenon.\n\nHowever, different aging\\index{aging!rate} rates $z(T)$ for different temperatures make the coherence length\\index{coherence length} $\\xi \\sim \\ensuremath{t_\\mathrm{w}}\\xspace^{1\/z(T)}$ a much more convenient quantity to describe the ``aging\\index{aging} state'' of the system. The meaning of the aging\\index{aging!rate} rate $z(T)$ comes directly from the Arrhenius law. As we have mentioned many times along this thesis, the \\gls{SG}s exhibit extremely slow dynamics. If we assume that the slow dynamics are due to the presence of many valleys in a rugged free-energy\\index{free energy!landscape} landscape, it is natural to propose the Arrhenius law to characterize the typical time-scale that the system needs to overcome these free-energy barriers\\index{free energy!barrier} and explore different valleys\\index{free energy!valley}.\n\\begin{equation}\n\\ensuremath{t_\\mathrm{w}}\\xspace = t(\\xi) = \\tau_0 \\exp \\left[ \\beta \\Delta(\\xi) \\right] \\, , \\labeq{arrhenius_law}\n\\end{equation}\nbeing $\\tau_0 \\propto \\beta_\\mathrm{c}$ a microscopic time-scale and the exponent $\\Delta(\\xi)$ is the size of the energetic barriers\\index{free energy!barrier} in units of $1\/\\beta=k_{\\mathrm{B}}T$. Therefore, the aging\\index{aging!rate} rate\n\\begin{equation}\nz(T,\\xi) = \\dfrac{{\\mathrm{d}} \\log \\ensuremath{t_\\mathrm{w}}\\xspace}{{\\mathrm{d}} \\log \\xi} = \\dfrac{{\\mathrm{d}} \\left[ \\beta \\Delta(\\xi)\\right]}{{\\mathrm{d}} \\log \\xi} \\, , \\labeq{def_aging_rate}\n\\end{equation}\ngive us the information of the evolution of the free-energy\\index{free energy!barrier} barriers with $\\log \\xi$. Actually, as we will discuss in \\refsec{large_xi_limit}, different hypothesis about the specific form of $\\beta \\Delta(\\xi)$ will lead to very different behavior in the large-$\\xi$ limit.\n\nMoreover, as mentioned in the introduction of the chapter, experimental studies on thin-film geometry \\gls{SG}s have achieved accurate measurements of this, once elusive, quantity~\\cite{zhai:17}. Besides, it has been found that the experimental estimation of $\\xi$ and the numerical one matches~\\cite{janus:17b}.\n\nThe recent advances in the experimental determination of the coherence length\\index{coherence length} $\\xi$ have also brought a discrepancy in the estimation of the aging\\index{aging!rate} rate $z(T)$ from numerical simulations and experiments~\\cite{zhai:17}. To solve this discrepancy, it is fundamental to estimate $\\xi$ with unprecedented accuracy. Two main factors have allowed us to compute the data needed to perform this research.\n\nFirst, the dedicated hardware Janus\\index{Janus} II~\\cite{janus:14} has a central role in this work. The simulation of very large systems to very long times has been the result of thousands of computational hours with the largest special-purpose\\index{special-purpose computer} machine focused on \\gls{SG}s.\n\nSecond, our particular choice of the simulation parameters has turned out to be fortunate. The numerical effort is usually focused on increasing the number of samples\\index{sample} $\\ensuremath{N_{\\text{S}}}\\xspace$ as much as possible and simulating the minimum number of replicas\\index{replica} needed to compute the observables, typically $\\ensuremath{N_{\\text{Rep}}}\\xspace=2$ or $\\ensuremath{N_{\\text{Rep}}}\\xspace=4$. However, we had in mind to study the \\textit{temperature chaos}\\index{temperature chaos} phenomenon (see~\\refch{out-eq_chaos}), where the determination of the error of the observables of interest is greatly benefited by a maximization number of overlaps\\index{overlap} $\\ensuremath{N_{\\text{ov}}}\\xspace = \\ensuremath{N_{\\text{Rep}}}\\xspace (\\ensuremath{N_{\\text{Rep}}}\\xspace-1)\/2$. Unexpectedly, this has led to a dramatic increase in precision. We analyze this reduction of the error in~\\refsec{Nr_aging}.\n\nThis study is a clear demonstration of the importance of the high-precision results for the investigation of glassiness. Indeed, without the dramatic reduction of the error bars\\index{error bars}, we would not be able to solve the discrepancy between numerical simulations and experiments.\n\n\\section{Numerical simulation} \\labsec{numerical_simulation_aging}\nIn this work, we simulate in the FPGA-based\\index{FPGA} [\\gls{FPGA}] computer Janus\\index{Janus} II an \\gls{EA}\\index{Edwards-Anderson!order parameter} model in three-dimensional spin glasses (\\refsubsec{3D_EA_model}) for several temperatures $T$ in a lattice of linear size $L=160$, which is aimed to represent a system of infinite size. This assumption is sound, provided that $L\\gg\\xi$ (see~\\refsec{finite_size_effects}). Note that this condition limits the maximum time at which we can safely ignore finite-size effects\\index{finite-size effects}. The temperature remains constant throughout the whole simulation.\n\nWe shall perform direct quenches from configurations\\index{configuration} of spins randomly initialized (which corresponds to infinite temperature) to the working temperature $T<\\ensuremath{T_\\mathrm{c}}\\xspace$, where the system is left to relax for a time $\\ensuremath{t_\\mathrm{w}}\\xspace$. This relaxation\\index{relaxation} corresponds with the (very slow) growth of glassy magnetic domains\\index{magnetic domain} of size $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$.\n\nWe compute a total of $\\ensuremath{N_{\\text{S}}}\\xspace = 16$ different samples\\index{sample}. For each sample\\index{sample}, we shall consider $\\ensuremath{N_{\\text{Rep}}}\\xspace=256$ replicas\\index{replica}. As we have already said, this simulation had the original aim to study the temperature chaos\\index{temperature chaos} phenomenon under non-equilibrium conditions (see~\\refch{out-eq_chaos}), however, the reader may notice that in that study we use a total number of replicas\\index{replica} $\\ensuremath{N_{\\text{Rep}}}\\xspace=512$. Indeed, this study about the aging\\index{aging!rate} rate was performed much earlier and we had at our disposal ``only'' $\\ensuremath{N_{\\text{Rep}}}\\xspace=256$.\n\nThe main observable of this study is the coherence length\\index{coherence length} $\\xi(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$, estimated by $\\xi_{12}(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$, computed from integral estimators of the correlation\\index{correlation function!four point} function $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$. Both observables have been described with great detail in \\refsubsec{observables_introduction}. The large number of replicas\\index{replica} simulated has allowed us to follow the decay of $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$ over six decades (see inset of~\\reffig{growth_xi}). The $\\xi$ estimation used in our work is plotted in~\\reffig{growth_xi}.\n\n\\begin{figure}[t]\n\\includegraphics[width=0.8\\linewidth]{aging\/xi12_and_C4}\n\\caption[\\textbf{Growth of the coherence length \\boldmath $\\xi(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$.}]{\\textbf{Growth of the coherence length\\index{coherence length} \\boldmath $\\xi(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$.} Growth of the coherence length\\index{coherence length} $\\xi_{12}(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$ with the waiting time $\\ensuremath{t_\\mathrm{w}}\\xspace$ after a quench to temperature $T$ in a log-log  scale [the critical\\index{critical temperature} temperature is $\\ensuremath{T_\\mathrm{c}}\\xspace=1.102(3)$]. Given the smallness of the statistical errors, instead of error bars\\index{error bars} we have plotted two lines for each $T$, which enclose the error estimate.  At this scale, the curves seem linear for long times, indicating a power-law growth but, see~\\reffig{a2}, there is actually a measurable curvature. \\textbf{Inset:} Spatial four-point correlation\\index{correlation function!four point} function of the overlap\\index{overlap!field} field $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$, plotted as a function of distance at the last simulated time for several temperatures. Note the six orders of magnitude in the vertical axis.}\n\\labfig{growth_xi}\n\\end{figure}\n\n\\section{The controversy of the aging rate}\\labsec{controversy_aging}\nThe growth of the coherence length\\index{coherence length} has been a debated issue in the \\gls{SG} literature (see, e.g. \\cite{fisher:88,huse:91,marinari:96,joh:99,bernardi:01,bouchaud:01,berthier:02}). However, despite the existence of different proposals, the simplest functional form that was able to fit the data was the power law\n\\begin{equation}\n\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace,T) = A(T) \\ensuremath{t_\\mathrm{w}}\\xspace^{1\/z(T)} \\quad , \\quad z(T) \\approx z(\\ensuremath{T_\\mathrm{c}}\\xspace)\\dfrac{\\ensuremath{T_\\mathrm{c}}\\xspace}{T} \\, , \\labeq{xi_powerlaw}\n\\end{equation}\nbeing $z(T)={\\mathrm{d}} \\log \\ensuremath{t_\\mathrm{w}}\\xspace \/ {\\mathrm{d}} \\log \\xi$ the so-called aging\\index{aging!rate} rate. Indeed, the experimental measurements concerns to the renormalized aging\\index{aging!rate} rate\n\\begin{equation}\nz_c = z(T) \\dfrac{T}{\\ensuremath{T_\\mathrm{c}}\\xspace} \\, . \\labeq{renormalized_aging_rate}\n\\end{equation}\nExperiments performed in thin-film geometry systems~\\cite{zhai:17} has measured $z_c \\approx 9.62$, which is very far from the value predicted by numerical simulations $z_c = 6.86(16)$~\\cite{janus:08b} and $z_c=6.80(15)$~\\cite{lulli:16}. \n\nThose experiments are performed in \\gls{CuMn} films with $20$ nm of thickness which translates to a distance of 38 lattice spacings (the typical Mn-Mn distance is 5.3 \\r{A}). Therefore, we will need to extrapolate our results to $\\xi_{12} \\approx 38$ in order to confront the numerical simulations and the experiments.\n\n\\subsection[The growth of $\\xi$ does not follow a power law]{The growth of \\boldmath $\\xi$ does not follow a power law} \\labsubsec{no-powerlaw}\nThe increase of the precision of the data shows that the pure power law of~\\refeq{xi_powerlaw} is not a faithful description anymore. Indeed, in order to discern if our data of~\\reffig{growth_xi} presents a deviation from a power law, we propose a very naive ansatz\n\\begin{equation}\n\\log \\ensuremath{t_\\mathrm{w}}\\xspace(T,\\xi_{12}) = a_0(T) + a_1(T)\\log \\xi_{12} + a_2(T) \\log^2 \\xi_{12}  \\, , \\labeq{naive_divergent}\n\\end{equation}\nwhere $a_0(T)$, $a_1(T)$ and $a_2(T)$ are meaningless coefficients, only useful to interpolate our data. An absence of curvature [$a_2(T)=0$] would reduce~\\refeq{naive_divergent} to~\\refeq{xi_powerlaw}. On the contrary, $a_2(T)>0$ indicates a slowing down in the dynamics for increasing $\\xi_{12}(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$.\n\n\\reffig{a2} is telling us that $a_2 \\geq 0$ and only vanishes for $T=\\ensuremath{T_\\mathrm{c}}\\xspace$ with $z_c=6.69(6)$ which improves the accuracy of previous estimations. Therefore, the solution of the discrepancy of the results of $z_c$ seems to be the introduction of a (very mild) scale dependence in the \\textit{effective} dynamical exponent which is defined as\n\\begin{equation}\nz(T,\\xi_{12}(\\ensuremath{t_\\mathrm{w}}\\xspace)) = \\dfrac{{\\mathrm{d}} \\log \\ensuremath{t_\\mathrm{w}}\\xspace}{{\\mathrm{d}} \\log \\xi_{12}} \\, . \\labeq{definition_aging_rate}\n\\end{equation}\n\nThe reader may think about the possibility that our deviation from the power-law behavior might be due to the existence of finite-size effects\\index{finite-size effects}, however, two main reasons against this argument can be defended. First, the curvature decreases when increasing the temperature (see~\\reffig{a2}) and one would expect the opposite behavior in presence of finite-size effects\\index{finite-size effects}\\footnote{Actually, finite-size effects\\index{finite-size effects} would be controlled by $\\xi\/L$ which is smaller for the lower temperatures.}. Second, exhaustive checks have been done in order to establish our system size $L=160$ as a safe choice, see~\\refsec{finite_size_effects}.\n\n\\begin{figure}[h!]\n\\includegraphics[width=0.8\\linewidth]{aging\/a2_vs_T}\n\\caption[\\textbf{Deviation of \\boldmath $\\xi_{12}(\\ensuremath{t_\\mathrm{w}}\\xspace)$ from a simple power-law growth.}]{\\textbf{Deviation of \\boldmath $\\xi_{12}(\\ensuremath{t_\\mathrm{w}}\\xspace)$ from a simple power-law growth.} We plot the quadratic parameter $a_2$ in a fit to~\\refeq{naive_divergent}. This quantity is zero at the critical point, but  has a positive value at low temperatures, indicating that the growth  of $\\xi_{12}$ slows down over the simulated time range.}\n\\labfig{a2}\n\\end{figure}\n\nOf course, our naive ansatz of~\\refeq{naive_divergent} is only useful for interpolations. If we want to explore the growth of $\\xi_{12}(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$ in the large-$\\xi_{12}$ limit, we need some insight from theory.\n\n\\section[The large-$\\xi$ limit]{The \\boldmath large-$\\xi$ limit} \\labsec{large_xi_limit}\nIn this section, we explore the growth of $\\xi$ in the large-$\\xi$ limit. For that purpose, we need to propose extrapolations from our numerical data. We introduce and study two functional forms for the function $\\log \\ensuremath{t_\\mathrm{w}}\\xspace$ that will allow us to extrapolate $z(T,\\xi_{12}(\\ensuremath{t_\\mathrm{w}}\\xspace))$ in~\\refsubsec{ansatzs_aging}. However, before extrapolating our data, we need to compute the exponent $\\vartheta$ appearing in the long-distance decay of $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$ [see \\refeq{long_distance_C4}] because it is required by one of our ans\\\"atze. We recall the relation between $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$ and $\\vartheta$ for the reader's convenience and compute the exponent $\\vartheta$ in \\refsubsec{vartheta_aging}. Finally, we extrapolate our results and confront them with the experimental ones in~\\refsubsec{extrapolations_aging}.\n\\subsection[Computing $\\vartheta$]{Computing \\boldmath $\\vartheta$} \\labsubsec{vartheta_aging}\nThe correlation\\index{correlation function!four point} function $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$ presents the following long-distance decay\n\\begin{equation}\nC_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace) = r^{-\\vartheta} f(r\/\\xi(T,\\ensuremath{t_\\mathrm{w}}\\xspace)) \\, , \\labeq{recall_long_distance_C4}\n\\end{equation}\nwhere $f(x)$ is an unknown function which vanishes faster than exponentially. As we have already introduced in~\\refsubsec{observables_introduction}, the exponent $\\vartheta$ at $\\ensuremath{T_\\mathrm{c}}\\xspace$ is given by $\\vartheta = 1 + \\eta$ where $\\eta=-0.390(4)$~\\cite{janus:13} is the anomalous dimension. For $T<\\ensuremath{T_\\mathrm{c}}\\xspace$ the two main pictures, namely droplets\\index{droplet!picture} and \\gls{RSB}\\index{replica!symmetry breaking (RSB)}, have differing expectations: coarsening\\index{coarsening} domains\\index{magnetic domain!compact} with $\\vartheta=0$ is the \\index{droplet!picture}droplets' prediction while in the \\gls{RSB}\\index{replica!symmetry breaking (RSB)} theory, $\\vartheta$ is given by the replicon\\index{replicon} (see~\\cite{janus:10b} for a detailed discussion). The best previous numerical study of $\\vartheta$ found $\\vartheta=0.38(2)$~\\cite{janus:09b}.\n\nIf we recall the integral $I_k= \\int_0^{\\infty} r^k C_4(T,r,t) {\\mathrm{d}} r$ [see~\\refeq{integral_estimator_xi}] it is easy to prove that $I_2(T,\\xi_{12}) \\propto \\xi_{12}^{3-\\vartheta}$ and, therefore, we can estimate the value of $\\vartheta$ from the numerical derivative of $I_2$\n\\begin{equation}\n\\vartheta=3-\\dfrac{{\\mathrm{d}} \\log I_2}{{\\mathrm{d}} \\log \\xi_{12}} \\, .\n\\end{equation}\n\nOur estimations of $\\vartheta$ show that, for $T=\\ensuremath{T_\\mathrm{c}}\\xspace$, its value is compatible with $1+\\eta$, as expected. However, for $T<\\ensuremath{T_\\mathrm{c}}\\xspace$ we found a slow decrease as $\\xi$ increases or $T$ decreases, i.e. $\\vartheta \\to \\vartheta(T,\\xi_{12})$.\n\nThis result is unsatisfactory because we expect $\\vartheta$ to be constant (possibly 0) in the large-$\\xi$ limit. For the sake of clarity, we will call that theoretically expected value $\\vartheta_{\\infty}$. The simplest explanation is that the values of $\\vartheta(T,\\xi_{12})$ are affected by critical effects of the fixed point at $T=\\ensuremath{T_\\mathrm{c}}\\xspace$ with $\\vartheta(\\ensuremath{T_\\mathrm{c}}\\xspace,\\xi_{12}) \\approx 0.61$. For large $\\xi$, we expect those critical effects to vanish and the value of $\\vartheta(T,\\xi_{12}\\to \\infty)$ should be dominated by the $T=0$ fixed point, i.e. $\\vartheta(T,\\xi_{12}\\to \\infty) = \\vartheta_{\\infty}$.\n\nIn analogy with the ferromagnetic phase\\index{phase!ferromagnetic} of the $O(N)$ model, we study this crossover from $\\vartheta(T=\\ensuremath{T_\\mathrm{c}}\\xspace)$ to the unknown $\\vartheta(T=0)$ in terms of the Josephson length $\\ell_J$~\\cite{josephson:66}. The Josephson length is expected to behave as $\\ell_J \\sim (\\ensuremath{T_\\mathrm{c}}\\xspace - T)^{-\\nu}$ with $\\nu=2.56(4)$~\\cite{janus:13} for temperatures $T \\to \\ensuremath{T_\\mathrm{c}}\\xspace$. The scaling corrections for $\\ell_J(T)$ at lower temperatures are explained in~\\refsec{Josephson_length}.\n\nWe can test the crossover hypothesis by considering the ratio of two different estimations of $\\xi$: $\\xi_{23}\/\\xi_{12}$. We plot this ratio against the scaling variable $x=\\ell_J\/\\xi_{12}$. This ratio should be scale-invariant in the large-$\\xi_{12}$ limit because different determinations of $\\xi$ should grow with the same rate but with a different prefactor (see~\\cite{janus:09b} and~\\refsubsec{observables_introduction}).\n\n\\reffig{testing_crossover} shows us that the ratio grows towards the critical value (represented with a gray line) for the curves with the largest temperatures when $x$ is large, i.e. for a given curve, which is $T$-constant, when $\\xi_{12}$ is small. Then, it relaxes to the value corresponding to the $T=0$ fixed point (small $x$). The lowest temperatures, namely $T=0.55$, $T=0.625$ and $T=0.7$, seem to be free of critical effects.\n\n\\begin{figure}[h!]\n\\includegraphics[width=0.8\\linewidth]{aging\/josephson_crossover_inset}\n\\caption[\\textbf{Testing the crossover hypothesis with \\boldmath $\\xi_{23}\/\\xi_{12}$.}]{\\textbf{Testing the crossover hypothesis with \\boldmath $\\xi_{23}\/\\xi_{12}$.} We consider the ratio $\\xi_{23}\/\\xi_{12}$ between two definitions of the coherence length\\index{coherence length}, which should be constant in the large-$\\xi_{12}$ (or $x\\to0$) limit. For $T$ close to $\\ensuremath{T_\\mathrm{c}}\\xspace$, this ratio initially grows, approaching the $T=\\ensuremath{T_\\mathrm{c}}\\xspace$ value (represented by the thick gray line) and eventually relaxes towards the $T=0$ fixed point.}\n\\labfig{testing_crossover}\n\\end{figure}\n\nThis positive result encourages us to perform a similar analysis for $\\vartheta(T,\\xi_{12})$. If the hypothesis of the crossover is correct, we should observe a collapse of the $\\vartheta(T,\\xi_{12})$ values when we plot them in terms of the scaling variable $x=\\ell_J\/\\xi_{12}$. \n\nThe functional form of $\\vartheta(T,\\xi_{12})$ should be, in the \\gls{RSB}\\index{replica!symmetry breaking (RSB)} picture\n\\begin{equation}\n\\vartheta(x) = \\vartheta_{\\infty} + b_2 x^{2-\\vartheta_{\\infty}} + b_3 x^{3-\\vartheta_{\\infty}} + \\cdots \\, . \\labeq{vartheta_RSB}\n\\end{equation}\nThe reader may find a derivation of this expression in~\\refsec{Josephson_length}. On the contrary, for the droplets\\index{droplet!picture} picture, it should be \n\\begin{equation}\n\\vartheta(x)=C x^\\zeta \\, , \\labeq{vartheta_droplet}\n\\end{equation}\nwhere $\\zeta\\approx 0.24$~\\cite{boettcher:04} is the stiffness coefficient\\index{stiffness!coefficient}.\n\n\\reffig{vartheta_collapse} shows us a nice collapse of the $\\vartheta(x)$ values. Moreover, we have to keep in mind that our goal is to extrapolate the values of the aging\\index{aging!rate} rate to the experimental $\\xi$ length-scale which roughly corresponds to $\\xi_{\\mathrm{films}}\\approx 38$. \n\nFor the \\gls{RSB}\\index{replica!symmetry breaking (RSB)} picture, a fit of the $\\vartheta(x)$ values to~\\refeq{vartheta_RSB} gives us the value $\\vartheta_{\\infty} \\approx 0.30$, although we take $\\vartheta^{\\mathrm{upper}}=0.35$ as our upper bound for $\\vartheta_{\\infty}$. For the droplets\\index{droplet!picture} picture, a fit to~\\refeq{vartheta_droplet} can be performed only by considering\\footnote{Similar results were found in~\\cite{janus:10}} $\\zeta \\approx 0.15$. It is worthy to note that we find this exponent very sensitive to the fitting range. We extrapolate the droplet\\index{droplet!picture} $\\vartheta(x)$ to $\\xi = \\xi_{\\mathrm{films}}=38$ and we obtain $\\vartheta(\\xi_{\\mathrm{films}}=38) \\approx 0.28$.\n\nHowever, because our determination of $\\xi$ through the estimator $\\xi_{12}$ may differ from the experimental estimation of $\\xi$ by a small constant factor, we consider also $\\xi_{\\mathrm{films}}=76$ and we obtain $\\vartheta(\\xi_{\\mathrm{films}}=76) \\approx 0.25$.\n\nThe same conclusions found in~\\cite{janus:10,janus:10b} stands here: for the experimental relevant scales, the physics is well described by a non-coarsening\\index{coarsening} picture with $0.25<\\vartheta(\\xi_{\\mathrm{films}})<0.35$ depending on the theory we use to extrapolate the data and the exact value chosen for the experimental scale $\\xi_{\\mathrm{films}}$.\n\n\\begin{figure}[h!]\n\\includegraphics[width=0.8\\linewidth]{aging\/josephson_crossover_main}\n\\caption[\\textbf{Crossover between the \\boldmath $T=\\ensuremath{T_\\mathrm{c}}\\xspace$ and the $T=0$ fixed points controlled by a Josephson length $\\ell_\\text{J}(T)$.}]{\\textbf{Crossover between the \\boldmath $T=\\ensuremath{T_\\mathrm{c}}\\xspace$ and the $T=0$ fixed points controlled by a Josephson length $\\ell_\\text{J}(T)$.} We plot the evolution of the replicon\\index{replicon} exponent $\\vartheta$ for several temperatures against the relevant scaling variable $x=\\ell_\\text{J}(T)\/\\xi_{12}$. We show two possible extrapolations (dashed lines) to infinite $\\xi_{12}$: one with finite $\\vartheta$, as expected in the RSB picture, and one with $\\vartheta=0$, as expected in the droplet\\index{droplet!picture} picture. For the latter, we also show the extrapolated value for the experimental scale corresponding to experiments in \\gls{CuMn} films~\\cite{zhai:17}, which we estimate between $\\xi_{12}=38$ and $\\xi_{12}=76$.}\n\\labfig{vartheta_collapse}\n\\end{figure}\n\n\\subsection{The convergent and the divergent ansatz} \\labsubsec{ansatzs_aging}\nNow, we discuss the possible extrapolations of $z(T,\\xi_{12})$ to the large-$\\xi_{12}$ limit. The most natural assumption is to consider that $z(T,\\xi_{12} \\to \\infty) = z_{\\infty}(T)$ with a convergence $z(T,\\xi_{12}) - z_{\\infty}(T) \\propto \\xi_{12}^{-\\omega}$. Taking into account the~\\refeq{definition_aging_rate}, the expresion of $\\ensuremath{t_\\mathrm{w}}\\xspace$ should be\n\\begin{equation}\n\\log \\ensuremath{t_\\mathrm{w}}\\xspace = C_1(T) + z_{\\infty}(T) \\log \\xi_{12} + C_2(T)\\xi_{12}^{-\\omega} \\, , \\labeq{convergent_ansatz}\n\\end{equation}\nwhere $\\omega$ is the exponent that controls finite-$\\xi_{12}$ corrections. The value of $\\omega$ for the critical\\index{critical temperature} temperature $T_c$ is $\\omega=1.12(10)$~\\cite{janus:13,fernandez:15,lulli:16}. In the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass}, the leading behavior is given by $\\omega=\\vartheta$, see \\cite{janus:10b} for a detailed discussion.\n\nThe effective exponent in the \\textit{convergent} ansatz would be, therefore\n\\begin{equation}\nz_{\\mathrm{conv}}(T,\\xi_{12}) = \\dfrac{{\\mathrm{d}} \\log \\ensuremath{t_\\mathrm{w}}\\xspace}{{\\mathrm{d}} \\log \\xi_{12}} = z_{\\infty}(T) - \\omega C_2(T) \\xi_{12}^{-\\omega} \\, . \n\\end{equation}\n\nThe fits to~\\refeq{convergent_ansatz} have two main sources of error. First, the value of $\\vartheta$ has associated some uncertainty. We choose $\\vartheta=0.35$, as explained above. Second, we consider possible systematic effects due to the fitting range. We follow objective criteria to select a minimum $\\xi_{12}^{\\min}$ for the fitting range. Further details can be found in~\\refsec{parameters_aging}.\n\nAn alternative approach was proposed in~\\cite{bouchaud:01,berthier:02}. In those works, the authors proposed a crossover to activated dynamics. This approach is a refinement of the droplet\\index{droplet!picture} proposal that we expose at the beginning of the chapter [$\\xi \\sim (\\log \\ensuremath{t_\\mathrm{w}}\\xspace)^{1\/\\psi}$]. In this case, the authors propose\n\\begin{equation}\n\\ensuremath{t_\\mathrm{w}}\\xspace = \\tau_0 \\xi^{z_c} \\exp \\left(\\dfrac{\\Upsilon(T) \\xi^{\\psi}}{k_{B}T} \\right) \\, , \\labeq{original_divergent}\n\\end{equation}\nbeing $\\Upsilon(T) = \\Upsilon_0(1-T\/T_c)^{\\psi\\nu}$ and $k_{B}$ the Boltzmann\\index{Boltzmann!constant} constant. Here, we express the original ansatz in logarithmic form with generic coefficients\n\\begin{equation}\n\\log \\ensuremath{t_\\mathrm{w}}\\xspace = D_1(T) + z_c \\log \\xi_{12} + D_2(T) \\xi_{12}^{\\psi} \\, . \\labeq{divergent_ansatz}\n\\end{equation}\nThe exponent $\\psi$ has been used before in experiments~\\cite{schins:93} and numerical simulations~\\cite{rieger:93} with values $\\psi \\approx 1$. Moreover, the reader may noticed that $D_2(T) \\sim (T_c-T)^{\\psi\\nu}$ [see~\\refeq{original_divergent}]. This can be regarded as another way to present the crossover between the $T=T_c$ fixed point and the $T=0$ fixed point. We need $\\xi_{12}(T_c-T)^{\\nu} \\gg 1$ in order to perceive deviations from the pure power-law with an aging\\index{aging!rate} rate $z_c$ equal to the critical one.\n\nThe effective exponent in the \\textit{divergent} ansatz would be, therfore\n\\begin{equation}\nz_{\\mathrm{div}}(T,\\xi_{12})=\\dfrac{{\\mathrm{d}} \\log \\ensuremath{t_\\mathrm{w}}\\xspace}{{\\mathrm{d}} \\log \\xi_{12}} =z_c + D_2(T) \\psi \\xi_{12}^{\\psi} \\, .\n\\end{equation}\n\nWe find fair fits to our simulated data for both ans\\\"atze,~\\refeq{convergent_ansatz} and~\\refeq{divergent_ansatz}. Indeed, for~\\refeq{divergent_ansatz} we find $\\psi \\approx 0.4$. The next step is clear, we need to test both proposals at the experimental length-scales.\n\n\\subsection{Extrapolation to experimental regime} \\labsubsec{extrapolations_aging}\nWe perform extrapolations of the renormalized aging\\index{aging!rate} rate $z_c(T,\\xi) = z(T,\\xi) T\/T_c$ at the experimental length-scale $\\xi_{\\mathrm{films}}$. The value of $z(T,\\xi)$ is computed for both $z_{\\mathrm{conv}}(T,\\xi_{\\mathrm{films}})$ and $z_{\\mathrm{div}}(T,\\xi_{\\mathrm{films}})$.\n\nIn a similar way as we did in the previous analysis of~\\refsubsec{vartheta_aging}, we use safe estimations of $\\xi_{\\mathrm{films}}$ and we consider $\\xi_{\\mathrm{films}}=38$ and $\\xi_{\\mathrm{films}}=76$. All the relevant data from the fits can be found in~\\refsec{parameters_aging}, however, we plot in~\\reffig{zc} the relevant information.\n\nThe main plot shows the renormalized aging\\index{aging!rate} rate $z_c(T,\\xi_{\\mathrm{films}})$ plotted against the reduced temperature $T\/\\ensuremath{T_\\mathrm{c}}\\xspace$. We see that the convergent ansatz of~\\refeq{convergent_ansatz} is very successful in reproducing the experimental behavior for both $\\xi_{12}=38$ and $\\xi_{12}=76$.  Compatible results with experiments are found for a wide range of temperatures.\n\nThe inset shows the divergent ansatz of~\\refeq{divergent_ansatz}. In this case this proposal for $z_c(T,\\xi_{\\mathrm{films}})$ is not able to reproduce the constant value found in experiments for $T\/T_c < 0.8$.\n\n\n\\begin{figure}[h!]\n\\includegraphics[width=0.8\\linewidth]{aging\/zc_vs_T_inset}\n\\caption[\\textbf{Numerical and experimental aging rate.}]{\\textbf{Numerical and experimental aging rate.} Value of the experimental aging\\index{aging!rate} rate for \\gls{SG}s $z_c(T,\\xi_{\\mathrm{films}}) = z(T,\\xi_\\text{films}) T\/\\ensuremath{T_\\mathrm{c}}\\xspace$, extrapolated from our data for values of the coherence length\\index{coherence length} corresponding to thin  \\gls{CuMn} films. The main plot considers an ansatz~\\refeq{convergent_ansatz} with a finite $z_{\\infty}(T)T\/\\ensuremath{T_\\mathrm{c}}\\xspace$, which agrees very well with the experimental value of $z_c(T)\\approx 9.62$~\\cite{zhai:17}, indicated by the straight line, whose width represents the experimental temperature range. Notice that critical effects are only visible for $T>0.7$. \\textbf{Inset:} Same plot but now considering a crossover to activated dynamics~\\refeq{divergent_ansatz}, as in~\\cite{bouchaud:01}. This is less successful at reproducing the roughly constant $z_c(T)$ observed in experiments.}\n\\labfig{zc}\n\\end{figure}\n\n\n\\section*{Agradecimientos}\n\n\n\\section*{Agradecimientos}\n\nEsta tesis ha sido posible gracias a la ayuda de muchas personas que han puesto su granito de arena para que yo pudiera avanzar en la consecuci\u00f3n de los objetivos (no s\u00f3lo acad\u00e9micos) que me he impuesto a lo largo de los cuatro \u00faltimos a\u00f1os.\n\nA pesar de que mi tesis se ha desarrollado principalmente en la Universidad de Zaragoza, he tenido la suerte de poder colaborar estrechamente con la Universidad Complutense de Madrid. Quiero agradecer en primer lugar a mis directores de tesis, David \u00cd\u00f1iguez Dieste de la Universidad de Zaragoza y V\u00edctor Mart\u00edn Mayor de la Universidad Complutense de Madrid, su ayuda y dedicaci\u00f3n a lo largo de estos cuatro a\u00f1os.\n\nDavid \u00cd\u00f1iguez me recibi\u00f3 con los brazos abiertos en mi primera toma de contacto con Zaragoza y siempre me ha dado todas las facilidades del mundo, tanto al principio para establecerme en mi puesto de trabajo y en la ciudad (nueva para m\u00ed) como a lo largo de toda la tesis donde sab\u00eda que cruzando el pasillo y llamando a su puerta siempre lo encontrar\u00eda dispuesto a echar una mano.\n\nA V\u00edctor Mart\u00edn Mayor lo conoc\u00ed en la realizaci\u00f3n del m\u00e1ster de F\u00edsica Te\u00f3rica en la Universidad Complutense de Madrid. Con V\u00edctor ha sido con la persona que m\u00e1s he aprendido a lo largo de esta tesis, ya fuera en sesudas sesiones de trabajo en su despacho, por videoconferencias o escap\u00e1ndonos en alg\u00fan hueco entre charlas en una conferencia en Argentina para hacer una reuni\u00f3n improvisada. Su labor como director ha ido mucho m\u00e1s all\u00e1 de sus obligaciones, y su dedicaci\u00f3n y esfuerzo ha sido todav\u00eda mayor teniendo en cuenta que la mayor\u00eda de veces tuvimos que trabajar a distancia.\n\nQuiero agradecer tambi\u00e9n toda la ayuda que me han prestado a Alfonso Taranc\u00f3n y a Luis Antonio Fern\u00e1ndez. Alfonso Taranc\u00f3n oficialmente figura como mi ``Tutor de Tesis'', un t\u00edtulo que no hace justicia a toda la ayuda que he recibido de su parte. Al igual que me ocurr\u00eda con David, sab\u00eda que cruzando el pasillo y llamando a su puerta encontrar\u00eda su ayuda para cualquier problema que se me presentase. Fue la persona que me introdujo a la docencia y creo que eso no se olvida. Podr\u00eda decirse que ha sido un director no oficial de mi tesis. Y si Alfonso Taranc\u00f3n ha sido mi director de tesis no oficial en Zaragoza, sin duda, Luis Antonio ha sido mi director de tesis no oficial en Madrid. Desde luego, Luis Antonio ha tenido la dedicaci\u00f3n de un director tesis sin tener ninguna obligaci\u00f3n a ello. Siempre ha estado dispuesto a ayudarme con cualquier cosa y recuerdo con especial cari\u00f1o las largas tardes delante de su ordenador, explic\u00e1ndome las tripas de alg\u00fan programa que me servir\u00eda de base para desarrollar uno nuevo, en las que yo muchas veces sal\u00eda asustado con un mont\u00f3n de notas y con las cosas no del todo claras, pero que m\u00e1s tarde comenzaban a cobrar sentido cuando me met\u00eda en harina y descubr\u00eda que esa extra\u00f1a l\u00ednea de c\u00f3digo era uno de los checks exhaustivos que ahora son tambi\u00e9n imprescindibles en todos mis programas.\n\nA mis directores de tesis y tambi\u00e9n a Alfonso y a Luis Antonio a quienes considero mis ``directores no oficiales'', muchas gracias. Sin vuestra ayuda esta tesis no habr\u00eda sido posible.\n\nTambi\u00e9n me gustar\u00eda mencionar en estos agradecimientos a la gente con la que realic\u00e9 mis estancias de investigaci\u00f3n fuera de Espa\u00f1a. Mi primera estancia de investigaci\u00f3n la realic\u00e9 en Roma, en la Universidad La Sapienza, donde fui recibido por Enzo Marinari y Andrea Maiorano. En el trabajo del d\u00eda a d\u00eda con quien m\u00e1s tiempo pas\u00e9 fue con Andrea, que en aquella \u00e9poca estaba prepar\u00e1ndose unas oposiciones y a\u00fan as\u00ed sacaba tiempo de donde no lo hab\u00eda para echarme una mano e interesarse por mi trabajo. A Enzo, Federico, Giorgio, y especialmente a Andrea, me gustar\u00eda agradecerles su c\u00e1lida acogida durante mi estancia en Roma. \n\nMi segunda estancia de investigaci\u00f3n la realic\u00e9 en Paris, en la Universidad de Paris Sud. Debido a la pandemia las condiciones fueron mucho peores por razones obvias, sin embargo agradezco enormemente a Cyril y a Aur\u00e9lien tanto su ayuda (y m\u00e1s teniendo en cuenta la situaci\u00f3n tan delicada que hab\u00eda) como la posibilidad que me brindaron de comenzar a investigar en algo tan apasionante como el Machine Learning.\n\nTambi\u00e9n quiero agradecer a todos los investigadores de los cuales he aprendido mucho en estos cuatro a\u00f1os. Especialmente me gustar\u00eda mencionar a Juan Jes\u00fas Ruiz que me introdujo en el mundo de los espines en mi TFG, que siempre me prest\u00f3 su ayuda desde Badajoz y que incluso organiz\u00f3 un curso intensivo sobre el grupo de renormalizaci\u00f3n con el \u00fanico objetivo de ayudarnos en nuestra tesis, sacrificando\nmuchas horas sin tener ninguna obligaci\u00f3n a ello. Tambi\u00e9n a David Yllanes, que siempre est\u00e1 al pie del ca\u00f1\u00f3n, y al que le tengo en gran estima. A Sergio P\u00e9rez Gaviro, que me ayud\u00f3 much\u00edsimo en mi primera toma de contacto en Zaragoza ense\u00f1\u00e1ndome todos los detalles t\u00e9cnicos, d\u00e1ndome programas para ayudarme en mi tarea hasta abrumarme y bajando conmigo al CPD para verle las tripas a los Janus. Y tambi\u00e9n a Ilaria e Isidoro, con los que he compartido director de tesis y a los que espero que les vaya realmente bien en su carrera investigadora y en su vida.\n\nPor supuesto, no puedo olvidarme de toda la gente de Zaragoza con la que he trabajado estos \u00faltimos cuatro a\u00f1os. Isabel Vidal me ayud\u00f3 con todo el tema administrativo. Nunca he visto a una persona hacer su trabajo tan bien como lo hac\u00eda ella, con tanto cuidado, cari\u00f1o y siempre con una sonrisa. El grupo de Yamir Moreno, que me acogi\u00f3 desde el primer d\u00eda con los brazos abiertos y con los que he vivido grandes momentos trabajando juntos. Toda la gente del BIFI, especialmente Pedro, Daniel, Sergio y Rub\u00e9n que me ayudaron cada vez que tuve alg\u00fan problema con \\textit{Cierzo}.\n\nDejo para la parte final a aquellos a los que conoc\u00ed mucho antes de iniciar el doctorado y que s\u00e9 que seguir\u00e1n formando parte de mi vida mucho despu\u00e9s de completarlo. A mis amigos Kike, \u00c1lvaro, Pilar, Carlos, Helena, Cienfu, Gordillo, Kubicki, Luiso y V\u00e1zquez, muchas gracias por estar ah\u00ed estos a\u00f1os y los que vendr\u00e1n. Por supuesto, muchas gracias a mi familia, a Tito y a Ariadna. Tambi\u00e9n a Toni, Candela, Lolo y Ana. Sin vuestra ayuda, que va mucho m\u00e1s all\u00e1 de la tesis, nada de esto hubiera salido adelante.\n\nNo me olvido de Sarita, con la que llevo media vida y con la que espero que me queden muchos a\u00f1os por delante. La persona que ha sido mi apoyo emocional durante toda esta tesis y a la que le debo tanto que me sabe a poco dedicarle solo estas pocas l\u00edneas. Sin ti, no s\u00f3lo no hubiera sido posible esta tesis, tampoco muchas otras cosas en mi vida. Te quiero. A la Juani y al Ram\u00f3n, los perros m\u00e1s fant\u00e1sticos del mundo. Que comparten habitaci\u00f3n conmigo (son perros investigadores) y cuya alegr\u00eda al verme siempre me levanta el \u00e1nimo (te echar\u00e9 much\u00edsimo de menos Juanita). Y por \u00faltimo a mam\u00e1, que lo ha hecho todo por m\u00ed. Probablemente la persona con la que m\u00e1s haya discutido, y tambi\u00e9n la \u00fanica persona que me ha acompa\u00f1ado en cada paso que he dado y que no me ha dejado caerme nunca. Sin ella no es que no hubiese sido posible esta tesis, es que no hubiera podido hacer nada en esta vida. Much\u00edsimas gracias mam\u00e1, te quiero.\n\nEstos agradecimientos pretend\u00edan ser unas pocas l\u00edneas pero sois muchos los que me hab\u00e9is ayudado y me siento afortunado por teneros. A todas las personas que he nombrado, a todas aquellas que mi terrible memoria no me haya permitido incluir y a mi yaya Paula, a la que le dedico esta tesis, muchas gracias. Hab\u00e9is dejado huella en m\u00ed y hab\u00e9is contribuido a que esta tesis llegase a buen puerto.\n\nMuchas gracias a todos.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\chapter{Multispin Coding} \\labch{AP_multispin_coding}\n\\setlength\\epigraphwidth{.7\\textwidth}\n\\epigraph{\\textit{Premature optimization is the root of all evil.}}{-- Donald Knuth, \\textit{The Art of Computer Programming} }\n\nParallel computing\\index{parallel!computing} has turned out to be essential in scientific computing to achieve the computational power that we enjoy nowadays. The tasks that usually tackle the computational hardware do not need a completely sequential work-flow. Instead, many computational tasks can be cut up into independent smaller tasks that can be performed at the same time.\n\nThe evolution of the hardware through the years has reflected this fact. From the most general-purpose hardware, the \\gls{CPU}, to the (originally) game-oriented hardware, the \\gls{GPU}, the mainstream-design tends to increase the independent cores with great benefits for the parallel\\index{parallel!computation} computation. Of course, there exists a lot of hardware capable of performing parallel tasks. In addition to the above-mentioned \\gls{CPU} and \\gls{GPU}, the \\gls{FPGA}\\index{FPGA} stands as a great example of parallel hardware.\n\nIn this appendix, we focus on the \\gls{CPU} parallel\\index{parallel!computation} computation. Specifically, we will focus on the streaming extensions proposed for the first time by Intel with the MultiMedia eXtension (MMX)~\\cite{yu:97} and that was subsequently improved until the most recent iteration of this technology, the Advanced Vector Extensions (AVX, AVX-2, and AVX-512)~\\cite{intel_avx}. This technology allows one-clock-cycle boolean\\index{boolean} operations of registers of 128 or 256 bits, i.e. we can perform 128 or 256 boolean\\index{boolean} operations at the same time in one cycle of the \\gls{CPU}'s clock.\n\nWe first introduce the Multispin Coding\\index{Multispin Coding} as a general concept in~\\refsec{what_is_multispin_coding} and then we explain with great detail one implementation that takes advantage of the \\gls{CPU} streaming extensions AVX in \\refsec{overlap_spheres_multispin_coding}. Last, we explain how we use Multispin Coding\\index{Multispin Coding} for more general and complex simulation programs in \\refsec{musa_musi_multispin_coding}.\n\n\\section{What is the Multispin Coding?} \\labsec{what_is_multispin_coding}\nThe \\gls{MSC}\\index{Multispin Coding}~\\cite{friedberg:70,jacobs:81} is a method that is born out of the necessity of performing simulations with limited computational resources. The basic idea is that, for Ising\\index{Ising} spins, we are wasting a lot of memory and computational resources if we encode each spin as an integer number. Indeed, in a 32-bit processor\\footnote{Actually, nowadays the standard of 64-bit processors is imposing.} we can encode a variable with $2^{32}$ possible values, and an Ising\\index{Ising} spin needs only $2$ values to be encoded.\n\nFor binary variables, the solution is clear. By using an integer, we can store $32$ spins at the same time. However, our problem is not the memory but the performance. Our main task is not to be more efficient by storing data, but by running our algorithm. Here is where the streaming extensions (specifically we will focus on the family AVX) take center stage.\n\nThe new Intel and AMD processors are able to execute the AVX set of instructions, which allows us to perform one-clock-cycle instructions for registers of $128$ or $256$ bits. We are interested in perform boolean\\index{boolean} operations like the AND boolean\\index{boolean} operation \\\\\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n0 & 0 & 1 & $\\cdots$ & 1 \\\\\n\\hline\n\\end{tabular} $\\&$ \\begin{tabular}{|c|c|c|c|c|}\n\\hline\n1 & 0 & 1 & $\\cdots$ & 0 \\\\\n\\hline\n\\end{tabular} $=$ \\begin{tabular}{|c|c|c|c|c|}\n\\hline\n0 & 0 & 1 & $\\cdots$ & 0\\\\\n\\hline\n\\end{tabular} ,\n\\end{center}\nbut also in performing rotations of the bits in our registers\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|} \n\\hline\n$s_0$ & $s_1$ & $s_2$ & $s_3$ & $s_4$ & $s_5$ & $s_6$ & $s_7$ \\\\\n\\hline\n\\end{tabular} $\\to$ \\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n$s_7$ & $s_0$ & $s_1$ & $s_2$ & $s_3$ & $s_4$ & $s_5$ & $s_6$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\nAll the operations available for each one of the instructions sets can be found on the official page of Intel (\\href{https:\/\/software.intel.com\/sites\/landingpage\/IntrinsicsGuide\/}{Intel Intrinsics Guide}).\n\nNow, we know how to efficiently compute a large number of boolean\\index{boolean} operations but, how can we use this in a real situation? The next two sections are devoted to explain specific implementations in programs developed during this thesis. All the programs showed here have been coded in the programming language C.\n\n\\section{An easy example: computing overlaps inside spheres} \\labsec{overlap_spheres_multispin_coding}\nWe start with a simple implementation of \\gls{MSC}\\index{Multispin Coding}. The program that we describe here was coded to compute the chaotic parameter of a given sphere (see~\\refch{out-eq_chaos}). We will briefly describe the program and focus on the implementation of the \\gls{MSC}.\n\nIn the program, we receive as an input two $L=160$ three-dimensional cubic lattices for each of the $512$ replicas\\index{replica}, where the nodes correspond to the spins and the edges to the couplings\\index{couplings}. One of the lattices has been simulated with a thermal reservoir at temperature $T_1$ and the other one with $T_2$, both at a time $\\ensuremath{t_\\mathrm{w}}\\xspace$ such that the coherence length\\index{coherence length} of both systems $\\xi_1=\\xi_2=\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$. \n\nOnce we have stored the lattices, we select (randomly) 8000 centers in the dual lattice\\footnote{The dual lattice of a cubic lattice with \\gls{PBC}\\index{boundary conditions!periodic} is another cubic lattice of the same size, and with \\gls{PBC}\\index{boundary conditions!periodic} as well. The nodes of the dual lattice are the centers of the elementary cells of the original lattice. See \\refch{out-eq_chaos}.} and we build for each center a sphere of radius $r$. The number of spins inside the sphere is $N_r$, for example, the smallest sphere $r=1$ has $N_r=8$ spins inside.\n\nWe want to compute the chaotic parameter of the sphere, defined in \\refeq{def_chaotic_parameter} and repeated here for the reader's convenience\n\\begin{equation} \nX^{s,r}_{T_1,T_2}(\\xi) = \\dfrac{\\langle [q_{T_1,T_2}^{s,r}(\\xi)]^2\\rangle_T}{\\sqrt{\\langle[q_{T_1,T_1}^{s,r}(\\xi)]^2\\rangle_T \\,\\langle[q_{T_2,T_2}^{s,r}(\\xi)]^2\\rangle_T}} \\, . \n\\end{equation} \n\nThe square of the overlaps\\index{overlap} $[q_{T_1,T_2}^{s,r}(\\xi)]^2$ has to be averaged over the thermal noise. As long as we are in an out-of-equilibrium simulation, our estimation of that thermal average would be an average over the replicas\\index{replica}. Therefore, if we focus now on a given sphere, we have a total of $N_r \\times \\ensuremath{N_{\\text{Rep}}}\\xspace$ number of spins at each temperature. Furthermore, we have to repeat this procedure for the 8000 spheres, for different sizes of spheres $r$, for different coherence lengths\\index{coherence length} $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ and for different pairs of temperatures $T_1$ and $T_2$. \n\nJust to put the reader in context, if we fix $T_1$, $T_2$ and $\\xi$, we have to perform $2.5526747136 \\cdot 10^{14}$ overlaps\\index{overlap}. This is, indeed, a huge amount of computations that are completely independent of each other. Hence, we can greatly benefit from the \\gls{MSC} in this situation. \n\nFirst, to each sphere, we associate a vector of pairs of $256$-bit registers. To this purpose, we coded an specific function \\textit{void fill\\_sphere(int, int*, int, int)}\n\\begin{lstlisting}[language=C,style=mystyle]\n\n#define NR_PIECES 2 \/\/The number of words of 256 bits\ntypedef __m256i MY_WORD; \/\/MY_WORD is a register of 256 bits (see Intel Intrinsics for further information)\n\nMY_WORD replicas[NR_PIECES][V],replicas2[NR_PIECES][V]; \/\/Full lattice for temperature T1 and T2 respectively\nMY_WORD sphere_replicas[NR_PIECES][V], sphere_replicas2[NR_PIECES][V]; \/\/Spins inside the sphere for the replica 1 and replica 2.\n\nvoid fill_sphere(int size,int* sphere_index, int size_word, int temperature_flag){\n  int is; \/\/loop variable to run over the spins of the sphere\n  int i512; \/\/loop variables to run over the two words of 256-bits\n  \n  \/\/Auxiliar variables  \n  MY_WORD* aux1[NR_PIECES];\n  MY_WORD* aux2[NR_PIECES];\n  \n  \/\/The value of the flag is arbitrary, but this specific set of values\n  \/\/allows us to iterate over it in a loop\n  \n  for(i512=0;i512<NR_PIECES;i512++){\n    if(temperature_flag==0){ \/\/Means T1T1\n      aux1[i512]=replicas[i512];\n      aux2[i512]=replicas[i512];\n    }else if(temperature_flag==1){ \/\/Means T2T2\n      aux1[i512]=replicas2[i512];\n      aux2[i512]=replicas2[i512];\n    }else if(temperature_flag==2){ \/\/Means T1T2\n      aux1[i512]=replicas[i512];\n      aux2[i512]=replicas2[i512];\n    }else{ \/\/There are no more options\n      print_and_exit(\"Temperature flag cannot be differente from 0,1 or 2 -> temperature_flag =\n    }\n  }\n\n  if(size_word==512){\n    for(is=0;is<size;is++){\n      for(i512=0;i512<NR_PIECES;i512++){\n\t\tsphere_replicas[i512][is]=aux1[i512][sphere_index[is]];\n\t\tsphere_replicas2[i512][is]=aux2[i512][sphere_index[is]];\n      }\n    }\n  }\n}\n\\end{lstlisting}\nThis function is really easy, it only associates each spin of the sphere (\\textit{sphere\\_replicas} and \\textit{sphere\\_replicas2}) with its corresponding value of the full lattice. Now, we have built our spheres. The variable \\textit{temperature\\_flag} is selecting the proper temperature of the replicas\\index{replica} to construct $[q_{T_1,T_1}^{s,r}(\\xi)]^2$, $[q_{T_2,T_2}^{s,r}(\\xi)]^2$ or $[q_{T_1,T_2}^{s,r}(\\xi)]^2$.\n\nFrom now on, we focus on the $T_1 \\neq T_2$ case for the sake of simplicity, but for the $T_1=T_2$ case, the situation is almost the same (below, the reader can find a minor discrepancy). The next step is to compute the overlap\\index{overlap} between the $512$ replicas\\index{replica}. The code for this operation, using the AVX instructions, is really easy:\n\\begin{lstlisting}[language=C,style=mystyle]\nMY_WORD q[NR_PIECES][V]; \/\/ Matrix (2 x V) of registers of size 256-bits\nint i512,j; \/\/Loop variables\n\nfor (j=0;j<sphere_size;j++){\n   for(i512=0;i512<NR_PIECES;i512++){\n      q[i512][j]=_mm256_xor_si256(sphere_replicas[i512][j],sphere_replicas2[i512][j]);\n   }\n}\n\n\\end{lstlisting}\n\nNow, the variable \\textit{q[NR\\_PIECES][V]} contains, for each site inside the sphere, a set of $512$ overlaps\\index{overlap}. We can imagine it as a $(512 \\times n_{\\mathrm{sph}})$ matrix, where $n_{\\mathrm{sph}}$ is the number of spins in the sphere. To compute $[q_{T_1,T_2}^{s,r}(\\xi)]^2$ we fix one of the $512$ bits of the register and compute the square of the boolean\\index{boolean} sum over the $n_{\\mathrm{sph}}$ elements. At the end of this process, we will have computed $512$ values of $[q_{T_1,T_2}^{s,r}(\\xi)]^2$.\n\nHowever, we can compute much more values of $[q_{T_1,T_2}^{s,r}(\\xi)]^2$. Indeed, we can rotate the 512 replica-spins\\index{replica} in one of the spheres (as done in the example of~\\refsec{what_is_multispin_coding}) and compute again the overlap\\index{overlap}. This code makes the rotations of the bits\n\n\\begin{lstlisting}[language=C,style=mystyle]\ntypedef union {\n    __m256i sse;\n    uint64_t vec[4];\n} VectorUnion64;\n\nvoid rota(int size, MY_WORD *sphere1, MY_WORD *sphere2)\n{\n\n  VectorUnion64 w[2];\n  uint64_t resto[8];\n\n  int i,iw; \/\/Loop variables\n  \n  for (i=0;i<size;i++){\n     w[0].sse=sphere1[i];\n     w[1].sse=sphere2[i];\n     for (iw=0;iw<8;iw++){\n        resto[iw]=w[iw\/4].vec[i\n\t    w[iw\/4].vec[i\n     }\n    \n     for (iw=0;iw<8;iw++){\n\t    w[iw\/4].vec[i\n     }\n     sphere1[i]=w[0].sse;\n     sphere2[i]=w[1].sse;\n  }\n}\n\\end{lstlisting}\n\nThis rotation process can be repeated $511$ times until we reach again the initial disposition of spins. Therefore, we compute a total of $512^2$ values of $[q_{T_1,T_2}^{s,r}(\\xi)]^2$ that will allow us to estimate the thermal average $\\langle [q_{T_1,T_2}^{s,r}(\\xi)]^2\\rangle_T$. Here is where the small discrepancy between the $T_1 \\neq T_2$ and the $T_1 = T_2$ cases appears. For the $T_1=T_2$ case, we can compute only $512(512-1)\/2$ overlaps\\index{overlap}.\n\nWith this simple code, we are able to reduce by a factor $512$ the total number of operations we have to compute. We have considered explaining with detail this simple case because it is very illustrative with a relatively small amount of code and technical details. Nonetheless, the same idea sketched here can be extended to more complex simulations as we will discuss in the next section.\n\n\\section{Spin glass simulations by using Multispin Coding} \\labsec{musa_musi_multispin_coding}\n\nThe equilibrium simulations performed for the works appearing in \\refch{metastate} and \\refch{equilibrium_chaos} required a huge numerical effort. Because those simulations were performed in \\gls{CPU}s, we could take advantage of the \\gls{MSC}\\index{Multispin Coding} method to save a great amount of time. Here, we explain two different optimizations that are based on the \\gls{MSC}\\index{Multispin Coding} technique.\n\n\\subsection{Multisample Multispin Coding} \\labsubsec{multisample_multspin_coding_appendix}\nAs said in~\\refch{equilibrium_chaos}, is known for around 20 years that we can perform the Metropolis update of a single spin by using a sequence of boolean\\index{boolean} operations \\cite{newman:99}. Thus, we can take advantage of the \\gls{MSC}\\index{Multispin Coding} technique to simulate several independent \\gls{EA}\\index{Edwards-Anderson!model} models with Ising\\index{Ising} spins and $\\pm 1$ couplings\\index{couplings}. This method is widely used in computational physics \\cite{newman:99,leuzzi:08,fernandez:09f,banos:12,janus:13,manssen:15,lulli:15,billoire:17,billoire:18,gonzalez-adalid-pemartin:19,fernandez:19} and it is known as \\gls{MUSA}\\index{Multispin Coding!Multisample}. Because \\gls{MUSA} is well known (see e.g. Appendix B.1 in~\\cite{seoane:13}) we will only briefly describe it here.\n\nThe idea is to take a set of $N$ independent systems (in our particular case, $N$ different samples\\index{sample}) that can be simulated in a parallel\\index{parallel!computation} way and encode, for each site of the lattice, the spins of the $N$ systems in a bit-register of size $N$. Then, all the computations can be done by performing boolean\\index{boolean} operations and rotations of spins, just like we have exemplified in \\refsec{overlap_spheres_multispin_coding}. In our particular case, with the available hardware (Intel Xeon E5-2680 and AMD Opteron Processor 6272 processors), we found that the $128$ bits version was the most efficient one.\n\nHowever, as introduced in~\\refch{equilibrium_chaos}, this method is not efficient for all the samples\\index{sample}. Indeed, if only a few of the 128 samples\\index{sample} coded in a computer word are not yet thermalized, continuing the simulation of the already equilibrated samples\\index{sample} is a waste of computer time. In \\reffig{all_L_prob_tau} one can understand that the width of the autocorrelation time distribution increases with $L$, which makes this problem more important for large sizes. For those samples\\index{sample}, we need an extra level of optimization.\n\n\\subsection{Multisite Multispin Coding} \\labsubsec{musi}\nThe \\gls{MUSI}\\index{Multispin Coding!Multisite} method provides a solution for the problem of slow equilibration mentioned at the end of \\refsubsec{multisample_multspin_coding_appendix}. In this case, we found that the $256$ version was more efficient. The idea is to use the 256 bits in a computer word to code 256 distinct spins of a single replica\\index{replica} of a single sample\\index{sample}~\\cite{fernandez:15}. In this way, we execute the Metropolis algorithm in $L^3\/256$ steps. The major problem of the \\gls{MUSI}\\index{Multispin Coding!Multisite} method is the generation of random numbers. Because we are performing $256$ updates of spins at the same time, we need $256$ independent random numbers. In our realization of the \\gls{MUSI}\\index{Multispin Coding!Multisite} we circumvent this problem as explained in~\\cite{fernandez:15}.\n\nThe geometry used to encode the lattice with $L=16$ is the same that appears in~\\cite{fernandez:15}. Here, we briefly recall it.\n\nThe physical lattice of Cartesian coordinates $0\\leq x,y,z<L$ is mapped to a \\emph{super-spin} lattice. Each super-spin is coded in a 256-bits computer word (of course, the 256 bits correspond to 256 physical spins which are updated in parallel). The crucial requirement is that spins that are nearest-neighbors in the physical lattice are coded into nearest-neighbors super-spins. In particular, our super-spins are placed at the nodes of a cubic lattice with the geometry of a parallelepiped of dimensions $L_x=L_y=L\/8$, and $L_z=L\/4$. The relation between physical coordinates $(x,y,z)$ and the coordinates in the super-spin lattice $(i_x,i_y,i_z)$ is\n\\begin{eqnarray}\nx&=& b_x L_x + i_x\\,,\\ 0\\leq i_x<L_x\\,,\\ 0\\leq b_x < 8\\,,\\nonumber\\\\\\labeq{MSC-lattice}\ny&=& b_y L_y + i_y\\,,\\ 0\\leq i_y<L_y\\,,\\ 0\\leq b_y < 8\\,,\\\\\nz&=& b_z L_z + i_z\\,,\\ 0\\leq i_z<L_z\\,,\\ 0\\leq b_z < 4\\,.\\nonumber\n\\end{eqnarray}\nIn this way, exactly 256 sites in the physical lattice are given the same super-spin coordinates $(i_x,i_y,i_z)$. We distinguish between them by means of the bit index:\n\\begin{equation}\ni_b=64 b_z+8b_y+b_x\\,,\\ 0\\leq i_b\\leq 255\\,.\n\\end{equation}\nSince we have to simulate $N_T$ independent system copies in our \\gls{PT} simulation (see~\\refch{equilibrium_chaos} and \\refsec{thermalizing_PT}), we simply carry out successively the simulation of the $N_T$ systems. \n\nThe alert reader will note that the above geometric construction is very anisotropic (we start with a cube, but end-up with a parallelepiped). Fortunately, this unsightly feature can be easily fixed by noticing that the single-cubic lattice is bipartite. Indeed the lattice splits into the \\emph{even} and \\emph{odd} sub-lattices according to the parity of $x+y+z$. The two sub-lattices contain $L^3\/2$ sites. Furthermore, odd spins interact only with even spins and vice versa. It follows that the update ordering is irrelevant, provided that our full-lattice sweep first updates all the (say) odd sites and next to all the even sites. Now, provided that $L_x$, $L_y$ and $L_z$ are all \\emph{even}, the parity of $x+y+z$ and $i_x+i_y+i_z$ coincide.  This implies that all the spins coded in a single super-spin share the same parity, making irrelevant the super-spin lattice asymmetry. For $L=16$ one finds that $L_x=L_y=2$ and $L_z=4$, the three of them even numbers, and hence the above geometric construction works smoothly. \n\nDespite the success for $L=16$, we found that changes in the geometry were needed in order to correctly simulate the $L=24$ system because one has $L_x=L_y=3$ and $L_z=6$ which implies that the super-spin lattice cannot be split into even and odd sub-lattices.\n\nOur solution consisted in introducing \\emph{logical} super-spins of $512$ physical spins, that were later on coded into two computer words of 256 bits each. The geometrical correspondence was ($L_x=L_y=L_z=L\/8$)\n\\begin{eqnarray}\nx&=& \\tilde b_x L_x + j_x\\,,\\ 0\\leq j_x<L_x\\,,\\ 0\\leq \\tilde b_x < 8\\,,\\nonumber\\\\\\labeq{MSC-lattice-L24}\ny&=& \\tilde b_y L_y + j_y\\,,\\ 0\\leq j_y<L_y\\,,\\ 0\\leq \\tilde b_y < 8\\,,\\\\\nz&=& \\tilde b_z L_z + j_z\\,,\\ 0\\leq j_z<L_z\\,,\\ 0\\leq \\tilde b_z < 8\\,.\\nonumber\n\\end{eqnarray}\nIn this way, exactly 512 sites in the physical lattice are given the same super-spin coordinates $(j_x,j_y,j_z)$. We distinguish between them by means of the bit index:\n\\begin{equation}\nj_b=64 \\tilde b_z+8\\tilde b_y+\\tilde b_x\\,,\\ 0\\leq i_b\\leq 511\\,.\n\\end{equation}\nNow, the crucial observation is that (because $L_x=L_y=L_z=3$ for $L=24$) the parity of $x+y+z$ coincides with that of $j_x+j_y+j_z$ if (and only if) the parity of $\\tilde b_x+\\tilde b_y+\\tilde b_z$ is even. In other words, given super-spin coordinates $(j_x,j_y,j_z)$, the 512 spins coded in the super-spin split into 256 even spins and 256 odd spins. Because same-parity spins are guaranteed to be mutually non-interacting, we decided to code the 256 bits with the same parity in the same computer word, with the corresponding bit index being the integer part of $j_b\/2$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\chapter{Parallel Tempering. Technical details.} \\labch{AP_PT}\n\n\\section{On the selection of relevant parameters of the simulation} \\labsec{selection_parameters}\nIn this section, we try to clarify some of the choices made for the parameters of the simulation that can be obscure from the reader's point of view.\n\n\\subsection{Setup-independence of the results}\nAt the beginning of the chapter, we assured that our results do not depend on the particular setup of our simulations, motivated by the metastate\\index{metastate} study exposed in~\\refch{metastate}. Now, it is time to quantitatively justify our statement.\n\nIf we focus on the relevant observable from the dynamical point of view, $\\tintf=\\tau$ we can check in~\\reffig{selection_samples_dinamica} that the internal disorder\\index{disorder} is affecting almost nothing the probability distribution of our observable. The average over the outer disorder\\index{disorder} (which we can call the metastate\\index{metastate} average, as introduced in~\\refch{metastate}) dramatically reduces the fluctuations due to the internal disorder\\index{disorder}.\n\nThe same discussion is straightforwardly applicable to the chaos integral in~\\reffig{selection_samples_estatica}.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{eq_chaos\/dist_tau_interiores_g}\n\\caption[\\textbf{The integrated time is independent on the simulation setup.}]{\\textbf{The integrated time is independent on the simulation setup.} Empirical probability distribution function of $\\tau$ represented for the $10$ inner samples\\index{sample} separately.  $L=16$ case (top) and $L=24$ case (bottom). Averaging over the metastate\\index{metastate} (i.e. the outer samples\\index{sample}) with fixed inner couplings\\index{couplings} strongly reduces the fluctuations between the inner samples\\index{sample}.}\n\\labfig{selection_samples_dinamica}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{eq_chaos\/dist_I_interiores_g}\n\\caption[\\textbf{The chaos integral is independent on the simulation setup.}]{\\textbf{The chaos integral is independent on the simulation setup.} Empirical probability distribution function of the integrated chaotic parameter. \\textbf{Top} We compare the distribution (labeled as \"Metastate\"\\index{metastate}) obtained with our particular choice of samples\\index{sample} with the distribution obtained from $4000$ fully independent samples\\index{sample} (data from Janus\\index{Janus}). \\textbf{Bottom:} Distributions obtained for the $10$ inner samples\\index{sample} plotted separately. Averaging over the metastate\\index{metastate} (over the outer couplings\\index{couplings}) strongly reduces the fluctuations between the inner samples\\index{sample}.}\n\\labfig{selection_samples_estatica}\n\\end{figure}\n\n\\subsection{Temperature-mesh's choice}\nOur main results are mainly concerned about the $L=16$ and $L=24$ simulations. The reader may find it surprising that we have two simulations for the $L=16$ case, one with $\\Tmin=0.479$ and the other one with $\\Tmin=0.698$. The latter is obvious since is the $\\Tmin$ needed to compare with the other lattice sizes in our scaling result of~\\refsec{scaling_eq_chaos}. However, the choice of $\\Tmin=0.479$ is not capricious. We carefully choose this $\\Tmin$ with the aim of having similar $\\tint$ for the most chaotic samples\\index{sample} in both simulations ($L=24$, $N=24$, $\\Tmin=0.698$) and ($L=16$, $N=16$, $\\Tmin=0.479$). This is shown in~\\reffig{comparison_dynamics_L24_L16}.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{eq_chaos\/dist_tau_comparative_g}\n\\caption[\\textbf{Empirical probability distribution function of $\\mathbf{\\tau}$.}]{\\textbf{Empirical probability distribution function of $\\mathbf{\\tau}$.} Comparison of results for the simulations ($L=24$,$T_\\mathrm{min}=0.698$) and ($L=16$,$T_\\mathrm{min}=0.479$). Note that at the high-end of very difficult samples\\index{sample}, these two simulations are similarly challenging.}\n\\labfig{comparison_dynamics_L24_L16}\n\\end{figure}\n\n\\section{Thermalizing with Parallel Tempering} \\labsec{thermalizing_PT}\nThe reader used to deal with thermalization\\index{thermalization} protocols might find surprising that we focus on the temperature-random-walk\\index{random walk!temperature} to establish our particular thermalization\\index{thermalization} protocol. This appendix is devoted to justify our decision (which is not pioneer and can be find in other works, see for example~\\cite{janus:10}).\n\nWe have introduced in~\\refsubsec{Monte_Carlo} several properties and theorems of the Markov\\index{Markov chain} chains. Specifically, there exists one theorem~\\cite{sokal:97} that assures, for an aperiodic\\index{aperiodicity} Markov\\index{Markov chain} chain that fulfills the balance\\index{balance condition} condition, that the time evolution of the Markov\\index{Markov chain} chain eventually reaches a stationary distribution (which is unique).\n\nIn the case of \\gls{PT}, that stationary distribution appears in~\\refeq{PT_eq_distribution} but we recall it here for the reader's convenience.\n\\begin{equation}\nP_{\\mathrm{eq}}(\\{ \\pi, \\{s^{(\\alpha)}\\}_{\\alpha=1}^N\\}) = \\dfrac{1}{N!}\\prod_{\\alpha=1}^N \\dfrac{\\exp \\left[-\\beta_{\\alpha} \\mathcal{H}(\\{s^{\\pi^{-1}(\\alpha)}\\})\\right]}{Z_{\\beta_{\\alpha}}} \\, , \n\\end{equation}\nIn the previous expression, $N$ is the number of temperatures in the temperature-mesh of our \\gls{PT} method, the index $\\alpha$ labels a given clone and, here, $\\pi(\\alpha)$ is the permutation of the clone $\\alpha$ at a given time and $\\pi^{-1}$ is its inverse.\n\nThis equilibrium configuration\\index{configuration} for the \\gls{PT} is telling us two main things. On the one hand, the marginal probability for the clones permutation is uniform as the factor $1\/N!$ implies. On the other hand if we focus on the marginal probability for the spins of one clone at temperature $T_{\\alpha'}$, namely $\\pi(\\alpha) = \\alpha'$, we find that the equilibrium probability at a given temperature is\n\\begin{equation}\nP_{\\mathrm{eq}}(\\{\\{s^{(\\alpha)}\\} | \\pi(\\alpha)=\\alpha' ) = \\dfrac{\\exp \\left[-\\beta_{\\alpha'} \\mathcal{H}(\\{s^{\\alpha}\\})\\right]}{Z_{\\beta_{\\alpha'}}} \\, , \n\\end{equation}\nwhich is none but the Boltzmann-Gibbs\\index{Boltzmann!-Gibbs distribution} distribution.\n\nThe message is clear, when the random-walks\\index{random walk} equilibrate, the Boltzmann-Gibbs\\index{Boltzmann!-Gibbs distribution} distribution is reached by all the $N$ clones simultaneously.\n\nThe theorem ensuring the equilibration in the Markov\\index{Markov chain} chains is fundamental because it legitimates its use to study equilibrated systems. However, that theorem only assures that we will reach, at some point, our desired equilibrium distribution, but it says nothing about the convergence to that limit. We have succinctly explained in~\\refsec{time_scales_eq_chaos} the usual way to study this convergence through the computation of time autocorrelation functions\\index{correlation function!time auto-} $\\hat{C}_f(t)$. \n\nNevertheless, the reader familiarized with those methods may be used to build that time autocorrelation functions\\index{correlation function!time auto-} from spin-dependent functions instead of focusing on the temperature random-walk\\index{random walk!temperature} of \\gls{PT}. Actually, this assumption is customary rather than necessary. All the statements done concerning the time autocorrelation function\\index{correlation function!time auto-} $\\hat{C}_f(t)$ are valid from any function $f$ related to the Markovian\\index{Markovian} dynamics. The reader can check that no special condition, restricting the function $f$, was imposed in~\\refsec{time_scales_eq_chaos}. In our particular case, it is really convenient to study the random-walk\\index{random walk!temperature} in the temperature. We expose here an example to illustrate our statements.\n\nWe shall consider an $L=24$ sample\\index{sample} chosen from the simulated samples\\index{sample} with $N=24$ and $T_{\\min}=0.698$ (see~\\refsec{numerical_simulations_eq_chaos}). Specifically, the chosen sample\\index{sample} will be more difficult (i.e. will have a larger autocorrelation time) than the $90\\%$ of the simulated samples\\index{sample}.\n\nNow, we consider two different \\gls{PT} setups. The first one corresponds to the setup used in the simulations, namely a temperature mesh with $N=24$ temperatures and $T_{\\min}=0.698$. The second one will consist only of the four lowest temperatures: $T_1=T_{\\min}=0.698$, $T_2 = 0.735$, $T_3=0.771$ and $T_4=0.808$.\n\nThe $N=24$ setup is found to need $2\\cdot10^9$ Metropolis sweeps to meet our thermalization\\index{thermalization} criteria and we decided to run both \\gls{PT} setups for that number of Metropolis sweeps. We know that the $N=24$ system will thermalize and the natural expectation for the $N=4$ system is that it will not reach our thermalization\\index{thermalization} criteria. The reason for that expectation is that, while the standard simulation will spend around $1\/2$ of the total time above the critical\\index{critical temperature} temperature, the truncated simulation ($N=4$) will not. \n\nActually, we know that, for the highest temperature in the (standard) mesh, $T=1.6$, the exponential autocorrelation time\\index{autocorrelation time!exponential} for Metropolis dynamics is about $10^4$ lattice sweeps~\\cite{ogielski:85}, which corresponds to $1000$ times more lattice sweeps that each clone will spend at the highest temperature ($2\\cdot 10^9\/24 \\approx 8 \\cdot 10^7$). On the contrary, for the truncated system with $N=4$ the highest temperature is still in the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass} and the time that each clone spends at the highest temperature is not enough to decorrelate the spins.\n\nNow, we need a function $f$ depending on the spin configurations\\index{configuration} in order to compare with our function $f$ depending on the temperature random-walk\\index{random walk!temperature} of the \\gls{PT}. Using the fact that we have already equilibrated this sample\\index{sample}, we have selected randomly four equilibrium spin configurations\\index{configuration} at our lowest temperature $T_{\\min}=0.698$. We denote those configurations\\index{configuration} as $\\{ \\tau_{x,a}\\}$ with $a=$1, 2, 3, 4. Our four proposed functions (for each clone $\\alpha$) $f_{a,\\alpha}$ will be the time-dependent overlap\\index{overlap} between the spin configurations\\index{configuration} of the clone $\\alpha$, and each of the four selected equilibrium configurations\\index{configuration} $\\{ \\tau_{x,a} \\}$\n\\begin{equation}\nf_{a,\\alpha} = q_{a,\\alpha}(t) = \\dfrac{1}{L^3} \\sum_x \\tau_{x,a} s_x^{(\\alpha)}(t) \\, . \\labeq{spin-dependent_observable}\n\\end{equation}\n\nWe compute those overlaps\\index{overlap} for a set of $10$ new standard simulations (i.e. $N=24$) and for the truncated simulations $N=4$. Because we are particularly interested in the thermalization\\index{thermalization} process, we measure $q_{a,\\alpha}(t)$ very often (every $5\\cdot 10^4$ Metropolis sweeps\\footnote{This suppose a total of $40000$ measure from $t=0$ to $t=2\\cdot 10^9$ Metropolis sweeps.}). \n\nThe global spin-flip symmetry of the \\gls{EA}\\index{Edwards-Anderson!Hamiltonian} Hamiltonian\\index{Hamiltonian} implies that the equilibrium distribution for $q_{a,\\alpha}$ is symmetric under overlap\\index{overlap} inversions $q_{a,\\alpha} \\leftrightarrow -q_{a,\\alpha}$. It is important to check this symmetry, since it is believed that the largest dynamical barriers are related to global spin-flips~\\cite{billoire:01}.\n\nOnce we simulate both setups, we notice that the truncated simulation is not able to thermalize within the time span of our simulations. To illustrate our findings, we show~\\reffig{historia_q}.\n\n\\begin{figure}[t!]\n\\includegraphics[width=0.8\\textwidth]{eq_chaos\/historia_q}\n\\caption[\\textbf{Overlap history. Exploring the thermalization process.}]{\\textbf{Overlap\\index{overlap} history. Exploring the thermalization\\index{thermalization} process.}\\textbf{ Top:} Monte\\index{Monte Carlo} Carlo history for the overlap\\index{overlap} $q_4(t)$. Note that our simulation time is too short to expose the symmetry $q_4 \\leftrightarrow -q_4$. As a consequence, we know for sure that thermal equilibrium has not been reached for the truncated simulation. \\textbf{Bottom:} as in the top panel, for the first four clones in one of our standard simulations with $N = 24$ temperatures (there were 10, completely independent, standard simulations). For each clone, the overlap\\index{overlap} $q_4(t)$ changes sign many times along the simulation (as it is to be expected for a well-equilibrated simulation). Note that, with small probability, each clone reaches a state where $|q_4| \\sim 0.8$. These events, which are not observed for the other three overlaps\\index{overlap} $q_a$ with $a=$1, 2, 3, make it particularly interesting to study the dynamics of $q_4$.}\n\\labfig{historia_q}\n\\end{figure}\n\nIn the top panel of this figure, we represent the time-dependent overlap\\index{overlap} $q_4(t)$ for each of the four clones in the truncated simulation. Furthermore, in the bottom panel, we represent the same quantity for the clones corresponding to the four lowest temperatures in the standard simulation. We clearly observed that nor the overlap\\index{overlap} symmetry neither the clone equivalence is respected in the truncated simulation. On the contrary, the standard simulation with $N=24$ displays the expected overlap\\index{overlap} symmetry. The Monte\\index{Monte Carlo} Carlo histories (in the standard simulation) for $q_{a,\\alpha}$ with $a =$ 1, 2, 3 (not shown) are symmetric as well. However, only $q_4$ uncovers a state that arises with small probability, characterized by $|q_4| \\sim 0.8$. This feature suggests that $q_4$ is the most interesting overlap\\index{overlap} to look at.\n\nIt is clear, from the above discussion, that thermalization\\index{thermalization} is not reached. But we have focused on the overlaps\\index{overlap} $q_a$ with $a=$ 1, 2, 3, 4. One could wonder if looking at the functions based on the temperature random-walk\\index{random walk!temperature} of the \\gls{PT} we would infer a different result, after all, the truncated simulation is composed only of $4$ copies of the system and one could think that it should not be so difficult to equilibrate the clone permutation. The answer is negative as can be seen in~\\reffig{histogramas}. The fact that the spins are out of equilibrium makes it also impossible to equilibrate the clone permutations.\n\n\\begin{figure}[t!]\n\\includegraphics[width=0.8\\textwidth]{eq_chaos\/histogramas}\n\\caption[\\textbf{Time spent in each temperature by the four clones of the truncated Parallel Tempering simulation.}]{\\textbf{Time spent in each temperature by the four clones of the truncated  Parallel Tempering simulation.} For each of the four clones in the truncated simulation, we show the histogram of temperature (i.e. the number of times that each clone can be found at $T_1$, $T_2$, \\dots). The temperature state was sampled every $5\\times 10^4$ Metropolis sweeps (per clone). Had the simulation equilibrated, we would have expected the occupation histograms to be uniform.}\n\\labfig{histogramas}\n\\end{figure}\n\nNow, we have exemplified that if the equilibrium is not reached by spin-related quantities it would neither be reached by clone-permutations-related quantities but, are the autocorrelation timescales of the spin-related quantities and the clone-permutations-related quantities equivalent?\n\nTo answer this question we focus on the standard setup with $N=24$, which is the only one that equilibrates and, therefore, the only one in which we can compute the equilibrium autocorrelation functions\\index{correlation function!time auto-}. In~\\reffig{correlaciones} we plot the time autocorrelation functions\\index{correlation function!time auto-} for the four quantities $q_a$ with $a=$ 1, 2, 3, 4, and also the time autocorrelation function\\index{correlation function!time auto-} for the temperature random-walk\\index{random walk!temperature} (labeled as $T$-correlation), specifically the piecewise linear function with $T^* = T_3$ (see~\\refch{equilibrium_chaos}).\n\n\\begin{figure}[t!]\n\\includegraphics[width=0.8\\textwidth]{eq_chaos\/correlaciones}\n\\caption[\\textbf{Equilibrium time dependent correlation functions, as computed from the standard simulation with \\boldmath $N_T=24$.} ]{\\textbf{Equilibrium time dependent correlation functions, as computed from the standard simulation with \\boldmath $N_T=24$.}  We consider five observables, one related to temperature (computed from the piece-wise linear function with $T^*=T_3$, see \\refch{equilibrium_chaos}), and the overlaps\\index{overlap} $q_a$ with $a=1,2,3,4$. The fact that the $T$ and $q_4$ correlations become parallel in this semi-logarithmic scale indicates that we are safely computing the exponential auto-correlation time (which is independent of the observable). Instead, the $q_{a=1,2,3}$ correlations do not become parallel to the other curves, at least not within the range we can measure, which probably indicates that the amplitudes $A_{n=1,q_{a=1,2,3}}$, see~\\refeq{autocorrelation_function_decomposition}, are much smaller for these observables.}\n\\labfig{correlaciones}\n\\end{figure}\n\nWe observe that the long-time behavior of both the $q_4$-correlation function and the $T$-correlation function are equivalent. Recall that this long-time behavior is giving us the exponential autocorrelation time\\index{autocorrelation time!exponential} (which is independent of the particular quantity we are studying). This is indicating us that we are safely computing the exponential autocorrelation time\\index{autocorrelation time!exponential} and, furthermore, that both timescales are equivalent. Specifically, and measuring time in Metropolis sweeps, we find $\\texp = 3.0(4)\\cdot 10^{7}$ from $q_4$, while we find the fully compatible value $\\texp = 3.1(6) \\cdot 10^7$ from the $T$ random-walk\\index{random walk!temperature}. \n\nA final warning should be made in this respect. The reader might wonder, since we have shown the equivalence of both methods, if it is not simpler to study $q_4$ rather than the clone permutations. The answer is negative. In order to discover the $q_4$-clone-permutation equivalence we had to thermalize the system and find the equilibrium configuration\\index{configuration} $\\{\\tau_{x,4}\\}$. Moreover, we randomly pick four different configurations\\index{configuration} and only one of them was useful. It is not guaranteed that one can identify an interesting overlap\\index{overlap} by randomly picking a small number of equilibrated configurations\\index{configuration}.\n\n\\chapter{Statistical tools} \\labch{AP_statistics}\n\\setlength\\epigraphwidth{.5\\textwidth}\n\\epigraph{\\textit{Most people use statistics like a drunk man uses a lamppost; more for support than illumination.}}{-- Andrew Lang}\n\nIn this appendix, we expose the main statistical tools that we have used throughout this thesis. The huge amount of data (often correlated) and the non-linear nature of many quantities in our analysis are the main reasons for our interest in non-traditional statistics. We will discuss each of the methods used in this thesis and we provide examples.\n\n\\section{Estimating error bars} \\labsec{estimating_errorbars}\nLet us consider a random variable $x$ which follows a \\gls{pdf} with mean $\\mu_x$ and variance $\\sigma_x$. The common situation in both, numerical and real experiments, is that the \\gls{pdf} of $x$ is completely unknown for us. Thus, we can only \\textit{sample} the \\gls{pdf} with a finite number $N$ of measurements of the variable $x$: $\\{x_1,x_2,\\dots,x_N\\}$. The usual estimators for $\\mu_x$ and $\\sigma_x$ are the arithmetic mean of the data $m(x)$ and the sample variance $s^2(x)$\n\\begin{equation}\nm(x) = \\dfrac{1}{N} \\sum_{i=1}^N x_i \\quad , \\quad s^2(x) = \\dfrac{1}{N-1} \\sum_{i=1}^N \\left[x_i - m(x)\\right]^2 \\, .\n\\end{equation}\n\nThese estimators are perfectly suitable for random variables and a linear combination of them. However, it is usual to be interested in a non-linear combination of random variables where the previous simple method does not work. In this case, we need a more advanced analysis in order to estimate the mean and the variance.\n\nThe traditional method is the well-known \\textit{error propagation}. This method is based on Taylor expansions and is unbiased (up to arbitrary precision, determined by the number of terms we retain in the Taylor expansion). Nonetheless, it is dependent on the specific expression of the non-linear quantity studied and, therefore, a different computation (involving derivatives) has to be performed for each particular case.\n\nHere, we introduce two alternatives that consist of different strategies to resample the data and compute the error bars\\index{error bars}: the Jackknife\\index{Jackknife} method and the Bootstrap\\index{Bootstrap} method.\n\n\\subsection{The Jackknife method} \\labsubsec{jk_method}\nJust as before, suppose we have $N$ independent measures $\\{x_1,x_2,\\dots,x_N\\}$. We define the Jackknife\\index{Jackknife} variables as\n\\begin{equation}\nx^{\\mathrm{JK}}_i = m(x) + \\dfrac{1}{N-1} \\left[ m(x) - x_i \\right] = \\dfrac{1}{N-1} \\sum_{j \\neq i} x_j \\, . \\labeq{jackknife_variable_definition}\n\\end{equation}\nNow, suppose we have a set of random variables $x_{\\alpha}$, $x_{\\beta}$, $x_{\\gamma}$, $\\dots$ (each one with $N$ independent measures $x_{\\alpha,i}$, $x_{\\beta,i}$, $\\dots$ with $i=1,\\dots,N$) and a non-linear combination of them $f(x_{\\alpha},x_{\\beta},x_{\\gamma},\\dots)$. We define the Jackknife\\index{Jackknife} variable (or Jackknife\\index{Jackknife} block) of that non-linear function as\n\\begin{equation}\nf^{\\mathrm{JK}}_i = f(x^{\\mathrm{JK}}_{\\alpha,i},x^{\\mathrm{JK}}_{\\beta,i},x^{\\mathrm{JK}}_{\\gamma,i},\\dots) \\, .\n\\end{equation}\nOur aim is to estimate $\\mu_f=f(\\mu_{x_{\\alpha}},\\mu_{x_{\\beta}},\\dots)$ and the standard deviation of the quantity $f$, $\\sigma_f$. To estimate that quantity, we introduce the mean of the Jackknife\\index{Jackknife} blocks\n\\begin{equation}\nm(f^{\\mathrm{JK}}) = \\dfrac{1}{N} \\sum_{i=1}^N f^{\\mathrm{JK}}_i \\, .\n\\end{equation}\nFor non-linear quantities is easy to show~\\cite{young:12} that there is a bias in the estimation of $f(\\mu_{x_{\\alpha}},\\mu_{x_{\\beta}},\\dots)$ with the quantity $m(f^{\\mathrm{JK}})$. Indeed, the elimination of the leading bias leads to the following estimator $f_{\\mathrm{JK,estimator}}$ of $f(\\mu_{x_{\\alpha}},\\mu_{x_{\\beta}},\\dots)$:\n\\begin{equation}\nf_{\\mathrm{JK,estimator}} = N f\\left(m(x_{\\alpha}),m(x_{\\beta}),\\dots\\right) - (N-1) m(f^{\\mathrm{JK}}) \\, .\n\\end{equation}\n\nIn this case, the bias is of order $1\/N^2$. If we just take the quantity $m(f^{\\mathrm{JK}})$ as an estimator of $f(\\mu_{x_{\\alpha}},\\mu_{x_{\\beta}},\\dots)$ the bias is of order $1\/N$. As we will discuss below, this is not a big problem since the bias in the statistical error is of order $1\/\\sqrt{N}$ which is much bigger than $1\/N$ for large $N$.\n\nWe can also define the Jackknife\\index{Jackknife} variance as\n\\begin{equation}\ns^2(f^{\\mathrm{JK}}) = m\\left((f^{\\mathrm{JK}})^2\\right) - \\left[m(f^{\\mathrm{JK}})\\right]^2 \\, .\n\\end{equation}\n\nAfter some easy mathematical steps~\\cite{young:12}, we can reach an expression for the estimator $\\sigma_{\\mathrm{JK,estimator}}$ of the variance $\\sigma_f$\n\\begin{equation}\n\\sigma^2_{\\mathrm{JK,estimator}} = (N-1) s^2(f^{\\mathrm{JK}}) \\, .\n\\end{equation}\nThis is the great success of the Jackknife\\index{Jackknife} method, the estimator of the variance can be obtained from the raw data, and no derivatives are needed to be computed. Hence, the error bars\\index{error bars} of our desired quantity, that usually corresponds to the standard deviation of $f(\\mu_{x_{\\alpha}},\\mu_{x_{\\beta}},\\dots)$, would be \n\\begin{equation}\n\\sigma_{\\mathrm{JK,estimator}} = \\sqrt{N-1} \\, s(f^{\\mathrm{JK}}) = \\sqrt{N-1} \\left[  m\\left((f^{\\mathrm{JK}} \\right)^2) - \\left[m(f^{\\mathrm{JK}})\\right]^2 \\right]^{1\/2} \\, . \\labeq{errorbar_estimator}\n\\end{equation}\n\nThe reader can found an example of a specific application of this method in~\\refsec{procedure}. \n\nAnother advantage of this method is that the Jackknife\\index{Jackknife} blocks can be easily propagated to perform further computations. This propagation usually involves non-linear operations, fortunately, we have just discussed that the Jackknife\\index{Jackknife} method can deal with them. We illustrate our statement with one example.\n\n\\subsubsection{Computing the error of the coherence length}\nAs we have said, the coherence length\\index{coherence length} is fundamental in order to characterize the off-equilibrium dynamics. Let us sketch the procedure to compute its error bars\\index{error bars}.\n\nFirst, we recall that our off-equilibrium simulation consists of an \\gls{EA}\\index{Edwards-Anderson!model} model with Ising\\index{Ising} spins for which we have simulated $\\ensuremath{N_{\\text{S}}}\\xspace=16$ different samples\\index{sample} and, for each sample\\index{sample}, $512$ replicas.\n\nBefore explaining the method, we briefly recall the definition of the estimator of the coherence length\\index{coherence length} $\\xi_{12}(t)$ from the four-point\\index{correlation function!four point} correlation function $C_4(T,r,t)$\n\\begin{align}\nC_4(T,r,t) & =  \\overline{\\braket{q^{\\sigma,\\tau}_{\\vec{x}}(t)q^{\\sigma,\\tau}_{\\vec{x}+\\vec{r}}(t)}} \\, , \\\\\nI_k(t) & = \\int_0^{\\infty} r^k C_4(T,r,t) {\\mathrm{d}} r \\, , \\\\\n\\xi_{12}(t) & = \\dfrac{I_{2}(t)}{I_1(t)} \\, .\n\\end{align}\n\nWe start by computing $C_4(T,r,t)$. The Jackknife\\index{Jackknife} method will allow us to estimate the disorder\\index{disorder!average} average $\\overline{(\\cdots)}$. For each sample\\index{sample} $\\mathcal{J}_i$, we compute\n\\begin{equation}\nC_4^{\\mathcal{J}_i}(T,r,t) = \\braket{q^{\\sigma,\\tau}_{\\vec{x}}(t)q^{\\sigma,\\tau}_{\\vec{x}+\\vec{r}}(t)}_{\\mathcal{J}_i} \\, ,\n\\end{equation}\nwhere the label $\\mathcal{J}_i$ stresses the fact that the quantity is sample-dependent. Then we can build the Jackknife\\index{Jackknife} blocks for this quantity\\footnote{The details on the computation of $ C_{4,i}^{\\mathrm{JK}}(T,r,t)$ can be found in~\\refsec{finite_size_effects}.}\n\\begin{equation}\nC_{4,i}^{\\mathrm{JK}}(T,r,t) = \\dfrac{1}{\\ensuremath{N_{\\text{S}}}\\xspace-1} \\sum_{i \\neq j} C_4^{\\mathcal{J}_j}(T,r,t)\n\\end{equation}\n\nIf we were interested only in computing the four-point\\index{correlation function!four point} correlation function, we would simply estimate the errors of $C_4(T,r,t)$ by using~\\refeq{errorbar_estimator}. However, we are interested in the quantity $\\xi_{12}(t)$. The \\textit{propagation} of the Jackknife\\index{Jackknife} method to derived quantities is quite simple. We now compute $I_k(t)$ [in our case we are interested in $I_1(t)$ and $I_2(t)$] for each Jackknife\\index{Jackknife} block of the four-point\\index{correlation function!four point} correlation function. Therefore, we will obtain\n\\begin{equation}\nI^{\\mathrm{JK}}_{k,i}(t) = \\int_0^{\\infty} r^k C_{4,i}^{\\mathrm{JK}}(T,r,t) {\\mathrm{d}} r \\, .\n\\end{equation}\nFinally we can, again, propagate the Jackknife\\index{Jackknife} block by computing\n\\begin{equation}\n\\xi^{\\mathrm{JK}}_{12,i}(t) = \\dfrac{I^{\\mathrm{JK}}_{2,i}(t)}{I^{\\mathrm{JK}}_{1,i}(t)} \\, .\n\\end{equation}\n\nIn the end, we will have $\\ensuremath{N_{\\text{S}}}\\xspace$ Jackknife\\index{Jackknife} blocks and we can estimate the error bars\\index{error bars} of $\\xi_{12}(t)$ by using~\\refeq{errorbar_estimator}. \n\n\n\\subsection{The Bootstrap method} \\labsubsec{bootstrap_method}\nAlthough most of the time we use the Jackknife\\index{Jackknife} method, there exists an alternative that also takes advantage of resampling the data: the Bootstrap\\index{Bootstrap} method. \n\nLet us consider again that we have a set of $N$ points $\\{x_1,x_2,\\dots,x_N\\}$. In the Jackknife\\index{Jackknife} method, we build $N$ blocks from all the points of the original data except one. On the contrary, in the Bootstrap\\index{Bootstrap} method, we build $N_{\\mathrm{boots}} \\neq N$ data sets, each one containing $N$ points picked at random from the original data. The probability of selecting one point of the original data is $1\/N$ and does not depend on whether it has been selected before.\n\nFor a given data set $k$, with $k$ running from 1 to $N_{\\mathrm{boots}}$, we can define the mean of the points inside this data set as\n\\begin{equation}\nx^{B}_k = \\dfrac{1}{N} \\sum_{i=1}^N n^{B}_{i,k} x_i \\, ,\n\\end{equation}\nwhere $n^{B}_{i,k}$ is the number of times that the point $x_i$ have been selected for the data set $k$. The mean of all the $x^{B}_k$ over the $N_{\\mathrm{boots}}$ data sets is\n\\begin{equation}\nm\\left(x^{B}\\right) = \\dfrac{1}{N_{\\mathrm{boots}}} \\sum_{k=1}^{N_{\\mathrm{boots}}} x_k^{B} \\, . \\labeq{def_mean_boots}\n\\end{equation}\n\nIt can be shown~\\cite{young:12}, that for $N_{\\mathrm{boots}} \\to \\infty$ the value of $m\\left(x^{B}\\right)$ corresponds to the value of $m(x)$, which is an unbiased estimator of $\\mu_x$. The variance of $x^{B}$ is\n\\begin{equation}\ns^2\\left(x^{B}\\right) = m \\left( (x^{B})^2\\right) - \\left[ m\\left( x^{B} \\right) \\right]^2 \\, .\n\\end{equation}\n\nTo address the more interesting scenario, suppose again that we have a set of random variables $x_{\\alpha}$, $x_{\\beta}$, $x_{\\gamma}$, $\\dots$ (each one with $N$ independent measures $x_{\\alpha,i}$, $x_{\\beta,i}$, $\\dots$ with $i=1,\\dots,N$) and a non-linear combination of them $f(x_{\\alpha},x_{\\beta},x_{\\gamma},\\dots)$.  We are again interested in estimating $\\mu_f= f(\\mu_{x_{\\alpha}},\\mu_{x_{\\beta}},\\dots)$ and its standard deviation $\\sigma_f$. The Bootstrap\\index{Bootstrap} variable will be\n\\begin{equation}\nf^{B}_k = f(x^{B}_{\\alpha,k},x^{B}_{\\beta,k},x^{B}_{\\gamma,k},\\dots) \\, . \\labeq{bootstrap_variable}\n\\end{equation}\n\nSimilarly to the Jackknife\\index{Jackknife} case, the mean and the variance of $f^{B}_k$ would be\n\\begin{equation}\nm\\left( f^{B} \\right) = \\dfrac{1}{N_{\\mathrm{boots}}} \\sum_{k=1}^{N_{\\mathrm{boots}}} f^{B}_{k} \\, ,\n\\end{equation}\n\\begin{equation}\ns^2 \\left( f^{B} \\right) = m\\left( \\left(f^{B} \\right)^2 \\right)- \\left[ m\\left(f^{B}\\right) \\right]^2 \\, .\n\\end{equation}\n\nAgain, the bias in $m\\left( f^{B} \\right)$ is of order $1\/N$, much smaller than the statistical error. Indeed, the Bootstrap\\index{Bootstrap} estimator $\\sigma_{\\mathrm{B,estimator}}$ of the standard deviation $\\sigma_f$ is\n\\begin{equation}\n\\sigma_{\\mathrm{B,estimator}} = \\sqrt{\\dfrac{N}{N-1}} \\left[ m\\left( \\left(f^{B} \\right)^2 \\right)- \\left[ m\\left(f^{B}\\right) \\right]^2 \\right]^{1\/2} \\, . \\labeq{standard_deviation_bootstrap}\n\\end{equation}\nIn the $N\\to \\infty $ limit, the prefactor tends to $1$.\n\nIt is worthy to note that a high enough number of data sets $N_{\\mathrm{boots}}$ is needed to ``achieve'' the $N_{\\mathrm{boots}} \\to \\infty$ limit in~\\refeq{def_mean_boots}. How large should be $N_{\\mathrm{boots}}$ to achieve that limit? We have not performed a systematic study but the relation $N_{\\mathrm{boots}} = 10N$ seems to be safe. Indeed, our use of the Bootstrap\\index{Bootstrap} method is limited and the computations involved do not require large computational times, but a much more detailed study should be done for a different situation.\n\nAs a final note, the reader may wonder why we would use the Bootstrap method if we can use the Jackknife method that involves less computations and provide similar results. The Jackknife method assumes a Gaussian behavior of the variables $x_{\\alpha}$, $x_{\\beta}$, $x_{\\gamma}$, $\\dots$, on the contrary, the Bootstrap method allows us to access to the probability distribution of $f(x_{\\alpha},x_{\\beta},x_{\\gamma},\\dots)$ in those cases in which $x_{\\alpha}$ does not follow a Gaussian distribution. \n\n\\subsubsection{Computing the Pearson coefficient}\nLet us consider a set of $N$ pairs of independent measures $\\{(x_1,y_1),(x_2,y_2),\\dots,(x_N,y_N)\\}$. In some situations, we want to test the linear dependence between the pairs of measures. To test that dependence, the proper tool is the Pearson coefficient $r$ which is between $-1$ and $1$. The extreme values are clear, $r=-1$ corresponds to a complete linear-anticorrelation between $x$ and $y$ and the value $r=1$ corresponds to a perfect correlation between $x$ and $y$.\n\nThe Pearson coefficient is defined as\n\\begin{equation}\nr = \\dfrac{\\sum_{i=1}^N x_i y_i - N m(x) m(y)}{(n-1) \\sqrt{m(x^2) - [m(x)]^2} \\sqrt{m(y^2) - [m(y)]^2}} \\, , \\labeq{pearson_coefficient}\n\\end{equation}\nwhere $m(x)=(1\/N)\\sum_i x_i$.\n\nThe reader may find one practical example of this computation in \\refch{equilibrium_chaos}. Indeed, the Bootstrap\\index{Bootstrap} estimation of the error for this quantity is very simple. From the complete cloud of $N$ points, we randomly (with uniform probability) select $N$ pairs with repetition (i.e. picking more than once the same pair is allowed). We repeat this procedure $N_{\\mathrm{boots}}$ times and, for each of the $N_{\\mathrm{boots}}$ clouds of $N$ points, we compute the Pearson coefficient $r$. \n\nThe correspondence between each of the $r_k$ computed values (with $k$ running from 1 to $N_{\\mathrm{boots}}$) and the values of $f^{B}_k$ appearing in~\\refeq{bootstrap_variable} is evident. Then, the estimation of the error bars\\index{error bars} is straightforward, we have just to apply~\\refeq{standard_deviation_bootstrap}.\n\n\n\\section{Improving the statistics} \\labsec{improving_statistics}\nHere, we explain how the correlation between variables (in a controlled situation) can reduce the error bars\\index{error bars} of the quantities we are interested in. We take advantage of the \\textit{control\\index{control variate} variates} for that purpose. The reader may find further information in~\\cite{fernandez:09c,ross:14}.\n\n\\subsection{Control Variates}\nThe starting point is the same of~\\refsec{estimating_errorbars}, consider a random variable $x$ which follows a \\gls{pdf} with mean $\\mu_x$ and standard deviation $\\sigma_x$ and a set $N$ measurements of the variable $x$: $\\{x_1,x_2,\\dots,x_N\\}$. Since we only have a finite number of measurements, we can only aspire to estimate the mean and the standard deviation through the arithmetic mean $m(x)$ and the sample standard deviation $s(x)$ (or equivalently, the sample variance $s^2(x)$)\n\\begin{equation}\nm(x) = \\dfrac{1}{N} \\sum_{i=1}^N x_i \\quad , \\quad s^2(x) = \\dfrac{1}{N-1} \\sum_{i=1}^N \\left[x_i - m(x)\\right]^2 \\, .\n\\end{equation}\n\nIn order to apply the control\\index{control variate} variates method, we need another variable $y$ (the control\\index{control variate} variate) from which we know exactly its mean $\\mu_y$. Hence, we define a new variable $z$\n\\begin{equation}\nz= x + c(y-\\mu_y) \\, , \\labeq{cv_new_variable}\n\\end{equation}\nwhere $c \\in \\mathbb{R}$ is a constant. It is clear that \n\\begin{equation}\nm(z) = m\\left(x + c(y-\\mu_y) \\right) = m(x) + c(m(y) - \\mu_y) = m(x) \\, .\n\\end{equation}\nTherefore, $m(z)$ is an unbiased estimator of $\\mu_x$. The situation is more interesting for the variance\n\\begin{equation}\ns^2(z) = s^2 \\left( x + c(y -\\mu_y)\\right) = s^2(x) + c^2 s^2(y) + 2c \\mathrm{Cov}(x,y) \\, ,\n\\end{equation}\nbeing $\\mathrm{Cov}(x,y)=m(xy) - m(x)m(y)$ the covariance between the random variables $x$ and $y$. As long as we are interested in reduce the variance of $z$ we can look for the constant value $c$ minimizing $s^2(z)$\n\\begin{equation}\n\\dfrac{\\partial s^2(z)}{\\partial c} = 0 \\implies c^* = - \\dfrac{\\mathrm{Cov}(x,y)}{s^2(y)} \\, ,\n\\end{equation}\nwhere $c^*$ is the value of the constant $c$ that minimizes the variance of $s^2(z)$. Then, the variance of $z$ will be\n\\begin{equation}\ns^2(z) = s^2\\left(x+c^*(y-\\mu_y)\\right) = s^2(x) - \\dfrac{\\left[ \\mathrm{Cov}(x,y) \\right]^2}{s^2(y)} \\, .\n\\end{equation}\nThe plan for reducing the error bars\\index{error bars} and improve the statistics is clear: we have to find a quantity $y$ with the smallest possible variance and the biggest possible covariance with $x$.\n\n\\subsubsection{Improving the energy statistics}\nOne practical example of the control\\index{control variate} variates has been already mentioned in \\refsec{numerical_simulation_mpemba}, here, we provide details of our particular implementation. To put the reader in context, we recall here that we are interested in computing the energy-density\\index{energy!density} $e(t,\\mathcal{J})$ to study the Mpemba\\index{Mpemba effect} effect.\n\nThe definition of the energy\\index{energy!density} density in our three-dimensional lattice of linear size $L$ is\n\\begin{equation}\ne(t,\\mathcal{J}) = \\dfrac{1}{L^3} \\braket{\\mathcal{H}_{\\mathcal{J}}(t)} \\quad , \\quad e(t) = \\overline{e(t,\\mathcal{J})} \\, .\n\\end{equation}\n\nIn order to apply the above-exposed method, we need to introduce a control\\index{control variate} variate with a large covariance with the energy-density\\index{energy!density}. Our proposal for the control\\index{control variate} variate is the following. For each of the $\\ensuremath{N_{\\text{S}}}\\xspace=16$ samples\\index{sample} simulated, we took a fully random initial configuration\\index{configuration}, placed suddenly at $T=0.7$. We perform an isothermal\\index{isothermal} simulation for $t=2^{21}$ Monte\\index{Monte Carlo} Carlo steps, and we store in our disk those configurations\\index{configuration} that fulfills $t = \\Floor{2^{n\/8}}$ with $n$ integer. Then, we choose all the configurations\\index{configuration} with times $\\mathcal{C} \\, \\in \\, [2^{19},2^{21}]$, the number of selected times is $N_{\\mathcal{C}}$. Then, we compute\n\\begin{equation}\n{\\varepsilon_{\\mathrm{cv}}(\\mathcal{J})} = \\dfrac{1}{L^3 N_{\\mathcal{C}}}  \\sum_{t \\in \\mathcal{C}} \\braket{\\mathcal{H}_{\\mathcal{J}}(t)} \\, . \\labeq{control_variate}\n\\end{equation}\n\nThe alert reader may wonder why we choose this quantity ${\\varepsilon_{\\mathrm{cv}}(\\mathcal{J})}$ if we do not know the real value of its mean $\\mu_{{\\varepsilon_{\\mathrm{cv}}(\\mathcal{J})}}$. At the first sight, it seems that we have made no improvements. However, although we do not know $\\mu_{{\\varepsilon_{\\mathrm{cv}}(\\mathcal{J})}}$ we can estimate it with great accuracy. Indeed, we perform a separated simulation with $1.6 \\cdot 10^5$ samples\\index{sample} and two replicas for each sample\\index{sample} (i.e. the energy\\index{energy!density} was averaged over $3.2 \\cdot 10^5$ systems). Thus, the estimation of $\\mu_{{\\varepsilon_{\\mathrm{cv}}(\\mathcal{J})}}$ would have negligible error compared with $e(t)$.\n\nThen, the final estimates of the internal energy\\index{energy!density} is\n\\begin{equation}\n\\tilde{e}(t,\\mathcal{J}) = e(t,{\\mathcal{J}}) - \\left[ {\\varepsilon_{\\mathrm{cv}}(\\mathcal{J})} - \\mu_{{\\varepsilon_{\\mathrm{cv}}(\\mathcal{J})}} \\right] \\quad \\, \\quad \\tilde{e}(t) = \\overline{\\tilde{e}(t,\\mathcal{J})} \\, . \n\\end{equation}\n\nThe benefits of using this control\\index{control variate} variate are obvious from~\\reffig{control_variate}. Moreover, the importance of the high covariance between the studied quantity [in our case, the energy-density $e(t,\\mathcal{J})$] and the control\\index{control variate} variate ${\\varepsilon_{\\mathrm{cv}}(\\mathcal{J})}$ is evident. In the three simulations showed in~\\reffig{control_variate}, the larger improvement in the error estimation is achieved for $T=0.7$ for a coherence length\\index{coherence length} $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ with $\\ensuremath{t_\\mathrm{w}}\\xspace$ roughly corresponding to $\\ensuremath{t_\\mathrm{w}}\\xspace \\in [2^{19},2^{21}]$. Because our control\\index{control variate} variate has been computed with great accuracy for those conditions, it is logical that we achieve the best performance in the method for that case, although, improvements are also achieved for different temperatures (see for example $T=0.9$).\n\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{mpemba\/control_variate.pdf}\n\\caption[\\textbf{Improving the accuracy with control variates.}]{\\textbf{Improving the accuracy with control\\index{control variate} variates.} The figure shows the ratio of statistical errors, as a function of $\\xi(t)$, for the naive $e(t)$ and improved $\\tilde{e}(t)$ estimates of the energy density. The data shown correspond to three different relaxations\\index{relaxation}. Two of them are isothermal\\index{isothermal} relaxations\\index{relaxation} starting from $\\xi=0$ at $t=0$. The third relaxation\\index{relaxation} corresponds to the preparation starting at $(T=0.9,\\xi=5)$ which is quenched to $T=0.7$ at $t=0$. The error reduction is largest for the isothermal\\index{isothermal} relaxation\\index{relaxation} at $T=0.7$ and $2^{19}\\leq t\\leq 2^{21}$, of course (after all, this is the temperature and time region defining the control\\index{control variate} variate), but the error reduction is also very significant at other times and temperatures.}\n\\labfig{control_variate}\n\\end{figure}\n\n\n\\chapter{The aging rate. Technical details.} \\labch{AP_technical_details_aging}\nIn this appendix, we expose the technical details that are fundamental to understand the procedure followed in~\\refch{aging_rate} but that would blur the mean message of the chapter and would make reading difficult.\n\nIn~\\refsec{Nr_aging} we explore the error reduction for the four-point\\index{correlation function!four point} correlation function when a high number of replicas\\index{replica} is simulated. We perform exhaustive checks to our $L=160$ system to be sure that our results are not affected by finite-size effects\\index{finite-size effects} in~\\refsec{finite_size_effects}. Then, in~\\refsec{Josephson_length}, we give additional details of the crossover between the $T=\\ensuremath{T_\\mathrm{c}}\\xspace$ fixed point and the $T=0$ one, which is controlled by the Josephson length. Finally, we discuss the choices of the parameters in our fits in~\\refsec{parameters_aging}.\n\n\\section{Error reduction for high number of replicas}\\labsec{Nr_aging}\nAs mentioned in~\\refch{aging_rate}, the choice of the number of replicas\\index{replica} ($\\ensuremath{N_{\\text{Rep}}}\\xspace$) and samples\\index{sample} ($\\ensuremath{N_{\\text{S}}}\\xspace$) was taken with the aim of improving the estimation of observables related to \\gls{TC}\\index{temperature chaos} in future work, where it is important to maximize the number of possible overlaps\\index{overlap} (pairs of replicas\\index{replica}) $N_\\text{ov}=\\ensuremath{N_{\\text{Rep}}}\\xspace(\\ensuremath{N_{\\text{Rep}}}\\xspace - 1)\/2$.\n\nUnexpectedly, this has led to a dramatic increase in precision.~\\reffig{error-NR} shows the reduction of the statistical error in the correlation function $C_4$ as a function of $1\/\\sqrt{\\ensuremath{N_{\\text{ov}}}\\xspace}$. Moreover, this effect is enhanced as $r$ increases, which leads to a qualitative improvement in the computation of the $I_k(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$ integrals (\\reffig{error-r2Cr}).\n\nThe reader should be alert that, in view of the above results, a previous study can be done for any quantity which involves disorder\\index{disorder!average} averages and thermal averages. The general form of the variance of such a quantity would be\n\\begin{equation}\n\\sigma^2 (\\ensuremath{N_{\\text{Rep}}}\\xspace,\\ensuremath{N_{\\text{S}}}\\xspace) = \\left[ \\sigma_{\\mathrm{S}}^2 + \\sigma_{\\mathrm{R}}^2 \\left( \\dfrac{2}{\\ensuremath{N_{\\text{Rep}}}\\xspace (\\ensuremath{N_{\\text{Rep}}}\\xspace-1)}\\right)^x \\right] \\dfrac{1}{\\ensuremath{N_{\\text{S}}}\\xspace} \\, ,\n\\end{equation}\nwhere $\\sigma_{\\mathrm{S}}^2$ and $\\sigma_{\\mathrm{R}}^2$ are the sample\\index{sample} and thermal contributions to the variance respectively and $x$ is the exponent controlling the error reduction due to the number of overlaps\\index{overlap} $\\ensuremath{N_{\\text{ov}}}\\xspace$ which should be between $0.5$ and $1$.\n\nA careful analysis in the context of thin-film \\gls{SG}s can be found in \\cite{fernandez:19b}. The great advantage of this analysis is that we can choose, for a given computational effort $E=\\ensuremath{N_{\\text{Rep}}}\\xspace \\ensuremath{N_{\\text{S}}}\\xspace$, the number of replicas\\index{replica} (or equivalently, of samples\\index{sample}) that minimizes the error instead of assuming that the maximum possible number of samples\\index{sample} should be simulated (as it was usual).\n\n\\begin{figure}[h!]\n\\includegraphics[width=0.8\\linewidth]{aging\/error-NR_2w}\n\\caption[\\textbf{Reduction of statistical error with the number of overlaps.}]{\\textbf{Reduction of statistical error with the number of overlaps.\\index{overlap}} Rapid reduction in the statistical error for the overlap\\index{overlap} autocorrelation function $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$ with increasing number of replicas\\index{replica} $N_\\text{R}$. We plot $\\Delta C(r)$ for $T=0.7$ and $\\ensuremath{t_\\mathrm{w}}\\xspace=2^{32}$ and several values of $r$ as a function of the number of possible overlaps\\index{overlap} [$N_\\text{ov} = N_\\text{R}(N_\\text{R}-1)\/2$]. For large values of $r$ the error is essentially linear in $1\/N_\\text{R}$. The inset shows the whole range from $N_\\text{R}=2$ (the minimum to define $C_4$), while the main plot is a close up of the large-$N_\\text{R}$ sector. The simulations reported in this paper have $N_\\text{R}=256$.}\n\\labfig{error-NR}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\includegraphics[width=0.7\\linewidth]{aging\/error-r2Cr}\n\\caption[\\textbf{Improving the determination of \\boldmath $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$.}]{\\textbf{Improving the determination of \\boldmath $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$.} Comparison of $r^2 C_4(r)$, function whose integral is used to estimate the coherence length\\index{coherence length}, as computed with $4$ and $256$ replicas\\index{replica} and the same number $N_\\text{S}=16$ of samples\\index{sample} ($T=0.7, \\ensuremath{t_\\mathrm{w}}\\xspace=2^{34}$). We have marked on the $x$ axis the resulting self-consistent cutoffs (see \\refsec{finite_size_effects}) used to estimate the $I_k$ integrals in both cases.}\n\\labfig{error-r2Cr}\n\\end{figure}\n\n\\section{Controlling finite-size effects} \\labsec{finite_size_effects}\nIn order to obtain an estimate for $\\xi_{k,k+1}(T,\\ensuremath{t_\\mathrm{w}}\\xspace) = I_{k+1}(T,\\ensuremath{t_\\mathrm{w}}\\xspace)\/I_k(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$ we need to compute the integrals \\begin{equation}\nI_k(T,\\ensuremath{t_\\mathrm{w}}\\xspace) = \\int_0^\\infty r^k C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace){\\mathrm{d}} r \\, . \\labeq{I_SI} \n\\end{equation}\nAs discussed in~\\cite{janus:09b}, the main difficulty  in this computation is handling the large-$r$ tail where relative errors [$\\Delta C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace) \/ C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$] are big. Our main goal in this part is to minimize the statistical errors and to check for finite-size effects\\index{finite-size effects} (which will appear when $\\xi\/L$ becomes relatively large).\n\nTo compute those $I_k(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$, we proceed in a similar way as in~\\cite{janus:09b,fernandez:18b}. We separate the integration range in two parts. From $0$ to $r_{\\mathrm{cutoff}}$, we compute the numerical integral of $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$. From $r_{\\mathrm{cutoff}}$ to infinity, we estimate the contribution of the tail with a smooth extrapolating function $F(r) \\sim r^{-\\vartheta} f\\bigl(r\/\\xi)$.\n\nIn short, the procedure is\n\\begin{enumerate}\n\\item Obtain $F(r)$ with fits of $C_4$ in a self-consistent region $[r_\\text{min},r_\\text{max}]$  where the signal-to-noise ratio is still good. \n\\item Integrate $C_4$ numerically up to some cutoff and add the analytical integral of $F(r)$  beyond the\ncutoff to estimate the tail contribution.\n\\end{enumerate}\nThe selection of the function $F(r)$ can be done in several ways, here, we propose two particular functions. First, we consider\n\\begin{equation}\nF_1(r) = A_1 r^{-\\vartheta} {\\mathrm{e}}^{-(r\/\\xi)^{\\beta_1}} \\, . \\labeq{F1}\n\\end{equation}\nWhere $\\vartheta$ is the replicon\\index{replicon} exponent and we fit for $A_1,\\beta_1$ and $\\xi$. This analytical form is motivated by the fact that this is the simplest choice that avoids a pole singularity in the Fourier transform of $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$ at finite $\\ensuremath{t_\\mathrm{w}}\\xspace$ and has been used in several previous works, for example, see~\\cite{janus:09b}. In order to check for finite-size effects\\index{finite-size effects}, we also consider a second function $F_2(r)$ resulting from fits that include the first-image term:\n\\begin{equation}\nF_2^*(r) = A_2 \\left[ \\frac{ {\\mathrm{e}}^{-(r\/\\xi)^{\\beta_2}}}{r^\\vartheta} + \\frac{ {\\mathrm{e}}^{-((L-r)\/\\xi)^{\\beta_2}}}{(L-r)^\\vartheta}\\right],\n\\end{equation}\nso we have a second extrapolating function\n\\begin{equation}\\labeq{F2}\nF_2(r) = A_2 r^{-\\vartheta} {\\mathrm{e}}^{-(r\/\\xi)^{\\beta_2}}.\n\\end{equation}\nFor these fits we used $\\vartheta=0.35$. However, this value has very little effect on the final computation of $\\xi_{k,k+1}$. We have checked this by recomputing the integrals with $\\vartheta = 0 $ and $\\vartheta = 1+\\eta\\approx 0.61$ \n(prediction of the droplet\\index{droplet!picture} theory and influence of the $T=T_\\mathrm{c}$ fixed point respectively). The different choices of $\\vartheta$  led to a systematic effect smaller than $20 \\%$ of the error bars\\index{error bars} in the worst case. \n\nOnce we have our two extrapolating functions $F_1$ and $F_2$, we can combine them with the $C_4$ data in several ways:\n\\begin{eqnarray}\nI^1_k &=& \\int_0^{r_{\\mathrm{max}}} {\\mathrm{d}} r~r^k C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace) + \\int_{r_{\\mathrm{max}}}^\\infty {\\mathrm{d}} r~r^k F_1(r) \\, , \\\\\nI^2_k &=& \\int_0^{r_{\\mathrm{max}}} {\\mathrm{d}} r~r^k C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace) + \\int_{r_{\\mathrm{max}}}^\\infty {\\mathrm{d}} r~r^k F_2(r) \\, , \\\\\nI^3_k &=& \\int_0^{r_{\\mathrm{min}}} {\\mathrm{d}} r~r^k C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace) + \\int_{r_{\\mathrm{min}}}^\\infty {\\mathrm{d}} r~r^k F_2(r) \\, . \\labeq{I3}\n\\end{eqnarray}\n\nThe difference between $I^2_k$ and $I^1_k$ is always under $1\\%$ in the error, so choosing between them has no effect in any computation. In contrast, $I^1_k$ and $I^3_k$ present measurable differences for long $\\ensuremath{t_\\mathrm{w}}\\xspace$, at least for our highest temperatures, where the faster dynamics allows us to reach higher values of $\\xi_{12}\/L$. As a (very conservative) cutoff we have discarded all the $\\ensuremath{t_\\mathrm{w}}\\xspace$ where $|I^1-I^3|$ is larger than $20\\%$ of the error bar\\index{error bars}, thus obtaining a $\\xi_{12}^\\text{max}(T)$ below which we are sure we do not have any finite-size effects\\index{finite-size effects} in our $L=160$ systems. The reader can find the values in~\\reftab{selected_ximax}.\n\n\\begin{table}[h!]\n\\centering\n\\begin{tabular}{cccccccc}\n\\toprule\n\\toprule\n$T$ & $0.55$ & $0.625$ & $0.7$ & $0.8$ & $0.9$ & $1.0$ & $1.1$  \\\\\n$\\xi_\\mathrm{max}$ & - & - & - & - & $18.1$ & $17.3$ & $17$ \\\\\n\\bottomrule\n\\end{tabular}\n\\caption[\\textbf{Cutoff values of \\boldmath $\\xi_{12}^\\mathrm{max}(T)$.}]{\\textbf{Cutoff values of \\boldmath $\\xi_{12}^\\mathrm{max}(T)$.} Cutoff values of $\\xi_{12}^\\mathrm{max}(T)$ below which we are sure that no finite-size effects\\index{finite-size effects} are present in our $L=160$ lattices. For $T<0.9$ the growth of $\\xi_{12}$ is very slow and we never reach the cutoff value.}\n\\labtab{selected_ximax}\n\\end{table}\n\nAs a final check that our data is not affected by finite-size effects\\index{finite-size effects}, we have compared our $\\xi_{12}(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$ with that of~\\cite{fernandez:15}. This reference considers shorter simulations but with $L=256$ and $50$ samples\\index{sample}. As shown in~\\reffig{xi256}, the $L=160$ and $L=256$ data coincide even  beyond our cutoff $\\xi_{12}^\\text{max}$.\n\n\\begin{figure}[h!]\n\\includegraphics[width=0.7\\linewidth]{aging\/xi12_conL256}\n\\caption[\\textbf{Comparision of \\boldmath $\\xi_{12}$.}]{\\textbf{Comparision of \\boldmath $\\xi_{12}$.} Comparison between the $\\xi_{12}(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$ computed in $L=160$ lattices with $N_\\text{S}=16$ samples\\index{sample} and $N_\\text{R}=256$ replicas\\index{replica}, and that of~\\cite{fernandez:15} ($L=256$; $N_\\text{S}=50$; $N_\\text{R}=4$  for $T=0.9,1.0$ and $N_\\text{R}=8$ for $T=1.1$). For the $L=160$ simulations, we plot with two parallel lines the error interval for $\\xi_{12}(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$. Only the \\ensuremath{t_\\mathrm{w}}\\xspace range depicted with  continuous lines is used in the paper, the  extension with dashed lines represents the discarded times with $\\xi_{12}>\\xi_{12}^\\text{max}$ (see~\\reftab{selected_ximax}). These curves were generated with the $I^3_k$ estimator for the  integrals~\\refeq{I3}. The values from the $L=256$ simulations are plotted with conventional error bars\\index{error bars}. Notice that both curves are compatible even beyond this cutoff.}\n\\labfig{xi256}\n\\end{figure}\n\n\\section{Josephson length} \\labsec{Josephson_length}\nThe Josephson length $\\ell_J$ is expected to grow as $\\ell_J(T) \\sim (\\ensuremath{T_\\mathrm{c}}\\xspace - T)^{-\\nu}$ with $\\nu = 2.56(4)$~\\cite{janus:13} for temperatures close to $\\ensuremath{T_\\mathrm{c}}\\xspace$. Scaling corrections are expected for lower temperatures\n\\begin{equation}\n\\ell_J(T) = (\\ensuremath{T_\\mathrm{c}}\\xspace - T)^{-\\nu} \\left[ a_0 + a_1(\\ensuremath{T_\\mathrm{c}}\\xspace-T)^{\\nu} + a_2(\\ensuremath{T_\\mathrm{c}}\\xspace-T)^{\\omega \\nu}   \\right] \\, , \\labeq{josephson_length}\n\\end{equation}\nwhere $a_0$, $a_1$ and $a_2$ are coefficients chosen to perform the best collapse in \\reffig{vartheta_collapse} aging\\index{aging} and $\\omega=1.12(10)$~\\cite{janus:13}.\n\nAssuming $\\xi(T,\\ensuremath{t_\\mathrm{w}}\\xspace) \\gg \\ell_J(T)$, the crossover from the fixed point at $T=\\ensuremath{T_\\mathrm{c}}\\xspace$ and the fixed point at $T=0$ is affecting our basic quantity $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$ in the following way\n\\begin{equation}\nC_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace) \\sim\n\\begin{cases}\n  \\displaystyle \\dfrac{1}{r^{D-2+\\eta}}\\,,  & r\\ll \\ell_\\text{J}(T)\\,, \\\\[3mm]\n  \\displaystyle \\dfrac{\\ell_\\text{J}^\\vartheta}{\\ell_\\text{J}^{D-2+\\eta}} \\dfrac{1}{r^\\vartheta} f(r\/\\xi)\\,, & r \\gg \\ell_\\text{J}(T) \\,.\n\\end{cases} \\labeq{C4_josephson}\n\\end{equation}\nThe prefactor $\\ell_J^{\\vartheta}\/\\ell_\\text{J}^{D-2+\\eta}$ is fixed by the condition that the two asymptotic expansions in $r$ connect smoothly at $r=\\ell_J$.\n\nFrom this expression, we arrive at an asymptotic expansion for the $I_k$ integrals\n\\begin{equation}\nI_k = \\int_0^{\\infty} r^k C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace) {\\mathrm{d}} r = \\dfrac{F_k}{\\ell_\\text{J}^{D-2+\\eta}} \\left(\\dfrac{\\xi}{\\ell_\\text{J}}\\right)^{k+1-\\vartheta} \\left[ 1 + a_k\\left(\\dfrac{\\xi}{\\ell_\\text{J}}\\right)^{k+1-\\vartheta}+\\ldots\\right]\\,, \\labeq{Ik_expansion}\n\\end{equation}\nwhere $F_k$ and $a_k$ are amplitudes.\n\nFinally, we need to eliminate the unknown $\\xi$ in favor of the computable $\\xi_{12}$,\n\\begin{equation}\n\\xi_{12}(T,\\xi) = \\dfrac{F_2}{F_1} \\xi \\left[ 1+ a_1'\\left(\\dfrac{\\xi}{\\ell_\\text{J}}\\right)^{2-\\vartheta} + a_2\\left(\\dfrac{\\xi}{\\ell_\\text{J}}\\right)^{3-\\vartheta}+\\ldots\\right]\\,,\n\\end{equation}\nwhere $a_1'$ considers contributions both from the numerator ($-a_1$) and  from the denominator.  The easiest way to obtain $\\vartheta$ is to study the evolution of $\\log I_2$ as a function of $\\log \\xi$. However, we have to settle for using $\\log\\xi_{12}$ as independent variable (see~\\reffig{I2xi}).\n\n\nWe can define an effective $\\vartheta(T,\\xi_{12})$ as\n\\begin{equation}\n\\vartheta(T,\\xi_{12}) = 3 - \\dfrac{{\\mathrm{d}} \\log I_2(T,\\xi_{12})}{{\\mathrm{d}} \\log \\xi_{12}} = \\vartheta + b_2 \\left(\\dfrac{\\xi_{12}}{\\ell_\\text{J}}\\right)^{\\vartheta-2}+b_3 \\left(\\dfrac{\\xi_{12}}{\\ell_\\text{J}}\\right)^{\\vartheta-3}+\\ldots\\, . \\labeq{fit}\n\\end{equation}\nTo estimate this derivative for a given $\\xi_{12}^*$, we  fit $\\log I_2$ to a quadratic polynomial in $\\log \\xi_{12}$  in a $[0.75\\xi_{12}^*,1.25\\xi_{12}^*]$ window. We then take the derivative of this polynomial at $\\xi^*$. The procedure, as well as the wiggles in the resulting values of $\\vartheta$ due to the extreme data correlation (see~\\reffig{replicon_sin_reescalar}), may remind the reader of Fig.~1 in~\\cite{janus:08b}.\n\nWe have computed a fit to the first two terms in~\\refeq{fit} in the range $0\\leq \\ell_\\text{J}\/\\xi_{12}\\leq 0.33$, resulting in the value of $\\vartheta\\approx 0.30$ reported in the main text.\n\nThe previous analysis solves the problem of the crossover between the $T=\\ensuremath{T_\\mathrm{c}}\\xspace$ and $T=0$ fixed points. However, in the framework of the droplet\\index{droplet!picture} picture, one would also need to consider corrections to scaling at the $T=0$ fixed point. This is precisely what the droplet\\index{droplet!picture} fit in the main text to $\\vartheta(x) \\simeq Cx^\\zeta$ does.\n\n\\begin{figure}\n\\includegraphics[width=0.8\\linewidth]{aging\/I2xi}\n\\caption[\\textbf{Integral \\boldmath $I_2$ as a function of $\\xi_{12}$ in a logarithmic scale, for all our $T<\\ensuremath{T_\\mathrm{c}}\\xspace$ temperatures.}]{\\textbf{Integral \\boldmath $I_2$ as a function of $\\xi_{12}$ in a logarithmic scale, for all our $T<\\ensuremath{T_\\mathrm{c}}\\xspace$ temperatures.} We use the numerical derivative of this curve to compute the replicon\\index{replicon} exponent $\\vartheta$.}\n\\labfig{I2xi}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=0.8\\linewidth]{aging\/theta-xi12}\n\\caption[\\textbf{The replicon \\boldmath $\\vartheta(T,\\xi_{12})$.}]{\\textbf{The replicon\\index{replicon} \\boldmath $\\vartheta(T,\\xi_{12})$.} Value of the replicon\\index{replicon} exponent $\\vartheta(T,\\xi_{12})$ computed from a numerical derivative of $\\log I_2$ as a function of $\\log \\xi_{12}$, nicely illustrating the crossover between the $T=\\ensuremath{T_\\mathrm{c}}\\xspace$ and  $T=0$ fixed points.}\n\\labfig{replicon_sin_reescalar}\n\\end{figure}\n\n\n\n\\section{Parameter choices in our fits}\\labsec{parameters_aging}\nWe will discuss separately the choice of $\\xi_{\\mathrm{min}}$ for different\ntemperatures and the choice of the value of $\\omega$.\n\n\\subsection[Selection of $\\xi_{12}^{\\mathrm{min}}$ for each temperature]{Selection of \\boldmath $\\xi_{12}^{\\mathrm{min}}$ for each temperature}\nWe have reported fits of our data to three different functional forms\n\n\\begin{align}\n\\log \\ensuremath{t_\\mathrm{w}}\\xspace&= a_0(T) + a_1(T) \\log \\xi_{12} + a_2(T) \\log^2 \\xi_{12},\\\\\n\\log t_\\mathrm{w} &= C_1(T) + z_\\infty(T) \\log \\xi_{12} + C_2(T) \\xi_{12}^{-\\omega} \\, , \\labeq{RSB}\\\\\n\\log t_\\mathrm{w} &= D_1(T) + z_\\mathrm{c} \\log \\xi_{12} + D_2(T) \\xi_{12}^{\\psi} \\, .\n\\labeq{Bouchaud}\n\\end{align}\n\nIn these fits we have used $z_\\text{c}=6.69$ and $\\omega=0.35$ ($T<\\ensuremath{T_\\mathrm{c}}\\xspace$), $\\omega=1.12$ ($T=\\ensuremath{T_\\mathrm{c}}\\xspace$), as discussed in the~\\refch{aging_rate}. Full results for the fits to~\\refeq{RSB} and~\\refeq{Bouchaud} can be seen in~\\reftab{RSB_omega_0.35} and~\\reftab{Saclay_6.69}, for different fitting ranges. We include for both cases the extrapolated values of $z(T,\\xi)$ for the experimental scale (as explained in the~\\refch{aging_rate} we use both $\\xi_{12}=38$ and $\\xi_{12}=76$) and for~\\refeq{RSB} also the value of the $\\xi\\to\\infty$ aging\\index{aging!rate} rate $z_\\infty$.\n\nIn order to make the choice of the fitting range, we have followed two criteria. Firstly we require the parameters of the fit to be stable inside the error when we increase $\\xi_{12}^\\mathrm{min}$. Secondly, we impose $\\xi_\\mathrm{min}$ to be monotonically increasing in $T$ (with the exception of \\ensuremath{T_\\mathrm{c}}\\xspace, which has different behavior).~\\reftab{selected_ximin} shows our final choices for $\\xi_{12}^\\text{min}(T)$, which is the same for all three fits.\n\n\\begin{table}[h!]\n\\centering\n\\begin{tabular}{cccccccc}\n\\toprule\n\\toprule\n$T$ & $0.55$ & $0.625$ & $0.7$ & $0.8$ & $0.9$ & $1.0$ & $1.1$  \\\\\n$\\xi_{12}^\\mathrm{min}$ & $4$ & $5$ & $6$ & $8$ & $8$ & $9$ & $5$ \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Values of $\\xi_{12}^\\text{min}(T)$ determining the common fitting range $\\xi_{12}\\geq\\xi_{12}^\\text{min}$ for our three different fits of $\\log\\ensuremath{t_\\mathrm{w}}\\xspace$ as a function of $\\log \\xi_{12}$.}\n\\labtab{selected_ximin}\n\\end{table}\n\n\\subsection[Selection of $\\omega$]{Selection of \\boldmath $\\omega$}\nFor our most important result, namely the extrapolation of the aging\\index{aging!rate} rate to the experimental scale of $\\xi_{12}=38,76$, we have repeated our fits with our upper and lower bounds for $\\omega=\\vartheta(\\xi_\\text{films})$ (\\gls{RSB}\\index{replica!symmetry breaking (RSB)} and droplet\\index{droplet!picture} extrapolations, respectively).  The results are completely compatible, as we can see in~\\reftab{omega_xi38}.\n\n\\begin{table}[h!]\n\\begin{tabular}{lccccc}\n\\toprule\n\\toprule\n& \\multicolumn{2}{c}{$z(T,\\xi_{12}=38)$} & & \\multicolumn{2}{c}{$z(T,\\xi_{12}=76)$} \\\\ \n       & $\\omega = 0.35$    & \\multicolumn{1}{c}{$\\omega = 0.28$} & \n       & $\\omega = 0.35$    & \\multicolumn{1}{c}{$\\omega = 0.25$} \\\\ \n\\hline\n$T=0.55$  & 19.80(20) & 20.08(22) &  & 20.75(24) & 21.41(27)                    \\\\\n$T=0.625$ & 16.90(19)  & 17.07(20)& & 17.69(24)  & 18.13(27)                   \\\\\n$T=0.7$   & 14.81(15) & 14.93(16)&  & 15.54(19) & 15.87(21)                   \\\\\n$T=0.8$   & 12.73(22) & 12.81(23)&  & 13.47(30) & 13.71(32)                    \\\\\n$T=0.9$   & 10.55(25) & 10.61(26)&  & 11.11(34) & 11.28(37)                   \\\\\n$T=1.0$   & 8.63(32) & 8.68(33)  &  & 8.98(44) & 9.02(42)                  \\\\ \n\\bottomrule\n\\end{tabular}\n\\caption[\\textbf{Estimations of the aging rate.}]{\\textbf{Estimations of the aging rate.} Comparison of our estimates of the experimental aging\\index{aging!rate} rate $z(T,\\xi_{12}=\\xi_\\text{films})$ for $\\xi_\\text{films}=38$ and $\\xi_\\text{films}=76$ using our lower and upper bounds for $\\omega=\\vartheta(\\xi_\\text{films})$. The choice of $\\omega$ is immaterial, since even in the worst case (lowest temperatures for $\\xi_\\text{films}=76$) there is  only a two-sigma difference.}\n\\labtab{omega_xi38}\n\\end{table}\n\n\n\\begin{table}[h!]\n\\resizebox{\\textwidth}{!}{\\begin{tabular}{lcccccccc}\n\\toprule\n\\toprule\n& & $\\xi_{\\mathrm{min}}$= 3.5 & $\\xi_{\\mathrm{min}}$= 4 & $\\xi_{\\mathrm{min}}$= 5 & $\\xi_{\\mathrm{min}}$= 6 & $\\xi_{\\mathrm{min}}$= 7 & $\\xi_{\\mathrm{min}}$= 8 & $\\xi_{\\mathrm{min}}$= 9 \\\\ \\hline\n\\multirow{4}{*}{{$T=0.55$}}&\n$z_\\infty$ & 23.61(28)   & \\bfseries 24.22(40)   & 25.30(86)   & 22.9(31)   & & &   \\\\ \n&$z (\\xi\\!=\\!38)$ & 19.49(15)   &  \\bfseries 19.80(20)   & 20.32(41)  & 19.2(14)   & & &   \\\\\n&$z (\\xi\\!=\\!76)$ & 20.38(18)   &  \\bfseries 20.75(24)   & 21.39(51)   & 20.0(18)   & & &   \\\\\n&$\\chi^2$\/dof &  $40(17)\/133$ &  \\bfseries 10.2(54)\/111 & 3.0(12)\/73 & 1.71(76)\/40  &  &  & \\\\ \n\\hline\n\\multirow{4}{*}{{$T=0.625$}}&\n$z_\\infty$ & 19.85(17)   & 20.26(23)   &  \\bfseries 20.60(41)   & 20.16(84)   &  &  &  \\\\ \n&$z (\\xi\\!=\\!38)$ & 16.538(91)   & 16.74(12)   &  \\bfseries 16.90(19)   & 16.71(37)   &  &  &  \\\\\n&$z (\\xi\\!=\\!76)$ & 17.25(11)   & 17.50(14)   &  \\bfseries 17.69(24)   & 17.45(47)   &  &  &  \\\\\n&$\\chi^2$\/dof & 81(34)\/167\n &  18(10)\/147\n &  \\bfseries  8.3(21)\/114\n&  5.1(19)\/86\n&  &  &  \\\\ \\hline\n\\multirow{4}{*}{{$T=0.7$}}&\n$z_\\infty$ & 17.04(18)   & 17.23(21)   & 17.61(27)   &  \\bfseries 18.23(35)   & 18.63(62)   &  &  \\\\ \n&$z (\\xi\\!=\\!38)$ & 14.295(88)   & 14.38(11)   & 14.55(13)   &  \\bfseries 14.81(15)   & 14.96(25)&  &  \\\\\n&$z (\\xi\\!=\\!76)$ & 14.89(11)   & 15.00(13)   & 15.21(16)   &  \\bfseries 15.54(19)   & 15.75(32)   &  &  \\\\ \n&$\\chi^2$\/dof & 116(40)\/190 & 66(36)\/173 & 33(24)\/144 &  \\bfseries 9.3(84)\/119 & 4.9(21)\/98\n &  &  \\\\ \\hline\n\\multirow{4}{*}{{$T=0.8$}}&\n$z_\\infty$ & 13.76(15)   & 14.06(19) & 14.53(26)   & 15.19(35)   & 15.68(42)   &  \\bfseries 16.18(58)   & 16.55(78)   \\\\ \n& $z (\\xi\\!=\\!38)$ & 11.787(73)   & 11.921(89)   & 12.11(12)   & 12.37(15)   & 12.55(17)   & \\bfseries  12.73(22)   & 12.85(28)   \\\\\n& $z (\\xi\\!=\\!76)$ & 12.211(93)   & 12.38(11)   & 12.63(15)   & 12.98(19)   & 13.23(23)   & \\bfseries  13.47(30)   & 13.65(39)   \\\\ \n& $\\chi^2$\/dof & 351(104)\/185 & 188(72)\/170& 93(41)\/146& 27(16)\/125& 12.4(82)\/107\n& \\bfseries  6.2(31)\/91\n& 4.9(21)\/77\n\\\\ \\hline\n\\multirow{4}{*}{{$T=0.9$}}&\n$z_\\infty$ & 11.00(13)  & 11.29(18)   & 11.54(24)   & 11.80(31)   & 12.55(41)   & \\bfseries  13.16(68)   & 12.3(13)   \\\\ \n& $z (\\xi\\!=\\!38)$ & 9.748(65)   & 9.883(93)   & 9.98(11)   & 10.08(13)   & 10.34(16)   & \\bfseries  10.55(25)   & 10.33(41)   \\\\\n& $z (\\xi\\!=\\!76)$ & 10.017(82)   & 10.18(11)   & 10.32(14)   & 10.45(17)   & 10.82(21)   &  \\bfseries 11.11(34)   & 10.80(60)   \\\\ \n& $\\chi^2$\/dof & 310(150)\/165 & 129(64)\/152 & 79(44)\/131 & 63(35)\/113 & 22(13)\/98\n & \\bfseries  5.9(21)\/84\n & 6.4(79)\/72\n \\\\ \\hline\n\\multirow{4}{*}{{$T=1.0$}}&\n$z_\\infty$ & 8.60(11)   & 8.69(15)   & 8.83(20)   & 8.97(53)   & 9.29(45)   & 9.36(46)   &  \\bfseries 10.28(89)   \\\\\n& $z (\\xi\\!=\\!38)$ & 8.041(59)   & 8.080(73)   & 8.132(93)   & 8.21(22)   & 8.27(18)   & 8.34(17)   &  \\bfseries 8.63(32)   \\\\ \n& $z (\\xi\\!=\\!76)$ & 8.162(74) & 8.210(86) & 8.28(11) & 8.38(29) & 8.46(24) & 8.57(24) & \\bfseries  8.98(44)   \\\\ \n& $\\chi^2$\/dof & 43(30)\/137\n & 27(18)\/126\n & 16(13)\/107\n & 12(25)\/91\n & 10.4(91)\/78\n & 8.4(60)\/66\n &  \\bfseries 2.9(21)\/55\n\\\\ \\hline\n\\multirow{4}{*}{{$T=1.1$}}&\n$z_\\infty$ & 6.672(44) & 6.671(41) & \\bfseries  6.689(63) & 6.751(84) & 6.80(12) & 7.00(18) & 7.02(21)  \\\\\n& $z (\\xi\\!=\\!38)$ & 6.682(32) & 6.673(41) & \\bfseries  6.694(50) & 6.732(68) & 6.77(10) & 6.92(14) & 6.94(16)   \\\\\n& $z (\\xi\\!=\\!76)$ & 6.677(33) & 6.671(41) & \\bfseries  6.691(54) & 6.742(72) & 6.79(11) & 6.96(16) & 6.99(16)   \\\\\n& $\\chi^2$\/dof & 32(19)\/119\n & 31(20)\/109\n &  \\bfseries 26(16)\/92\n & 19(10)\/78\n & 16.8(74)\/66\n & 5.9(20)\/55\n & 6.3(27)\/46 \\\\ \n \\bottomrule\n\\end{tabular}}\n\\caption[\\textbf{Parameters for the convergent ansatz.}]{\\textbf{Parameters for the convergent ansatz.}Parameters of the fits to~\\refeq{RSB} for different fitting ranges $\\xi_{12}\\geq\\xi_{12}^\\text{min}$.  We use $\\omega = 0.35$ ($\\omega = 1.12$ for $T = T_\\mathrm{c}$). The fitting range that we choose for our final values is highlighted in boldface.}\n\\labtab{RSB_omega_0.35}\n\\end{table}\n\n\\begin{table}[h!]\n\\resizebox{\\textwidth}{!}{\\begin{tabular}{lcccccccc}\n\\toprule\n\\toprule\n& & $\\xi_{\\mathrm{min}}$= 3.5 & $\\xi_{\\mathrm{min}}$= 4 & $\\xi_{\\mathrm{min}}$= 5 & $\\xi_{\\mathrm{min}}$= 6 & $\\xi_{\\mathrm{min}}$= 7 & $\\xi_{\\mathrm{min}}$= 8 & $\\xi_{\\mathrm{min}}$= 9 \\\\ \\hline\n\n\\multirow{4}{*}{{$T=0.55$}}& \n$z(\\xi\\!=\\!38)$ & 24.07(41)   & \\bfseries24.25(55)  & 24.6(11)& 24.7(81)  &  &  & \\\\\n&$z(\\xi\\!=\\!76)$ & 28.86(69)   & 2\\bfseries9.18(95)  & 29.9(19)   & 30(15)  &  &  & \\\\\n& $G(T)$ & 13.78(65)   & \\bfseries13.45(92) & 12.8(18)  & 18(13) &  &  &  \\\\ \n& $\\psi$ & 0.3512(92)   & \\bfseries0.355(21)   & 0.372(33)   & 0.29(24)  &  &  & \\\\\n& $\\chi^2$\/dof & 13.3(47)\/133\n & \\bfseries6.8(20)\/111\n & 3.2(15)\/73\n & 1.7(27)\/40\n &  &  &  \\\\ \\hline\n\\multirow{4}{*}{{$T=0.625$}}& \n $z(\\xi\\!=\\!38)$ & 19.73(22)   & 19.72(28)   & \\bfseries19.36(45)   & 18.53(77)  &  &  & \\\\\n& $z(\\xi\\!=\\!76)$ & 23.33(38)   & 23.31(49)   & \\bfseries22.66(79)   & 21.1(13)  &  &  & \\\\\n& $G(T)$ & 10.36(37)   & 10.39(52)   & \\bfseries11.3(11)   & 14.0(30)  &  &  & \\\\\n& $\\psi$ & 0.354(14)   & 0.352(12)   & \\bfseries0.334(21)   & 0.290(39)  &  &  &  \\\\\n& $\\chi^2$\/dof & 19(10)\/167\n & 15(10)\/147\n & \\bfseries8.5(33)\/114\n & 4.5(14)\/86\n &  &  &  \\\\ \\hline\n\\multirow{4}{*}{{$T=0.7$}}& \n$z(\\xi\\!=\\!38)$ & 16.58(22)   & 16.44(23)   & 16.35(27)   &\\bfseries 16.51(32)   & 16.55(52) &  &   \\\\ \n& $z(\\xi\\!=\\!76)$ & 19.40(37)   & 19.14(40)   & 18.98(47)   &\\bfseries 19.29(58)   & 19.4(10)  &  &  \\\\\n& $G(T)$ & 7.32(34)   & 7.63(43)   & 7.84(59)   &\\bfseries 7.41(75)   & 7.3(14)   &  & \\\\ \n& $\\psi$ & 0.364(13)   & 0.354(12)   & 0.354(24)   &\\bfseries 0.358(23)   & 0.360(39)   &  & \\\\\n& $\\chi^2$\/dof & 49(38)\/190\n & 28(20)\/173\n & 10.5(83)\/144\n & \\bfseries6.3(31)\/119\n & 5.7(29)\/98\n &  &  \\\\ \\hline\n\\multirow{4}{*}{{$T=0.8$}}& \n$z(\\xi\\!=\\!38)$ & 13.37(18)   & 13.39(21) & 13.45(25) & 13.68(31) & 13.80(35) & \\bfseries13.94(45) & 14.1(17)   \\\\\n& $z(\\xi\\!=\\!76)$ & 15.44(33) & 15.48(37) & 15.60(46) & 16.06(59)   & 16.31(68)   & \\bfseries16.59(93)   & 17.1(38)   \\\\ \n& $G(T)$ & 4.16(25)   & 4.13(29)   & 4.01(38)   & 3.57(43)   & 3.36(48)   & \\bfseries3.13(66)   & 3.0(19)   \\\\\n& $\\psi$ & 0.392(13) & 0.390(19) & 0.395(18)   & 0.421(27) & 0.443(31) & \\bfseries0.447(50) & 0.46(18) \\\\ \n& $\\chi^2$\/dof & 31(19)\/185\n & 29(19)\/170\n & 22(17)\/146\n & 10.0(60)\/125\n & 7.5(34)\/107\n & \\bfseries5.5(21)\/91\n & 5(11)\/77\n \\\\ \\hline\n\\multirow{4}{*}{{$T=0.9$}}& \n$z(\\xi\\!=\\!38)$ & 10.76(17) & 10.86(21) & 10.82(24) & 10.86(27) & 11.23(35) & \\bfseries11.49(54) & 11.13(56) \\\\\n& $z(\\xi\\!=\\!76)$ & 12.12(30) & 12.31(39) & 12.24(45) & 12.31(53) & 13.12(73) & \\bfseries13.7(12)   & 12.9(12)  \\\\\n& $G(T)$ & 2.15(19) & 2.01(23) & 2.07(31) & 2.00(39) & 1.52(31)   & \\bfseries1.18(45)   & 1.67(77)  \\\\\n& $\\psi$ & 0.417(23)  & 0.430(34) & 0.431(28) & 0.427(41) & 0.490(51) & \\bfseries0.546(88) & 0.47(10) \\\\\n& $\\chi^2$\/dof & 68(44)\/165\n & 46(25)\/152\n & 41(25)\/131\n & 38(24)\/113\n & 17(10)\/98\n & \\bfseries8.7(45)\/84\n & 4.8(25)\/72\n \\\\ \\hline\n\\multirow{4}{*}{{$T=1.0$}}& \n$z(\\xi\\!=\\!38)$ & 8.53(15) & 8.54(18) & 8.55(20) & 8.56(30) & 8.59(60) & 8.74(72) &\\bfseries 9.22(18)  \\\\\n& $z(\\xi\\!=\\!76)$ & 9.19(27) & 9.20(33) & 9.22(39) & 9.25(58) & 9.3(12) & 9.7(16) &\\bfseries 10.9(45)  \\\\\n& $G(T)$ & 0.85(14) & 0.84(18) & 0.83(22) & 0.79(40) & 0.7(10)   & 0.5(10) &\\bfseries 1.4(19)  \\\\\n& $\\psi$ & 0.440(36)   & 0.441(51)   & 0.444(64) & 0.45(10) & 0.49(25) & 0.55(30)   & \\bfseries0.34(71) \\\\ \n& $\\chi^2$\/dof & 12.6(95)\/137\n & 12.1(90)\/126\n & 10.0(87)\/107\n & 9.3(83)\/91\n & 8.2(97)\/78\n & 07(11)\/66\n & \\bfseries11(11)\/55\n \\\\ \\hline\n\\multirow{4}{*}{{$T=1.1$}}& \n$z(\\xi\\!=\\!38)$ & 6.684(12)   & 6.682(14)   &\\bfseries 6.684(11)   & 6.672(31)   & 6.683(32)& 6.694(31)& 6.712(41)   \\\\ \n& $z(\\xi\\!=\\!76)$ & 6.684(13)   & 6.681(11)   & \\bfseries 6.682(14)   & 6.674(41)   & 6.684(41)& 6.692(31)& 6.721(42)  \\\\ \n& $G(T)$ & 1.9(10)   & 1.71(91) & \\bfseries0.4(27) & 0.02(64)  & 0.0(26)   & 1.5(26)   & 1.2(10)   \\\\ \n& $\\psi$ & -0.0030(49)   & 0.0037(68)   &\\bfseries 0.03(15)   & 0.29(42)   & 0.37(55)   & 0.002(16)   & 0.023(51)   \\\\\n& $\\chi^2$\/dof & 34(20)\/119\n & 33(19)\/109\n &\\bfseries 27(18)\/92\n & 25(18)\/78\n & 23(15)\/66\n & 21(14)\/55\n & 11.0(26)\/46\n \\\\ \n \\bottomrule\n\\end{tabular}}\n\\caption[\\textbf{Parameters for the divergent ansatz.}]{\\textbf{Parameters for the divergent ansatz.}Parameters of the fits to ~\\refeq{Bouchaud} for different fitting ranges $\\xi_{12}\\geq\\xi_{12}^\\text{min}$.  We use $z_\\text{c} = 6.69$. The fitting range that we choose for our final values is highlighted in boldface.}\n\\labtab{Saclay_6.69}\n\\end{table}\n\n\n\n\n\\chapter{Temperature Chaos. Technical details.} \\labch{AP_technical_details_out-eq_chaos}\n\n\\section{Our procedure to obtain the distribution functions} \\labsec{procedure}\nHere, we explain the details in our computation of the distribution functions $F(X,T_1,T_2,\\xi,r)$ or, rather, the quantity we really compute, namely its inverse $X(F,T_1,T_2,\\xi,r)$. First, in~\\refsubsec{parameters} we provide the relevant parameters for the construction of the distribution functions. Next, in~\\refsubsec{construction} we explain how we compute the chaotic parameter for a given sphere and a given number of replicas\\index{replica} $\\ensuremath{N_{\\text{Rep}}}\\xspace$. In~\\refsubsec{sample_to_sample_fluctuations} we explain our computation of $X(F,T_1,T_2,\\xi,r)$ for a given $\\ensuremath{N_{\\text{Rep}}}\\xspace$, including our procedure for the computation of the error bars\\index{error bars}. Finally, in~\\refsubsec{extrapolation} we explain the process of the extrapolation to $\\ensuremath{N_{\\text{Rep}}}\\xspace \\to \\infty$.\n\n\\subsection{The parameters in our computation} \\labsubsec{parameters}\nThe computation of $X^{s,r}_{T_1,T_2}(\\xi)$, namely the chaotic parameter for a given sphere $s$ of radius $r$, recall~\\refsubsec{observables-locales}, is specified by five parameters: two temperatures $T_1$ and $T_2$, the radius $r$, the coherence length\\index{coherence length} $\\xi$ and the number of replicas\\index{replica} used to estimate the thermal noise $\\ensuremath{N_{\\text{Rep}}}\\xspace$. Our choice of the parameters has been the following:\n\n\\begin{itemize}\n\\item \\textbf{Temperatures:} we impose $T_1<T_2$ with $T_1 \\in \\{0.625,0.7,0.8\\}$ and $T_2 \\in \\{ 0.7,0.8,0.9,1.0\\}$.\n\\item \\textbf{Radius:} integer values of $r$ from $r=1$ to $r=15$ are simulated.\n\\item \\textbf{Coherence lengths\\index{coherence length}:}  from $\\xi_{\\min} = 3$ to $\\xi_{\\max,T_1}$ with intervals $\\Delta_\\xi=0.25$ where $\\xi_{\\max,T_1}$ is the maximum $\\xi$ simulated for temperature $T_1$\\footnote{This condition is imposed by~\\refeq{el_reloj_doble}. Indeed, we need $\\xi$ to be reached at both temperatures $T_1$ and $T_2$. So, as long as $\\xi_{\\max}$ increases with temperature (recall~\\reftab{xi_max}), the maximum $\\xi$ has to be $\\xi_{\\max,T_1}$}.\n\\item \\textbf{Number of replicas\\index{replica}:} in order to quantify the thermal noise, we have computed $X^{s,r}_{T_1,T_2}(\\xi)$ by using different number of replicas\\index{replica} $\\ensuremath{N_{\\text{Rep}}}\\xspace \\in \\{ 16,32,64,128,256,512\\}$ (see below).\n\\end{itemize}\n\n\\subsection{On the computation of $X^{s,r}_{T_1,T_2}(\\xi)$}\\labsubsec{construction}\nThe reader will recall from~\\refeq{def_chaotic_parameter} that the computation of $X^{s,r}_{T_1,T_2}(\\xi)$ requires an infinite number of replicas\\index{replica}. Unfortunately we only have $512$ replicas\\index{replica} at our disposal. Our choice has been to produce different estimates of $X^{s,r}_{T_1,T_2}(\\xi)$ by varying the number of replicas\\index{replica} $\\ensuremath{N_{\\text{Rep}}}\\xspace$. Specifically, our procedure has been the following:\n\n\\begin{enumerate}\n\\item For each $\\ensuremath{N_{\\text{Rep}}}\\xspace < \\ensuremath{N_{\\text{Rep}}}\\xspace^{\\max}$ we order randomly the $\\ensuremath{N_{\\text{Rep}}}\\xspace^{\\max}$ replicas\\index{replica} and divide them in $\\ensuremath{N_{\\text{Rep}}}\\xspace^{\\max}\/\\ensuremath{N_{\\text{Rep}}}\\xspace$ groups of $\\ensuremath{N_{\\text{Rep}}}\\xspace$ replicas\\index{replica}.\n\\item In this way, we get $\\ensuremath{N_{\\text{Rep}}}\\xspace^{\\max}\/\\ensuremath{N_{\\text{Rep}}}\\xspace$ independent estimates of $X^{s,r}_{T_1,T_2}(\\xi)$.\n\\item In order to eliminate the effect of the initial permutation of the $\\ensuremath{N_{\\text{Rep}}}\\xspace^{\\max}$ replicas\\index{replica}, we repeat this procedure $10$ times for all $\\ensuremath{N_{\\text{Rep}}}\\xspace < \\ensuremath{N_{\\text{Rep}}}\\xspace^{\\max}$.\n\\end{enumerate}\n\nIn a nutshell, for every sphere of radius $r$ we obtain $\\ensuremath{N_{\\text{thermal}}}\\xspace(\\ensuremath{N_{\\text{Rep}}}\\xspace)$ estimates of $X^{s,r}_{T_1,T_2}(\\xi)$ where\n\\begin{equation}\n\\ensuremath{N_{\\text{thermal}}}\\xspace(\\ensuremath{N_{\\text{Rep}}}\\xspace<\\ensuremath{N_{\\text{Rep}}}\\xspace^{\\max}) = 10 \\times \\dfrac{\\ensuremath{N_{\\text{Rep}}}\\xspace^{\\max}}{\\ensuremath{N_{\\text{Rep}}}\\xspace} \\, ,\n\\end{equation}\nor\n\\begin{equation}\n\\ensuremath{N_{\\text{thermal}}}\\xspace(\\ensuremath{N_{\\text{Rep}}}\\xspace=\\ensuremath{N_{\\text{Rep}}}\\xspace^{\\max}) = 1 \\, .\n\\end{equation}\n\n\\subsection{The computation of $X(F,T_1,T_2,\\xi,r)$}\\labsubsec{sample_to_sample_fluctuations}\nGiven that, for every radius $r$, we have computed the chaotic parameter for $8000$ different spheres in each sample\\index{sample}, we shall fix the probability level $F$ as\n\\begin{equation}\nF_i=\\dfrac{i}{8000} \\, , \\quad i=1,2,\\ldots,7999 \\, .\n\\end{equation}\nErrors have been computed with the Jackknife method (see~\\refsubsec{jk_method}). Let us briefly describe our procedure:\n\n\\begin{enumerate}\n\\item \\textbf{Our estimate of $X(F,T_1,T_2,\\xi,r)$ for each $\\ensuremath{N_{\\text{Rep}}}\\xspace$.}\n\t\\begin{enumerate}\n\t\\item We group together the $\\ensuremath{N_{\\text{S}}}\\xspace \\times 8000 \\times \\ensuremath{N_{\\text{thermal}}}\\xspace(\\ensuremath{N_{\\text{Rep}}}\\xspace)$ estimates of $X^{s,r}_{T_1,T_2}(\\xi)$.\n\t\\item We sort them in increasing order.\n\t\\item Our estimate of the central values of $X(F=F_i,T_1,T_2,\\xi,r)$ is the $X^{s,r}$ that occupies the position $F_i \\times \\ensuremath{N_{\\text{S}}}\\xspace \\times 8000 \\times \\ensuremath{N_{\\text{Rep}}}\\xspace$ in the sorted list. Note that our choice of $F_i$ ensures that $F_i \\times \\ensuremath{N_{\\text{S}}}\\xspace \\times 8000 \\times \\ensuremath{N_{\\text{Rep}}}\\xspace$ is an integer.\n\t\\end{enumerate}\n\n\\item \\textbf{Computation of the fluctuations.}\\\\\n\tFor the $j$-th Jackknife block, $j=1,2,\\ldots,\\ensuremath{N_{\\text{S}}}\\xspace$, we proceed as follows:\n\t\\begin{enumerate}\n\t\\item We group together the $(\\ensuremath{N_{\\text{S}}}\\xspace-1) \\times 8000 \\times \\ensuremath{N_{\\text{thermal}}}\\xspace(\\ensuremath{N_{\\text{Rep}}}\\xspace)$ estimates of $X^{s,r}_{T_1,T_2}(\\xi)$, obtained excluding the sample\\index{sample} $j$.\n\t\\item We sort them in increasing order.\n\t\\item Our estimate of $X(F=F_i,T_1,T_2,\\xi,r)$ for the $j$ Jackknife block, is the $X^{s,r}$ that occupies the position $F_i \\times (\\ensuremath{N_{\\text{S}}}\\xspace-1) \\times 8000 \\times \\ensuremath{N_{\\text{Rep}}}\\xspace$ in the sorted list.\n\t\\end{enumerate}\n\tThe errors $X(F,T_1,T_2,\\xi,r)$ are computed through the Jackknife blocks using the standard formula (see e.g.~\\cite{amit:05} and \\refch{AP_statistics}).\n\\end{enumerate}\nExamples of these computations can be seen in~\\reffig{linear_quadratic} and~\\reffig{exponente_libre}.\n\n\\subsection{Extrapolation to infinite number of replicas}\\labsubsec{extrapolation}\n\nThe computation of the thermal expectation values necessary to calculate the chaotic parameter, see~\\refeq{def_chaotic_parameter}, requires an infinite number of replicas\\index{replica}. As the maximum number of replicas\\index{replica} in our simulation is $\\ensuremath{N_{\\text{Rep}}}\\xspace^{\\max}=512$, we need an extrapolation to $\\ensuremath{N_{\\text{Rep}}}\\xspace \\to \\infty$. \n\nFixing $T_1$, $T_2$, $\\xi$, $r$ and $F$ we find a smooth behavior of $X(F,T_1,T_2,\\xi,r)$ with $\\ensuremath{N_{\\text{Rep}}}\\xspace$ (see~\\reffig{linear_quadratic} and~\\reffig{exponente_libre}). This smooth evolution with $\\ensuremath{N_{\\text{Rep}}}\\xspace$ makes feasible the extrapolation to the $\\ensuremath{N_{\\text{Rep}}}\\xspace \\to \\infty$ limit. Our main ansatz for the extrapolation is\n\\begin{equation}\nX_{\\ensuremath{N_{\\text{Rep}}}\\xspace} = X_{\\infty} + \\dfrac{A}{\\ensuremath{N_{\\text{Rep}}}\\xspace} \\>\\>\\> , \\labeq{extrapolacion_lineal}\n\\end{equation}\nwhere $A$ is an amplitude, $X_{\\ensuremath{N_{\\text{Rep}}}\\xspace}$ is a short hand for $X(F,T_1,T_2,\\xi,r;\\ensuremath{N_{\\text{Rep}}}\\xspace)$ and also a short hand for $X_\\infty=X(F,T_1,T_2,\\xi,r; \\ensuremath{N_{\\text{Rep}}}\\xspace=\\infty)$. As a check for the linear ansatz in~\\refeq{extrapolacion_lineal}, we consider two alternative functional forms for the extrapolation:\n\\begin{equation}\nX_{\\ensuremath{N_{\\text{Rep}}}\\xspace} = X_{\\infty} + \\dfrac{B}{\\ensuremath{N_{\\text{Rep}}}\\xspace} + \\dfrac{C}{\\ensuremath{N_{\\text{Rep}}}\\xspace^2} \\>\\>\\> , \\labeq{extrapolacion_cuadratica}\n\\end{equation}\n\\begin{equation}\nX_{\\ensuremath{N_{\\text{Rep}}}\\xspace} = X_{\\infty} + \\dfrac{D}{\\ensuremath{N_{\\text{Rep}}}\\xspace^\\gamma} \\>\\>\\> , \\labeq{extrapolacion_libre}\n\\end{equation}\nwhere $B$, $C$ and $D$ are amplitudes and $\\gamma$ is a free exponent. We perform independent fits to~\\refeq{extrapolacion_lineal},~\\refeq{extrapolacion_cuadratica} and~\\refeq{extrapolacion_libre} for every value of the parameters $(F,T_1,T_2,\\xi,r)$. We reject fits with a diagonal $\\chi^2\/\\text{d.o.f.}\\geq 1.1$.\\index{degree of freedom} Errors in $X_{\\infty}$ are computed from the fluctuations of the Jackknife blocks. Indeed, we perform separated fits for each Jackknife block (the fitting procedure consists in minimizing the diagonal $\\chi^2$, see~\\cite{yllanes:11}).\n\n\\begin{figure}[!h]\n  \\centering\n  \\includegraphics[width=0.9\\textwidth]{off-eq_chaos\/lineal_cuadratica_T0708_xi11_r08_nb000000_X.pdf}\n  \\caption[\\textbf{Equivalence of linear and quadratic extrapolations.}]{\\textbf{Linear and quadratic extrapolations,~\\refeq{extrapolacion_lineal} and~\\refeq{extrapolacion_cuadratica}, turn out to be equivalent for the tail of the distribution function.} The continuous lines are the linear (golden curves) and quadratic (blue curves) extrapolations to $\\ensuremath{N_{\\text{Rep}}}\\xspace \\to \\infty$ for $F(X,T_1,T_2,\\xi,r)$ as a function of $X$. The data shown correspond to the case $T_1=0.7$, $T_2=0.8$, $\\xi=11$ and $r=8$. The two curves shown for each extrapolation correspond to the central value plus or minus the standard error. We show horizontal error bars\\index{error bars} because we are computing the inverse distribution function $X(F,T_1,T_2,\\xi,r)$. We only show extrapolated data when $\\chi^2\/\\text{d.o.f.} < 1.1$\\index{degree of freedom} in the fits to~\\refeq{extrapolacion_lineal} or to~\\refeq{extrapolacion_cuadratica}. For comparison, we also plot the data corresponding to $\\ensuremath{N_{\\text{Rep}}}\\xspace=512$ and $\\ensuremath{N_{\\text{Rep}}}\\xspace=256$ (yellow and blue dots respectively) \\textbf{Inset:} As in the main plot, but with the vertical axis in log-scale.}\n\\labfig{linear_quadratic}\n\\end{figure} \n\n\\begin{figure}[!h]\n  \\centering\n  \\includegraphics[width=0.9\\textwidth]{off-eq_chaos\/extrapola_free_expo_T0708.pdf}\n  \\caption[\\textbf{Free-exponent extrapolation.}]{\\textbf{The exponent $\\gamma$ in~\\refeq{extrapolacion_libre}, remains close to one when it becomes a fit parameter.} The distribution function $F(X,T_1,T_2,\\xi,r)$ is plotted as a function of $X$ for $\\ensuremath{N_{\\text{Rep}}}\\xspace=\\{512, 256, 128 ,64, 32, 16\\}$ together with their extrapolation to $\\ensuremath{N_{\\text{Rep}}}\\xspace \\to \\infty$ as obtained from a fit to~\\refeq{extrapolacion_libre}. The data shown correspond to $T_1=0.7$, $T_2=0.8$, $\\xi=11$ and $r=8$. In order not to clutter the figure, we do not show error bars\\index{error bars} in $\\ensuremath{N_{\\text{Rep}}}\\xspace \\to \\infty$ extrapolation. \\textbf{Left inset:} exponent $\\gamma$, which is plotted against the probability $F$, remains close to $\\gamma=1$ for all $F$, with the exception of the unstable behavior at $F \\approx 0.35$, where curves for different $\\ensuremath{N_{\\text{Rep}}}\\xspace$ cross (see also top right inset). \\textbf{Bottom right inset:} $\\chi^2$ per degree of freedom\\index{degree of freedom} is plotted against $F$. The blue line corresponds to $\\chi^2\/\\text{d.o.f.} = 1$.\\index{degree of freedom} \\textbf{Top right inset:} Zoom of the main plot, emphasizing the crossing region at $F \\approx 0.35$. Note that at that particle value of $F$ data show almost no dependence with $\\ensuremath{N_{\\text{Rep}}}\\xspace$, which makes unstable the fit to~\\refeq{extrapolacion_libre}.}\n \\labfig{exponente_libre}\n\\end{figure}\n\nAs a first check, we compare the linear and the quadratic extrapolations (see~\\reffig{linear_quadratic} for an illustrative example). The figure shows that even for our largest $\\ensuremath{N_{\\text{Rep}}}\\xspace$, namely $\\ensuremath{N_{\\text{Rep}}}\\xspace=256$, and $\\ensuremath{N_{\\text{Rep}}}\\xspace=512$, we are still far from the extrapolation to the $\\ensuremath{N_{\\text{Rep}}}\\xspace \\to \\infty$ limit. Fortunately, the linear and the quadratic extrapolations provide compatible results in our region of interest, i.e. the tail of the distribution function. We remark that the consistency condition $\\chi^2\/\\text{d.o.f.} < 1.1$\\index{degree of freedom} is met in a larger range for the quadratic extrapolation ($F<0.9$) than in the linear extrapolation ($F<0.7$). However, because both coincide in the low-$F$ range we are interested in, we have kept the simpler linear extrapolation.\n\nOur second check in~\\refeq{extrapolacion_libre} seeks the natural exponent $\\gamma$ for the extrapolation, which is a fitting parameter. We have found, see~\\reffig{exponente_libre}, that the consistency condition $\\chi^2 \/ \\mathrm{d.o.f.}<1.1$\\index{degree of freedom} is met for $F<0.85$. Fortunately, $\\gamma$ turns out to be very close to the value $\\gamma=1$, with the exception of the instability in the crossing region around $F\\approx 0.35$. \n\nIn summary, the quadratic and the free-exponent extrapolations support our choice of~\\refeq{extrapolacion_lineal} as the preferred form for the $\\ensuremath{N_{\\text{Rep}}}\\xspace \\to \\infty$ extrapolation.\n\n\\section{On the most convenient variable to characterize the sphere size}\\labsec{cambio_de_r}\n\n\n\\begin{figure}[!h]\n  \\centering\n  \\includegraphics[width=0.48\\textwidth]{off-eq_chaos\/Nr_vs_r_example.pdf}\n  \\includegraphics[width=0.48\\textwidth]{off-eq_chaos\/Nr_vs_r_theoretical.pdf}\t\n  \\caption[\\textbf{$N_r^{1\/3}$ postulates as a better variable to describe short length scales.}]{\\textbf{$N_r^{1\/3}$ postulates as a better variable to describe short length scales.} \\textbf{Left:} complementary of \\gls{TC}\\index{temperature chaos} $1-X^{s,r}_{T_1,T_2}(\\xi)$ against the region size for the two discussed independent variables, namely $N_r^{1\/3}$ and the radius $r$. The continuous lines are fits to~\\refeq{functional_form} taking as variables $z=r$ (golden curve) and $z=N_r^{1\/3}$ (blue curve). The shown data correspond to $T_1=0.7$, $T_2=0.9$, $F=0.01$ and $\\xi=7$. We zoom the region of small spheres, where both independent variables most differ. \\textbf{Right:} the cubic root of the volume of a sphere (blue curve) is plotted as a function of the radius of the sphere $r$. The golden curve is $N_r^{1\/3}$, namely the cubic root of the number of lattice points contained in a sphere of radius $r$, centered at a node of the dual lattice corresponding to our cubic lattice. Values of $N_r^{1\/3}$ corresponding to integer $r$ are highlighted as black dots.}\n \\labfig{Nr_vs_r}\n\\end{figure}\n\nHere we explain our rationale for choosing the cubic root of the number of spins contained in the sphere $N_r^{1\/3}$, rather than its radius $r$, to characterize the size of the spheres considered in our analysis.\n\nWe asked ourselves this question because our first attempt to fit the peaks of $1-X$ to~\\refeq{functional_form} by using as an independent variable the radius of the spheres $r$ failed. Indeed, see~\\reffig{Nr_vs_r} left, $1-X$ is not a smooth function of $r$. After some reflection, we realized that the number of lattice points in our spheres is not a smooth function of $r$ either (see~\\reffig{Nr_vs_r} right). At that point, it was only natural to trade $r$ with $N_r^{1\/3}$ as the independent variable. In fact, see~\\reffig{Nr_vs_r} left, the new independent variable $N_r^{1\/3}$ solved our problem of fitting to~\\refeq{functional_form}. Of course, the difference between both independent variables becomes immaterial for very large spheres.\n\n\\section{Difficulties in peak characterization for the weak temperature chaos regime}\\labsec{peak_characterization}\n\n\\begin{figure}[!h]\n  \\centering\n  \\includegraphics[width=1.0\\textwidth]{off-eq_chaos\/T062507_F01.pdf}\n  \\caption[\\textbf{Too weak a temperature chaos makes it difficult to characterize the peak.}]{\\textbf{Too weak a temperature chaos makes it difficult to characterize the peak.} The complementary of the chaotic parameter $1-X^{s,r}_{T_1,T_2}(\\xi)$ is represented against $N_r^{1\/3}$ for the temperatures $T_1=0.625$ and $T_2=0.7$, the probability $F=0.01$ and different values of $\\xi$. A quick growth for small $N_r^{1\/(3}$ followed by a plateau is observed in all the plots.}\n \\labfig{T062507_F01}\n\\end{figure}\n\nHere we illustrate the difficulty of characterizing the peak of the function $f(z)$ defined in~\\refeq{functional_form} when \\gls{TC}\\index{temperature chaos} is extremely weak. Indeed, as~\\reffig{T062507_F01} shows, the size of the error bars makes\\index{error bars} data almost compatible with a plateau (rather than a peak). Moreover, mind the vertical scale in~\\reffig{T062507_F01}, \\gls{TC}\\index{temperature chaos} is almost nil, which suggests that this set of parameters ($T_1=0.625,T_2=0.7,F=0.01$) is not suitable to study \\gls{TC}\\index{temperature chaos}. Consequently, we have decided to exclude from the analysis the data obtained with the temperatures $T_1=0.625$ and $T_2=0.7$ and the probability $F=0.01$.\n\n\\chapter{Conclusions} \\labch{conclusions}\n\n\\setlength\\epigraphwidth{.5\\textwidth}\n\\epigraph{\\textit{Si el pasado y el presente\\\\\nSe reflejan y no mienten\\\\\nTengo que hacer algo \\\\\npor mi porvenir.}}{-- Evaristo P\u00e1ramos, \\textit{Ya no quiero ser yo}}\n\nThe traditional picture of a solitary theoretical physicist, working alone with paper and pencil as her only tools, has evolved through the years. Instead, computational research has been an important part of the development of physics for the last decades and the discoveries are often performed by groups of physicists working together. However, although the increase of computational power is not new, it has been only recently that we can collect, process, and analyze a huge amount of data at affordable times.\n\nThe increase of computational power has particularly benefited theoretical physics. Specifically for spin glasses, the development of special-purpose\\index{special-purpose computer} hardware has allowed simulating systems up to the experimental time-scale. This thesis is another iteration in the development of a field that takes the general path of science and embraces the interplay between experiments, theory, and computing as a fruitful relation to make relevant advances.\n\nThroughout this thesis, we have studied the spin glasses from a numerical point of view. The study of the metastate\\index{metastate} in~\\refch{metastate} has been focused on addressing a theoretical problem through the first construction of the equilibrium metastate\\index{metastate} in numerical simulations. In~\\refch{aging_rate} and \\refch{mpemba} the off-equilibrium dynamics of spin glasses has been studied. In the former, we introduce the coherence length\\index{coherence length} as the most relevant quantity characterizing the aging\\index{aging} state of a spin glass, and we solve a numerical discrepancy in the aging\\index{aging!rate} rate between experiments and previous numerical simulations. In the latter, we discuss the Mpemba\\index{Mpemba effect} effect, which is a memory\\index{memory effects} effect that takes place in the off-equilibrium dynamics, providing a good example of how the coherence length\\index{coherence length} governs many out-of-equilibrium phenomena. The final part of the thesis, \\refch{Introduction_chaos}, \\refch{equilibrium_chaos} and \\refch{out-eq_chaos}, is devoted to introduce and analyze the Temperature Chaos\\index{temperature chaos} phenomenon in spin glasses. Actually, we tackle this problem from the equilibrium point of view (\\refch{equilibrium_chaos}) and we also observe and characterize the phenomenon in the off-equilibrium dynamics (\\refch{out-eq_chaos}).\n\nIn this chapter we outline the main results of this thesis, revisiting all the parts and summarizing the relevant messages.\n\n\\section{A brief thought about the importance of the data}\nThis thesis is mainly focused on the numerical study of spin glasses. One idea that is central throughout this thesis is the fundamental importance of high-quality data. The reader may wonder what could we say in~\\refch{aging_rate} without our precise estimation of the coherence length\\index{coherence length}, or which conclusions could we extract from~\\refch{out-eq_chaos} without our accurate estimations of the chaotic parameter. None of these works would be possible without our high-quality data.\n\nThis section aims to emphasize the role of the special-purpose\\index{special-purpose computer} FPGA-based\\index{FPGA} hardware, Janus\\index{Janus} II, in this thesis. It is usual in the physicist work to spend a lot of time by using (or developing) sophisticated statistical methods to improve the quality of the data, and, it is actually, a very important task. Nevertheless, if we can combine the statistical methods with powerful hardware, we can, indeed, obtain data at the forefront of the field.\n\nSpecifically, in this thesis, we have taken advantage (in some works) of the largest spin-glass simulation in off-equilibrium dynamics. We have simulated an Edwards-Anderson\\index{Edwards-Anderson!model} model with Ising\\index{Ising} spins, for a lattice size $L=160$ for a modest number of samples\\index{sample} $\\ensuremath{N_{\\text{S}}}\\xspace=16$, but a huge number of replicas\\index{replica} $\\ensuremath{N_{\\text{Rep}}}\\xspace=512$ (a choice that turned to be fundamental). The simulations have been performed up to temperatures well deep in the spin-glass phase\\index{phase!low-temperature\/spin-glass} (for instance, $T=0.625$) to unprecedented long times.\n\nIt is worthy to mention also the fundamental role of other computers such as the Madrid's Cluster in the UCM and the supercomputer Cierzo in BIFI. They have made possible the analysis of the data in this thesis through thousands of hours of computational time.\n\n\n\\section{Conclusions on the metastate} \\labsec{conclusion_metastate}\nIn its origins, the Replica Symmetry Breaking\\index{replica!symmetry breaking (RSB)} theory aimed to explain the nature of the low-temperature phase\\index{phase!low-temperature\/spin-glass} in spin glasses assuming infinite-size systems. However, some mathematical procedures in that development were ill-defined. In particular, for disordered\\index{disorder!systems} systems, the thermodynamic limit\\index{thermodynamic limit} $L \\to \\infty$ for a Gibbs state may not exist. This problem is originated from a phenomenon known as \\textit{chaotic size dependence}. Mathematical physics offered a new approach in the context of disordered\\index{disorder!systems} systems and brought a solution to this problem: \\textit{the metastate}\\index{metastate}. This concept is a generalization of the concept of Gibbs states. Nonetheless, this discussion used to be limited to the theoretical work, without any numerical or experimental counterpart.\n\nIn~\\refch{metastate} we have shown that the state of the art in numerical simulations allows the construction of the Aizenman-Wehr metastate\\index{metastate!Aizenman-Wehr}. Indeed, our numerical data suggest that the $1 \\ll W \\ll R \\ll L$ limit required from the construction of this metastate\\index{metastate!Aizenman-Wehr} (see~\\refsubsec{aw_metastate}) can be relaxed to $W\/R \\approx 0.75$ and $R\/L \\approx 0.75$ without changing substantially the thermodynamic physical behavior.\n\nThe main quantitative result of our work is the numerical computation of the exponent $\\zeta$. According to Read~\\cite{read:14}, it is possible to partially discriminate between the competing pictures trying to describe the nature of the spin-glass phase\\index{phase!low-temperature\/spin-glass} in\\footnote{Recall that $d_{\\mathrm{L}}$ and $d_{\\mathrm{U}}$ are the lower and the upper critical dimension\\index{critical dimension!lower}\\index{critical dimension!upper} respectively.} $d_{\\mathrm{L}} < d < d_{\\mathrm{U}}$ by computing this exponent $\\zeta$. Indeed, $\\zeta$ is related to the number of different states that can be measured in a system of size $W$. This number behaves as $\\log n_{\\mathrm{states}} \\sim W^{d -\\zeta}$. Therefore, $\\zeta < d$ would lead, for $W \\to \\infty$, to a metastate\\index{metastate} of infinitely many states.\n\nHere, we find $\\zeta = 2.3 (3)$. We compare our result to previous estimations of this exponent, computed in different contexts, and we summarize our knowledge about the behavior of this exponent as a function of the dimension $d$ in~\\refsec{relating_numerical_theory_metastate}.\n\nAll the numerical evidence strongly supports the existence of a spin-glass metastate\\index{metastate!dispersed} dispersed over infinitely many states for $d=3$. This result probably holds for $d>d_{\\mathrm{L}}$. Our findings are incompatible with the droplet\\index{droplet!picture} model, while they are compatible with both the chaotic pair picture and the Replica Symmetry Breaking\\index{replica!symmetry breaking (RSB)} scenario.\n\nThe results concerning this work are published in~\\cite{billoire:17}.\n\n\\section{Conclusions on the study of the aging rate}\nThe off-equilibrium dynamics in spin glasses is of glaring importance since the experiments are always conducted out-of-equilibrium. Under these conditions, domains\\index{magnetic domain} of correlated spins start to grow at the microscopical level. The linear size of these domains\\index{magnetic domain} is the so-called \\textit{coherence length\\index{coherence length}} $\\xi$. The \\textit{aging\\index{aging!rate} rate} $z(T)$, which is none but the variation rate of the free-energy\\index{free energy!barrier} barriers with the logarithm of the coherence length\\index{coherence length} $\\xi$ (see \\refsec{how_can_study_aging}), is directly related to the growth of the coherence length\\index{coherence length} with the time. Previous numerical evidences pointed to a dependency of the form $\\xi \\sim \\ensuremath{t_\\mathrm{w}}\\xspace^{1\/z(T)}$. However, numerical discrepancies in the determination of $z(T)$ were recently found between experiments~\\cite{zhai:17} and numerical simulations~\\cite{janus:08,lulli:16}.\n\nIn~\\refch{aging_rate} we have taken advantage of the previously mentioned simulations performed in Janus\\index{Janus} II to study the growth of the coherence length\\index{coherence length} in the glassy phase\\index{phase!low-temperature\/spin-glass} and solved this discrepancy. Specifically, we found that the growth of the coherence length\\index{coherence length} is controlled by a time-dependent aging\\index{aging!rate} rate $z(T,\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace))$. \n\nIn this work, we have described the dynamics as governed by a crossover between a critical and a low-temperature fixed point. The characterization of that crossover has allowed us to quantitatively model the growth of the aging\\index{aging!rate} rate. In particular, we have considered two different ans\\\"atze for that growth, and we have found that, for the convergent one [\\refeq{convergent_ansatz}], the computation of the aging\\index{aging!rate} rate is consistent with the most recent experimental measures~\\cite{zhai:17}.\n\nBesides, we find clear evidence of non-coarsening dynamics at the experimental scale and find that temperatures $T \\lesssim 0.7$ are free of critical effects and therefore safe for numerical studies of the spin-glass phase\\index{phase!low-temperature\/spin-glass}.\n\nThe results concerning this work are published in~\\cite{janus:18}.\n\n\\section{Conclusions on the Mpemba effect}\nConsider two beakers of water that are identical to each other except for the fact that one is hotter than the other. If we put both of them in contact with a thermal reservoir (for example, a freezer) at some temperature lower than the freezing point of the water, under some circumstances, it can be observed that the initially hotter water freezes faster than the colder one. This phenomenon is known as the Mpemba\\index{Mpemba effect} effect~\\cite{mpemba:69}.\n\nIn~\\refch{mpemba} we have shown that the Mpemba\\index{Mpemba effect} effect is present in spin glasses, where it appears as an intrinsically non-equilibrium process, ruled by the spin-glass coherence length\\index{coherence length} $\\xi$.\n\nFirst, we have identified the relevant quantities to mimic the effect in spin glasses, namely the energy-density\\index{energy!density} (as the role of the temperature in the classical Mpemba effect), and the Monte\\index{Monte Carlo} Carlo time. However, the introduction of the coherence length\\index{coherence length} as a hidden quantity ruling the process turned out to be fundamental.\n\nIndeed, we have provided the first explanation of this phenomenon (in the spin-glass context), by using the relation between the energy density\\index{energy!density} and the coherence length\\index{coherence length} [\\refeq{energy_coherence_length_relation}] to characterize the effect. Although this description is approximate, it is accurate enough to describe the Mpemba\\index{Mpemba effect} effect.\n\nOur results explain how the most natural experimental setup (prepare two identical systems at $T_1,T_2>T_\\text{c}$ with an identical protocol, then quench them) can fail to observe the effect. Indeed, for spin glasses at least, a different starting $\\xi$ is required. \n\nFinally, we have investigated the inverse Mpemba\\index{Mpemba effect} effect (\\refsec{inverse_mpemba}). In the spin-glass phase\\index{phase!low-temperature\/spin-glass}, the inverse Mpemba\\index{Mpemba effect} effect is completely symmetrical to the classical Mpemba\\index{Mpemba effect} effect. However, we found that above the critical\\index{critical temperature} temperature $\\ensuremath{T_\\mathrm{c}}\\xspace$ the inverse Mpemba\\index{Mpemba effect} effect is strongly suppressed because our description of \\refeq{energy_coherence_length_relation} is not valid for $T> \\ensuremath{T_\\mathrm{c}}\\xspace$.\n\nThe Mpemba\\index{Mpemba effect} effect is peculiar among the many memory\\index{memory effects} effects present in spin glasses. Indeed, this phenomenon can be studied through quantities, such as the energy density\\index{energy!density}, which are just measured at one-time scale (rather than the usual two times~\\cite{young:98,jonason:98,janus:17b}). However, our setup poses an experimental challenge, because we are not aware of any measurement of the non-equilibrium temperature associated with the magnetic degrees of freedom\\index{degree of freedom}. Perhaps one could adapt the strategy of Ref.~\\cite{grigera:99}, connecting dielectric susceptibility\\index{susceptibility} and polarization noise in glycerol, to measurements of high-frequency electrical noise in spin glasses~\\cite{israeloff:89}.\n\nOur investigation of the Mpemba\\index{Mpemba effect} effect offers as well a new perspective into an important problem, namely the study of the glassy coherence length\\index{coherence length} in supercooled liquids and other glass formers~\\cite{cavagna:09}. Indeed, the identification of the right correlation function to study experimentally (or numerically) is still an open problem. Spin glasses are unique in the\ngeneral context of the glass transition\\index{phase transition}, in both senses: we know which correlation functions should be computed microscopically~\\cite{edwards:75,edwards:76}, while accurate experimental determinations of the coherence length\\index{coherence length} have been obtained~\\cite{guchhait:17}.\n\nThe results concerning this work are published in~\\cite{janus:19}.\n\n\\section{Conclusions on the equilibrium Temperature Chaos}\nIn a spin glass, the Temperature\\index{temperature chaos} Chaos phenomenon refers to the complete reorganization of the Boltzmann\\index{Boltzmann!weight} weights that determines the frequency with which each configuration\\index{configuration} of spins will appear, upon an arbitrary small change in the temperature $T$.\n\nIn~\\refch{equilibrium_chaos} we have studied the Temperature Chaos\\index{temperature chaos} phenomenon in equilibrium simulations and have proposed an efficient variational method\\index{variational method} to estimate the elusive exponential autocorrelation time\\index{autocorrelation time!exponential} of a Monte\\index{Monte Carlo} Carlo Markov\\index{Markov chain} chain, specific to the case of a Parallel Tempering\\index{parallel!tempering} simulation.\n\nThis variational method\\index{variational method} takes into account three parameters and performs a maximization of the estimation of the integrated autocorrelation time\\index{autocorrelation time!integrated} in the phase-space\\index{phase space} of these parameters. Since the exponential autocorrelation time\\index{autocorrelation time!exponential} is an upper bound of the parameter-dependent integrated autocorrelation time\\index{autocorrelation time!integrated}, our procedure leads to robust estimations.\n\nIn addition, we have studied the scaling properties of the probability distribution of the autocorrelation time, obtained using the proposed variational\\index{variational method} approach. In particular, we have shown that scaling holds for lattices of sizes $L \\geq 24$, consistently with previous studies using effective potentials.\n\nThe presence of Temperature Chaos\\index{temperature chaos} is related to the poor performance of the Parallel Tempering simulations in the spin-glass phase\\index{phase!low-temperature\/spin-glass}. Then, the exponential autocorrelation time\\index{autocorrelation time!exponential} provides us a \\textit{dynamic} characterization of this phenomenon.\n\nIn this work, we have also characterized the Temperature Chaos\\index{temperature chaos} from a \\textit{static} point of view by studying the equilibrium configurations\\index{configuration} of the system. The observable that allows us to quantitatively study the Temperature Chaos\\index{temperature chaos} is the so-called \\textit{chaotic parameter}. The empirical observation of the most chaotic samples\\index{sample} (from the dynamical point of view), led to the construction of observables derived from the chaotic parameter to characterize the Temperature Chaos\\index{temperature chaos} (see \\cite{fernandez:13,fernandez:16}). In our work, we introduce a new observable to characterize the phenomenon.\n\nFinally, we have checked the statistical correlations between these \\textit{static} chaotic indicators and the dynamical correlation times. The introduction of the new \\textit{static} indicator has improved previous results in the correlations.\n\nThe results concerning this work are published in~\\cite{billoire:18}.\n\n\\section{Conclusions on the Temperature Chaos in off-equilibrium dynamics}\nIn~\\refch{out-eq_chaos} we have studied an interesting phenomenon in off-equilibrium dynamics that closely mimics Temperature Chaos\\index{temperature chaos}. Indeed, as we have just discussed in the previous section, Temperature Chaos\\index{temperature chaos} is defined as an equilibrium phenomenon. Therefore, studying it in a non-equilibrium system is an open challenge that we have addressed here. \n\nFirst, we tried a naive approach to non-equilibrium Temperature Chaos\\index{temperature chaos} (see \\refsec{average_killed_chaos_signal}), which found only an exceedingly small chaotic signal. Fortunately, the statics-dynamics equivalence\\index{statics-dynamics equivalence}~\\cite{barrat:01,janus:08b,janus:10b,janus:17} combined with the rare events analysis performed in equilibrium simulations, see~\\cite{fernandez:13}, provide the crucial insight to approach the problem. Specifically, the statics-dynamics equivalence allows us to relate the non-equilibrium dynamics of a spin glass (of infinite size) with a finite coherence length\\index{coherence length} $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$, with small samples\\index{sample} of size $L\\sim\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ which can be equilibrated.\n\nOur numerical protocol considers spherical-like regions of radius $r$. We focus on the probability distribution function of the chaotic parameter as computed over the spheres. We find that only the spheres in the tail of the distribution exhibit a strong Temperature Chaos\\index{temperature chaos}.\n\nChoosing a suitable length scale $r$ for the spherical-like regions turns out to be instrumental in the study of dynamic Temperature Chaos\\index{temperature chaos}. This optimal length scale is proportional to the coherence length\\index{coherence length}. However, our data mildly suggests that the importance of choosing exactly the correct $r$ becomes less critical in the $\\xi \\to \\infty$ limit.\n\nA striking link emerges between the dynamic and the static faces of the Temperature Chaos\\index{temperature chaos} phenomenon. Indeed, we find a characteristic length scale $\\xi^*(T_2-T_1,F)$ at which the crossover between the weak chaos and the strong chaos regimes occurs. The physical meaning of the characteristic length $\\xi^*$ suggests that the equilibrium chaotic length $\\ell_c$~\\cite{fisher:86,bray:87b} is its equilibrium counterpart. In fact, both quantities depend on $T_2-T_1$ in the same way and the exponent\\footnote{The reader should be warned that this exponent $\\zeta$ is completely different from the exponent $\\zeta$ in \\refsec{conclusion_metastate}. We follow here the usual notation in the literature and denote both of them with the same letter despite their completely different meanings.} $\\zeta$, controlling the temperature-dependence of $\\ell_c$ turns out to be equal to $\\zeta_{\\mathrm{NE}}$, obtained from the off-equilibrium estimation, at the two-$\\sigma$ level. We regard this coincidence as new and important evidence for the statics-dynamics equivalence\\index{statics-dynamics equivalence}.\n\nIn the second part of the work we have performed temperature-varying\\index{temperature-varying protocol} simulations to address the cumulative aging\\index{aging!cumulative} problem~\\cite{jonsson:02,bert:04,komori:00,berthier:02,picco:01,takayama:02,maiorano:05,jimenez:05}. We have found small but clear violations of cumulative aging\\index{aging!cumulative}, which are stronger upon \\textit{cooling} than upon \\textit{heating}. Although both protocols display a memory-erasing process, the \\textit{cooling} process shows better memory, i.e. larger coherence-lengths are needed to lose the memory of the previous state at the higher temperature. \n\nThe results concerning this work are published in~\\cite{janus:21}.\n\n\n\\chapter{Conclusiones} \\labch{conclusiones_castellano}\nLa visi\u00f3n tradicional de un f\u00edsico te\u00f3rico solitario, trabajando s\u00f3lo con papel y l\u00e1piz como sus \u00fanicas herramientas, ha ido evolucionando a lo largo de los a\u00f1os. En su lugar, la investigaci\u00f3n computacional ha sido una parte importante del desarrollo de la f\u00edsica en las \u00faltimas d\u00e9cadas y los descubrimientos son normalmente hechos por grupos de f\u00edsicos trabajando juntos. Sin embargo, aunque el incremento de la capacidad computacional no es nuevo, ha sido recientemente cuando hemos conseguido recolectar, procesar y analizar una gran cantidad de datos en tiempos razonables.\n\nLa f\u00edsica te\u00f3rica ha sido una de las grandes beneficiadas de este incremento de la capacidad computacional. Si nos centramos en los vidrios de esp\u00edn, el desarrollo de hardware dedicado ha permitido simular sistemas hasta las escalas de tiempo experimentales. Esta tesis es otra iteraci\u00f3n en el desarrollo de un campo que ha tomado el camino general de la ciencia y ha abrazado la interacci\u00f3n entre experimentos, teor\u00eda y computaci\u00f3n como una relaci\u00f3n fruct\u00edfera para hacer avances relevantes.\n\nA lo largo de esta tesis, hemos estudiado los vidrios de esp\u00edn desde un punto de vista num\u00e9rico. El estudio del metaestado en el Cap\u00edtulo 2 ha estado enfocado a abordar un problema te\u00f3rico a trav\u00e9s de la primera construcci\u00f3n del metaestado en equilibrio en simulaciones num\u00e9ricas. En los cap\u00edtulos 3 y 4, hemos estudiado la din\u00e1mica fuera del equilibrio de los vidrios de esp\u00edn. En el Cap\u00edtulo 3, hemos introducido la longitud de coherencia como la cantidad m\u00e1s relevante para caracterizar el estado de envejecimiento de un vidrio de esp\u00edn y hemos resuelto una discrepancia num\u00e9rica en el ratio de envejecimiento entre experimentos y simulaciones num\u00e9ricas. En el cap\u00edtulo 4, hemos discutido el efecto Mpemba, que es un efecto de memoria que tiene lugar en la din\u00e1mica fuera del equilibrio, d\u00e1ndonos un buen ejemplo de c\u00f3mo la longitud de coherencia gobierna una multitud de fen\u00f3menos en este r\u00e9gimen. La parte final de esta tesis, los cap\u00edtulos 5, 6 y 7, est\u00e1n dedicados a introducir y analizar el fen\u00f3meno del Caos en Temperatura en los vidrios de esp\u00edn. De hecho, abordamos este problema desde un punto de vista de equilibrio (Cap\u00edtulo 6) y tambi\u00e9n caracterizamos este fen\u00f3meno en la din\u00e1mica de fuera del equilibrio (Cap\u00edtulo 7).\n\nEn este cap\u00edtulo, resaltamos los principales resultados de esta tesis, revisitando todas las partes de la misma y resumiendo los mensajes m\u00e1s relevantes.\n\n\\section{Una breve reflexi\u00f3n sobre la importancia de los datos}\nEsta tesis est\u00e1 principalmente enfocada al estudio num\u00e9rico de los vidrios de esp\u00edn. Una idea que es central a lo largo de esta tesis es la importancia fundamental de los datos de alta calidad. El lector puede preguntarse qu\u00e9 podr\u00edamos haber dicho en el Cap\u00edtulo 3 sin una estimaci\u00f3n precisa de la longitud de coherencia, o qu\u00e9 conclusiones podr\u00edamos sacar del Cap\u00edtulo 7 sin una estimaci\u00f3n precisa del par\u00e1metro ca\u00f3tico. Ninguno de estos trabajos ser\u00eda posible sin nuestros datos de alta calidad.\n\nEsta secci\u00f3n tiene por objetivo enfatizar el rol del hardware dedicado basado en FPGA Janus II en esta tesis.  Es usual en el trabajo de un f\u00edsico dedicar mucho tiempo a usar (o desarrollar) m\u00e9todos estad\u00edsticos sofisticados para mejorar la calidad de los datos y esta es una tarea muy importante. No obstante, si podemos combinar \u00e9stos m\u00e9todos estad\u00edsticos con un hardware potente, podremos obtener datos en la vanguardia del campo.\n\nEn concreto, en esta tesis hemos tenido acceso (para varios trabajos) a la simulaci\u00f3n m\u00e1s grande en vidrios de esp\u00edn fuera del equilibrio. Hemos simulado un modelo de Edwards-Anderson con espines de Ising para una red de tama\u00f1o $L=160$, para un modesto n\u00famero de samples $\\ensuremath{N_{\\text{S}}}\\xspace=16$, pero para un gran n\u00famero de r\u00e9plicas $\\ensuremath{N_{\\text{Rep}}}\\xspace=512$ (una elecci\u00f3n que ha resultado ser fundamental). Las simulaciones han sido llevadas a cabo a temperaturas bastante bajas, en la fase v\u00edtrea (por ejemplo $T=0.625$) hasta tiempos muy largos, sin precedentes.\n\nTambi\u00e9n cabe destacar el papel fundamental de otros ordenadores como el Cluster de Madrid de la UCM y el superordenador Cierzo en el BIFI. Estos ordenadores han hecho posible el an\u00e1lisis de los datos de esta tesis a trav\u00e9s de miles de horas de tiempo computacional.\n\n\\section{Conclusiones sobre el metaestado}\nEn sus or\u00edgenes, la teor\u00eda de \\textit{Replica Symmetry Breaking} ten\u00eda como objetivo explicar la fase de bajas temperaturas en los vidrios de esp\u00edn asumiendo sistemas de tama\u00f1o infinito. Sin embargo, algunos procedimientos matem\u00e1ticos involucrados en la teor\u00eda estaban mal definidos. En concreto, para sistemas desordenados, el l\u00edmite termodin\u00e1mico $L\\to\\infty$ para un estado de Gibbs puede no existir. Este problema est\u00e1 originado por un fen\u00f3meno conocido como la \\textit{dependencia ca\u00f3tica con el tama\u00f1o}. La irrupci\u00f3n de la f\u00edsica matem\u00e1tica en el contexto de los sistemas desordenados trajo la soluci\u00f3n a este problema: el metaestado. Este concepto es una generalizaci\u00f3n del concepto de estado de Gibbs. No obstante, esta discusi\u00f3n sol\u00eda estar limitada al trabajo te\u00f3rico, sin una contraparte num\u00e9rica o experimental.\n\nEn el cap\u00edtulo 2 hemos demostrado que el estado del arte en simulaciones num\u00e9ricas permite la construcci\u00f3n del metaestado de Aizenman-Wehr~\\cite{aizenman:90}. De hecho, nuestros datos num\u00e9ricos sugieren que el l\u00edmite $1 \\ll W \\ll R \\ll L$ requerido para la construcci\u00f3n del metaestado puede relajarse a $W\/R \\approx 0.75$ y $R\/L \\approx 0.75$ sin que haya cambios sustanciales en el comportamiento f\u00edsico del mismo.\n\nEl principal resultado cuantitativo de nuestro trabajo es la computaci\u00f3n num\u00e9rica del exponente $\\zeta$. Seg\u00fan Read \\cite{read:14}, es posible discriminar parcialmente entre las teor\u00edas contrapuestas que tratan de describir la naturaleza de la fase v\u00edtrea entre la dimensi\u00f3n cr\u00edtica inferior $d_{\\mathrm{L}}$ y la dimensi\u00f3n cr\u00edtica superior $d_{\\mathrm{U}}$, calculando este exponente $\\zeta$. De hecho, $\\zeta$ est\u00e1 relacionado con el n\u00famero de estados distintos que pueden ser medidos en un sistema de tama\u00f1o $W$. Este n\u00famero se comporta como $\\log n_{\\mathrm{states}} \\sim W^{d -\\zeta}$. Por lo tanto, si $\\zeta < d$ tendr\u00edamos que, para el l\u00edmite $W \\to \\infty$, el metaestado tendr\u00eda un n\u00famero infinito de estados.\n\nNosotros encontramos $\\zeta = 2.3(3)$. Comparamos nuestro resultado num\u00e9rico con estimaciones previas de este exponente, calculadas en contextos distintos, y resumimos todo nuestro conocimiento sobre el comportamiento de dicho exponente como funci\u00f3n de la dimensi\u00f3n $d$ en la secci\u00f3n 2.8.\n\nTodas las evidencias num\u00e9ricas apoyan firmemente la existencia de un metaestado disperso en vidrios de esp\u00edn, formado por infinitos estados, en $d=3$. Este resultado, probablemente es v\u00e1lido para todo $d>d_{\\mathrm{L}}$. Nuestros resultados son incompatibles con el modelo \\textit{droplet}, mientras que son compatibles tanto con el modelo \\textit{chaotic pair} como con el modelo \\textit{replica symmetry breaking}.\n\nLos resultados de este trabajo est\u00e1n publicados en~\\cite{billoire:17}.\n\n\\section{Conclusiones sobre el estudio del ratio de envejecimiento}\nLa din\u00e1mica fuera del equilibrio tiene una importancia palmaria en los vidrios de esp\u00edn puesto que los experimentos siempre son llevados a cabo en estas condiciones. En estos experimentos, dominios de espines correlacionados empiezan a crecer a nivel microsc\u00f3pico. La longitud de estos dominios es conocida como \\textit{longitud de coherencia} $\\xi$. La \\textit{tasa de envejecimiento} $z(T)$, que no es m\u00e1s que la tasa de variaci\u00f3n de las barreras de energ\u00eda libre con el logaritmo de la longitud de coherencia $\\xi$ (v\u00e9ase la Secci\u00f3n 3.2), est\u00e1 directamente relacionada con el crecimiento de la longitud de coherencia con el tiempo. La evidencia num\u00e9rica se\u00f1ala que el comportamiento de este crecimiento con el tiempo es, esencialmente, $\\xi \\sim \\ensuremath{t_\\mathrm{w}}\\xspace^{1\/z(T)}$. Sin embargo, discrepancias num\u00e9ricas en la determinaci\u00f3n de $z(T)$ han sido encontradas recientemente entre experimentos~\\cite{zhai:17} y simulaciones num\u00e9ricas~\\cite{janus:08,lulli:16}.\n\nEn el cap\u00edtulo 3 hemos aprovechado las simulaciones previamente mencionadas, llevadas a cabo por Janus II, para estudiar el crecimiento de la longitud de coherencia en la fase v\u00edtrea. En concreto, hemos descubierto que el crecimiento de la longitud de coherencia est\u00e1 controlado por un ratio de envejecimiento que depende del tiempo $z(T, \\xi(\\ensuremath{t_\\mathrm{w}}\\xspace))$.\n\nEn este trabajo hemos descrito la din\u00e1mica mediante el cruce entre la influencia del punto fijo a temperatura cr\u00edtica y el punto fijo a temperatura 0. La caracterizaci\u00f3n de ese cruce nos ha permitido modelizar cuantitativamente el crecimiento del ratio de envejecimiento. En concreto, hemos considerado dos hip\u00f3tesis para ese crecimiento y hemos encontrado que la hip\u00f3tesis convergente [\\refeq{convergent_ansatz}] es consistente con el ratio de envejecimiento obtenido por las medidas experimentales m\u00e1s recientes~\\cite{zhai:17}.\n\nAdem\u00e1s, hemos encontrado claras evidencias de din\u00e1mica \\textit{non-coarsening} a la escala de tiempo experimental y hemos encontrado que las temperaturas $T \\lesssim 0.7$ est\u00e1n libres de efectos cr\u00edticos y, por lo tanto, son seguras para el estudio num\u00e9rico.\n\nLos resultados correspondientes a este cap\u00edtulo est\u00e1n publicados en~\\cite{janus:18}.\n\n\\section{Conclusiones sobre el efecto Mpemba}\nConsidere dos recipientes de agua que son id\u00e9nticos entre ellos con la \u00fanica excepci\u00f3n de que el agua contenida en uno de ellos est\u00e1 m\u00e1s caliente que la del otro recipiente. Si ponemos ambos recipientes en contacto con un ba\u00f1o t\u00e9rmico (por ejemplo, un congelador) a una temperatura por debajo del punto de congelaci\u00f3n del agua, bajo ciertas circunstancias, puede observarse que el agua inicialmente m\u00e1s caliente se congela antes que la fr\u00eda. Este fen\u00f3meno es conocido como el efecto Mpemba \\cite{mpemba:69}.\n\nEn el Cap\u00edtulo 4 hemos demostrado que el efecto Mpemba est\u00e1 presente en los vidrios de esp\u00edn, donde es un proceso que ocurre fuera del equilibrio y que est\u00e1 gobernado por la longitud de coherencia $\\xi$.\n\nEn primer lugar, hemos identificado los observables relevantes para imitar el efecto Mpemba cl\u00e1sico en los vidrios de esp\u00edn. Esos observables han sido la densidad de energ\u00eda, cumpliendo el rol de la temperatura en el efecto Mpemba cl\u00e1sico, y el tiempo de Monte Carlo. Sin embargo, la introducci\u00f3n de la longitud de coherencia como el observable oculto que gobierna el proceso ha resultado ser fundamental.\n\nHemos dado una primera explicaci\u00f3n de este fen\u00f3meno en los vidrios de esp\u00edn usando la relaci\u00f3n entre la densidad de energ\u00eda y la longitud de coherencia [\\refeq{energy_coherence_length_relation}] para caracterizar dicho fen\u00f3meno. Aunque la descripci\u00f3n dada es aproximada, es suficientemente precisa para caracterizar el efecto Mpemba.\n\nNuestros resultados explican c\u00f3mo la configuraci\u00f3n experimental m\u00e1s habitual (preparar dos sistemas id\u00e9nticos a $T_1,T_2 > \\ensuremath{T_\\mathrm{c}}\\xspace$ con un protocolo id\u00e9ntico y despu\u00e9s enfriarlos) puede fallar para ver este efecto. De hecho, al menos para los vidrios de esp\u00edn, diferentes longitudes de coherencia $\\xi$ iniciales son necesarias.\n\nFinalmente, hemos investigado el efecto Mpemba inverso. En la fase v\u00edtrea, el efecto Mpemba inverso es completamente sim\u00e9trico respecto del efecto Mpemba cl\u00e1sico. Sin embargo, hemos encontrado que por encima de la temperatura cr\u00edtica $\\ensuremath{T_\\mathrm{c}}\\xspace$, el efecto Mpemba inverso se ve fuertemente menguado debido a que la descripci\u00f3n de \\refeq{energy_coherence_length_relation} no es v\u00e1lida para $T>\\ensuremath{T_\\mathrm{c}}\\xspace$.\n\nEl efecto Mpemba es peculiar dentro de los muchos efectos de memoria presentes en los vidrios de esp\u00edn. De hecho, este fen\u00f3meno puede estudiarse a trav\u00e9s de cantidades como la densidad de energ\u00eda que requieren una sola escala temporal para ser medidas (al contrario que las dos escalas temporales que suelen requerir estos fen\u00f3menos para ser estudiados y caracterizados~\\cite{young:98,jonason:98,janus:17b}). Sin embargo, nuestra configuraci\u00f3n para el experimento num\u00e9rico plantea un desaf\u00edo experimental, dado que no tenemos noticia de ninguna medida de temperatura fuera del equilibrio asociada con los grados de libertad magn\u00e9ticos. Quiz\u00e1s podr\u00eda adaptarse la estrategia de~\\cite{grigera:99}, que conecta la susceptibilidad diel\u00e9ctrica y el ruido de polarizaci\u00f3n en glicerol, con las mediciones de ruido el\u00e9ctrico de alta frecuencia en vidrios de esp\u00edn~\\cite{israeloff:89}.\n\nNuestro estudio del efecto Mpemba inverso sugiere una v\u00eda experimental m\u00e1s f\u00e1cil, donde los sistemas son calentados, en lugar de enfriados. En este caso, aunque la respuesta de la energ\u00eda es muy peque\u00f1a, el proceso va acompa\u00f1ado de un efecto de memoria dram\u00e1tico en la longitud de coherencia. Esta cantidad tiene una evoluci\u00f3n temporal no mon\u00f3tona al calentarse desde la fase v\u00edtrea a la fase paramagn\u00e9tica, antes de converger a la curva maestra (isoterma). Adem\u00e1s puede medirse con las t\u00e9cnicas experimentales actuales.\n\nNuestra investigaci\u00f3n del efecto Mpemba ofrece tambi\u00e9n una nueva perspectiva sobre un problema importante, a saber, el estudio de la longitud de coherencia v\u00edtrea en l\u00edquidos sobreenfriados y otros formadores de vidrio~\\cite{cavagna:09}. De hecho, la identificaci\u00f3n de la funci\u00f3n de correlaci\u00f3n correcta para el estudio experimental (o num\u00e9rico) sigue siendo un problema abierto. Los vidrios de esp\u00edn son \u00fanicos en el contexto general de la transici\u00f3n v\u00edtrea: sabemos qu\u00e9 funciones de correlaci\u00f3n deben calcularse microsc\u00f3picamente~\\cite{edwards:75,edwards:76} y se han obtenido determinaciones experimentales precisas de la longitud de coherencia~\\cite{guchhait:17}.\n\nLos resultados relacionados con este trabajo se han publicado en~\\cite{janus:19}.\n\n\\section{Conclusiones sobre el Caos en Temperatura en equilibrio}\nEn un vidrio de esp\u00edn, el efecto del \\textit{caos en temperatura} es la completa reorganizaci\u00f3n de los pesos de Boltzmann que determinan la frecuencia con la que cada configuraci\u00f3n de espines aparece, debido a un cambio arbitrariamente peque\u00f1o en la temperatura $T$.\n\nEn el cap\u00edtulo 6 hemos estudiado el fen\u00f3meno del caos en temperatura en simulaciones de equilibrio y hemos propuesto un eficiente m\u00e9todo variacional para estimar la elusiva cantidad del tiempo de autocorrelaci\u00f3n exponencial en una cadena de Markov, espec\u00edficamente para el caso del \\textit{Parallel Tempering}.\n\nEste m\u00e9todo variacional toma en cuenta tres par\u00e1metros y realiza una maximizaci\u00f3n de la estimaci\u00f3n del tiempo de autocorrelaci\u00f3n integrado en el espacio de fases de dichos par\u00e1metros. Puesto que el tiempo de autocorrelaci\u00f3n exponencial es una cota superior del tiempo de autocorrelaci\u00f3n integrado (que depende de los par\u00e1metros anteriormente mencionados), nuestro procedimiento es capaz de dar estimaciones robustas.\n\nAdem\u00e1s, hemos estudiado propiedades de escalado de la distribuci\u00f3n de probabilidad del tiempo de autocorrelaci\u00f3n obtenido a trav\u00e9s del m\u00e9todo variacional. En concreto, hemos demostrado que el escalado se mantiene para redes de tama\u00f1os $L \\geq 24$. Este resultado es consistente con resultados previos que usaban potenciales efectivos.\n\nLa presencia del caos en temperatura est\u00e1 relacionada con el pobre desempe\u00f1o del algoritmo de \\textit{Parallel Tempering} en la fase v\u00edtrea. Por lo tanto, el tiempo de autocorrelaci\u00f3n exponencial constituye una caracterizaci\u00f3n din\u00e1mica de este fen\u00f3meno.\n\nEn este trabajo, tambi\u00e9n hemos caracterizado el caos en temperatura desde un punto de vista est\u00e1tico mediante el estudio de configuraciones de equilibrio del sistema. El observable que nos permite estudiar cuantitativamente el caos en temperatura es el par\u00e1metro ca\u00f3tico. La observaci\u00f3n experimental de las \\textit{samples} m\u00e1s ca\u00f3ticas (desde un punto de vista din\u00e1mico) llev\u00f3 a la construcci\u00f3n de cantidades derivadas del par\u00e1metro ca\u00f3tico para estudiar el caos en temperatura (v\u00e9ase \\cite{fernandez:13,fernandez:16}). En nuestro trabajo, hemos introducido un nuevo observable para caracterizar este fen\u00f3meno.\n\nFinalmente, hemos comprobado las correlaciones estad\u00edsticas entre los estimadores est\u00e1ticos del caos y los estimadores din\u00e1micos. La introducci\u00f3n del nuevo estimador est\u00e1tico del caos ha mejorado considerablemente la correlaci\u00f3n con los estimadores din\u00e1micos con respecto a los estudios previos.\n\nLos resultados relacionados con este trabajo est\u00e1n publicados en~\\cite{billoire:18}.\n\n\n\\section{Conclusiones sobre el Caos en Temperatura fuera del equilibrio}\nEn el Cap\u00edtulo 7 hemos estudiado el interesante fen\u00f3meno del caos en temperatura en din\u00e1mica fuera del equilibrio que imita de cerca al caos en temperatura en equilibrio. De hecho, como hemos visto en la secci\u00f3n previa, el caos en temperatura est\u00e1 definido como un fen\u00f3meno de equilibrio y, por lo tanto, el estudio de este fen\u00f3meno fuera del equilibrio ha sido hist\u00f3ricamente un desaf\u00edo abierto que nosotros trataremos aqu\u00ed.\n\nEn primer lugar, intentamos una metodolog\u00eda \\textit{naive} para explicar el fen\u00f3meno (v\u00e9ase secci\u00f3n 7.2), con la cual encontramos un caos extremadamente d\u00e9bil. Afortunadamente, la correspondencia est\u00e1tica-din\u00e1mica ~\\cite{barrat:01,janus:08b,janus:10b,janus:17} junto con el an\u00e1lisis de eventos raros llevado a cabo en simulaciones de equilibrio~\\cite{fernandez:13}, nos proporcionaron el enfoque correcto para abordar el problema. En concreto, la equivalencia est\u00e1tica-din\u00e1mica nos permite relacionar la din\u00e1mica fuera del equilibrio de un vidrio de esp\u00edn (de tama\u00f1o infinito) con una longitud de coherencia $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$, con peque\u00f1as \\textit{samples} de tama\u00f1o $L\\sim\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ que pueden ser llevadas al equilibrio.\n\nNuestro protocolo num\u00e9rico considera regiones quasi-esf\u00e9ricas de radio $r$. En este trabajo, nos centramos en la funci\u00f3n densidad de probabilidad del par\u00e1metro ca\u00f3tico calculado para dichas esferas. Encontramos que s\u00f3lo para las esferas en la cola de la distribuci\u00f3n puede observarse un caos en temperatura fuerte.\n\nEscoger una escala de longitud $r$ adecuada para las regiones esf\u00e9ricas ha resultado ser fundamental para estudiar el caos en temperatura. Esta escala de longitud \u00f3ptima es proporcional a la longitud de coherencia. Merece la pena resaltar que nuestros datos sugieren que la importancia de escoger la escala de longitud $r$ correcta es menos cr\u00edtica para el l\u00edmite $\\xi \\to \\infty$.\n\nUna conexi\u00f3n llamativa surge entre la din\u00e1mica y la est\u00e1tica en lo que al caos en temperatura se refiere. Encontramos una longitud caracter\u00edstica $\\xi^*(T_2-T_1,F)$ que marca el l\u00edmite entre un r\u00e9gimen de caos d\u00e9bil y un r\u00e9gimen de caos fuerte. El significado f\u00edsico de esta longitud caracter\u00edstica $\\xi^*$ sugiere que la llamada \\textit{longitud ca\u00f3tica de equilibrio} $\\ell_c$~\\cite{fisher:86,bray:87b} podr\u00eda ser su contrapartida en los sistemas en equilibrio. De hecho, ambas cantidades dependen de $T_2-T_1$ de la misma forma, a trav\u00e9s del exponente $\\zeta$\\footnote{Advertimos al lector que este exponente $\\zeta$ es completamente diferente al exponente $\\zeta$ de la Secci\u00f3n 8.2. Nosotros seguimos la notaci\u00f3n habitual en la literatura, denotando ambas cantidades con la misma letra a pesar de sus significados completamente distintos.}, que ha resultado ser el mismo para el equilibrio, y para fuera del equilibrio $\\zeta_{\\mathrm{NE}}$ dentro del nivel de dos desviaciones est\u00e1ndar. Contemplamos esta coincidencia como una nueva e importante evidencia de la equivalencia est\u00e1tica-din\u00e1mica.\n\nEn la segunda parte de este trabajo hemos llevado a cabo simulaciones con protocolos de cambios de temperatura para referir el problema del \\textit{cummulative aging}~\\cite{jonsson:02,bert:04,komori:00,berthier:02,picco:01,takayama:02,maiorano:05,jimenez:05}. Hemos encontrado peque\u00f1as pero claras violaciones al \\textit{cummulative aging} que son m\u00e1s fuertes cuando se enfr\u00eda el sistema que cuando se calienta. Aunque ambos protocolos (calentar y enfriar), muestran procesos de borrado de memoria, el protocolo de enfriamiento muestra una mejor memoria, es decir, se necesitan tiempos de coherencia m\u00e1s largos para borrar la memoria del estado previo a mayor temperatura.\n\nLos resultados relativos a este trabajo est\u00e1n publicados en~\\cite{janus:21}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\chapter{Dynamic variational study of Temperature Chaos} \\labch{equilibrium_chaos}\n\nThe process of taking a \\gls{SG} sample to equilibrium in numerical simulations requires the use of dynamic Monte\\index{Monte Carlo} Carlo methods. Unfortunately, as we have already commented in \\refsubsec{Monte_Carlo}, the sluggish dynamics exhibited by \\gls{SG}s impedes the use of simple methods like the Metropolis-Hasting algorithm. One solution comes from the use of the \\gls{PT} method, which equilibrates at once a set of $N$ copies of the system running at different temperatures.\n\nHowever, the \\gls{TC}\\index{temperature chaos} phenomenon represents a major obstacle in the performance of \\gls{PT}~\\cite{fernandez:13}. In this chapter, we follow the ideas proposed in previous studies~\\cite{fernandez:13,martin-mayor:15,fernandez:16} and we take advantage of that fact by quantitatively characterizing the \\gls{TC}\\index{temperature chaos} through a careful study of the process of thermalization\\index{thermalization} of the system when using the \\gls{PT} method. This work also extends the study of \\gls{TC}\\index{temperature chaos} performed in previous papers~\\cite{fernandez:13,fernandez:16} by the development of a variational method\\index{variational method}.\n\nMoreover, we also focus on the very definition of \\gls{TC}\\index{temperature chaos} and we study it by comparing equilibrium configurations\\index{configuration} at different temperatures. Both characterizations, namely dynamic and static, are found to correlate very well~\\cite{fernandez:13,fernandez:16}. Here, we propose new observables to study the \\textit{static} chaos and we found large correlations between the main observables of both characterizations, static and dynamic.\n\nAll the results exposed in this chapter came from the original work~\\cite{billoire:18} which has been developed during this thesis.\n\n\\section{Numerical simulations} \\labsec{numerical_simulations_eq_chaos}\nIn order to keep clean the rest of the chapter of technical details and to focus on the physical results, we explain here the simulations performed.\n\nFirst of all, it is fundamental to mention that the data used here come from the study of the metastate\\index{metastate} (see~\\refch{metastate}) and, therefore, the structure of the couplings\\index{couplings} is not conventional. We briefly recall here, for the reader's convenience, the particularities of this simulation.\n\nThe system, composed of $L^3$ spins, is divided into an inner region of $(L\/2)^3$ spins and an outer region surrounding it. For each of the $10$ realizations of the inner disorder\\index{disorder}, we have a set of $1280$ realizations of the outer disorder\\index{disorder}. Hence, we have a total of $12800$ samples\\index{sample} and for each one, we have simulated $\\Nrep=4$ different replicas\\index{replica}. \n\nA natural question is whether this particular setup's choice is affecting the results. One could imagine that those samples\\index{sample} sharing the same inner disorder\\index{disorder} would be strongly correlated and, hence, the statistics coming from only 10 different inner realizations could be not enough to deal with the sample-to-sample fluctuations\\index{sample-to-sample fluctuations}. However, this choice is irrelevant for the studied observables in this work. The interested reader can find a detailed discussion in \\refsec{selection_parameters}.\n\n\\begin{table}[t!]\n\\centering\n\\begin{tabular}{cccccccc}           \n\\toprule\n\\toprule\n\\multicolumn{7}{c}{MUSA-MSC} \\\\\n\\hline\n$L$ & $L_{\\text{int}}$ & $N_T$ & $T_{\\mathrm{min}}$ & $T_{\\max}$ & $N_\\text{Met}$ ($ \\times 10^6$) & $\\text{ps\/s}$  \\\\\n\\hline\n24 & 12 & 24 & 0.698 & 1.538 & 500 & 104 \\\\ \n16 & 8 & 16 & 0.479 & 1.575 & 250 & 107  \\\\ \n16 & 8 & 13 & 0.698 & 1.575 & 250 & 119  \\\\ \n16 & 12 & 13 & 0.698 & 1.575 & 250 & 119 \\\\ \n14 & 12 & 13 & 0.698 & 1.575 & 500 & 120 \\\\\n12 & 6 & 13 & 0.698 & 1.575 & 250 & 119  \\\\ \n8 & 4 & 13 & 0.698 & 1.575 & 250 & 126 \\\\\n\\bottomrule\n\\end{tabular}\n\\\\[5mm]\n\n\\begin{tabular}{ccccccccc}  \n\\toprule\n\\toprule         \n\\multicolumn{8}{c}{MUSI-MSC} \\\\\n\\hline\n$L$ & $L_{\\text{int}}$ & $N_T$ & $N_\\text{samp}$ & $N_\\text{Met,min}$ & $N_\\text{Met,mean}$ & $N_\\text{Met,max}$ & $\\text{ps\/s}$ \\\\\n\\multicolumn{4}{c}{} & $\\times 10^6$ & $\\times 10^6$ & $\\times 10^6$ & \\\\\n\\hline\n24 & 12 & 24 & 2441 & 1000 & 4262 & 326000 & 57 \\\\ \n16 & 8 & 16  & 2898 & 500 & 5096 & 355500 & 304 \\\\ \n16 & 8 & 13 & 338 & 500 & 543 & 4000 & 306 \\\\ \n16 & 12 & 13 & 314  & 500 & 578 & 8000 & 306 \\\\ \n\\bottomrule\n\\end{tabular}\n\n\\caption[\\textbf{Parameters of the simulations MUSA-MSC and MUSI-MSC.}]{\\textbf{Parameters of the simulations MUSA-MSC and MUSI-MSC.} $L$ is the lattice size; $L_\\text{int}$ the size of the inner part of the lattice; $N_T$, $T_{\\mathrm{min}}$ and $T_{\\max}$ are the number of temperatures, the minimum and the maximum temperatures used in the \\gls{PT} method; $N_\\text{Met}$ is the number of Metropolis sweeps (at each temperature); $\\text{ps\/spin}$ is the average CPU time per spin-flip in MUSI-MSC, using an Intel Xeon CPU E5-2680 processors; $N_\\text{samp}$ denotes the number of bad samples\\index{sample} whose simulations had to be extended in order to thermalize and finally $N_\\text{Met,min}$, $N_\\text{Met,mean}$ and $N_\\text{Met,max}$ are the minimum, mean and maximum number of Metropolis sweeps per temperature needed to reach thermalization\\index{thermalization} (bad samples\\index{sample}). The set of temperatures used is clearly the same in the MUSI-MSC and MUSA-MSC parts of this Table. The number of Metropolis sweeps between two consecutive \\gls{PT} updates is always $N_\\text{MpPT} = 10$.  For the MUSI-MSC simulation of $L=24$ we parallelized\\index{parallel!computation}, using \\emph{Pthreads}, by distributing the $N_T=24$ system copies among 12 CPU cores in the Intel Xeon CPU E5-2680.}\n\\labtab{parameters_simulation_MUSA_MUSI}\n\\end{table}\n\nThe samples\\index{sample} have been equilibrated by using the \\gls{PT} method with Metropolis updates between two consecutive \\gls{PT} exchanges. We increase the performance of the Metropolis update via multispin coding and we apply two methods widely used in numerical simulations in statistical physics, namely the \\gls{MUSA}\\index{Multispin Coding!Multisample}~\\cite{newman:99} and the \\gls{MUSI}\\index{Multispin Coding!Multisite}~\\cite{fernandez:15}. The basic idea of these methods is the parallelization\\index{parallel!computation} of operations that are, indeed, independent from each other by taken advantage of the streaming extensions of the current computer processors. Our simulations were carried out using either Intel Xeon E5-2680 or AMD Opteron Processor 6272. Further details can be found in \\refch{AP_multispin_coding}.\n\nThe selection of the parameters of the simulation are in \\reftab{parameters_simulation_MUSA_MUSI} and the reason for the choice of some of them is explained in~\\refsec{selection_parameters}. Although all the simulations are included for completeness, some of them were only used in the metastate\\index{metastate} study (see~\\refch{metastate}). We focus here only in those simulations with $L_{\\mathrm{int}} = L\/2$. It is worthy to note that in most of this chapter, for the $L=16$ system, we are using the simulation with $N=16$ temperatures, barring the discussion on the impact of the minimum temperature of the \\gls{PT} mesh in the \\gls{TC}\\index{temperature chaos} (see \\refsec{correlation_dynamics_static}), where we will use the simulation with $N=13$.\n\n\n\n\n\\section{Monte Carlo, why have you forsaken me?}\\labsec{Monte_Carlo_forsaken}\nWe have already sketched the main idea motivating this work. Traditional Monte\\index{Monte Carlo} Carlo methods like the Metropolis-Hasting algorithm are not useful to study (at equilibrium) the low-temperature phase\\index{phase!low-temperature\/spin-glass} of a \\gls{SG} because the presence of many free-energy\\index{free energy!valley} local minima often causes the numerical simulation to get trapped and, as a consequence, the correct sampling of the phase space\\index{phase space} gets severely harmed.\n\nThe \\gls{PT} method solves this problem. The introduction of $N$ copies at different temperatures, and the possibility for each copy to exchange its temperature with another different copy, allows the copies to visit the high-temperature phase\\index{phase!high-temperature\/paramagnetic}, where it decorrelates very quickly from its previous state. When the copy \\textit{comes back} to the low-temperature phase\\index{phase!low-temperature\/spin-glass}, it visits another different free-energy\\index{free energy!valley} local minima and the performance of the thermalization\\index{thermalization} process boosts.\n\nThe \\gls{TC}\\index{temperature chaos} phenomenon dramatically decreases that performance. Intuitively one can understand why the \\gls{TC}\\index{temperature chaos} represents a major obstacle in the \\gls{PT} temperature flow~\\cite{janus:10,fernandez:13,martin-mayor:15,fernandez:16}. Imagine we have two sets of configurations\\index{configuration} (states) of two equilibrated systems at different temperatures $T_1<T_2$. If \\gls{TC}\\index{temperature chaos} occurs, then the typical configurations\\index{configuration} will be very different for both temperatures. In the simpler scenario, we have a level crossing in which at a given temperature the free-energy\\index{free energy} for both sets of configurations\\index{configuration} are almost equal but their specific heats\\index{specific heat} are rather different. Actually, this discrepancy in the value of the specific heat\\index{specific heat} is very harmful to the \\gls{PT} method (see, for example, \\cite{rathore:05,katzgraber:06b,sabo:08,malakis:13}). As a consequence, to place a $T_1$\\textit{-like} set of configurations\\index{configuration} at $T_2$ is possible, but, due to the specific heat\\index{specific heat} difference, it will have a hard traveling to higher temperatures (the inverse example is also true). In addition, equilibrium in \\gls{PT} implies equilibrium at all the temperatures, hence, even a single \\textit{dynamic chaotic event} as described above can ruin our expectations.\n\nBut these \\textit{a priori} inconveniences are not a real reason to be sad. Actually, generally in Markov\\index{Markov chain} chains, the exponential time $\\tau_{\\exp}$~\\cite{sokal:97} tells us how long we have to simulate the system in order to reach the equilibrium. Those samples\\index{sample} which suffer stronger \\gls{TC}\\index{temperature chaos} would also need longer times to thermalize and the study of the exponential time could help us to point those chaotic samples\\index{sample}~\\cite{fernandez:13,fernandez:16}. \n\nUnfortunately, $\\tau_{\\exp}$ has proven to be a very elusive quantity, and its accurate computation is not an easy task. It has been suggested that $\\tau_{\\exp}$ is best computed by studying the temperature-flow of the system copies of the \\gls{PT} simulation~\\cite{janus:10}. In the next section, we revisit the problem of the computation of $\\tau_{\\exp}$ and present a variational method\\index{variational method} that can potentially save a large amount of time.\n\n\\section{Time scales in a Markov chain}\\labsec{time_scales_eq_chaos}\nWe have stated in~\\refsubsec{Monte_Carlo} that a well-behaved Monte\\index{Monte Carlo} Carlo method, based on a Markov\\index{Markov chain} chain, assures the system to reach the equilibrium (i.e. the configurations\\index{configuration} of the system are distributed according to the Boltzmann-Gibbs\\index{Boltzmann!-Gibbs distribution} distribution), no matter which particular initial condition we choose [recall~\\refeq{thermalization_condition}]. However, knowing how long we need to simulate the system to reach that equilibrium is an important question.\n\nLet us consider an equilibrated system, for a given quantity $f$ its \\textit{autocorrelation function}\\index{correlation function!time auto-} would be\n\\begin{equation}\nC_f(t) = \\mcav{f(t_1)f(t_2)} - \\mcav{f(t_1)}^2 \\, , \\quad t=t_1 - t_2 \\, , \\labeq{unnormalized_autocorrelation_function}\n\\end{equation}\nwhere $\\mcav{\\cdots}$ stands for the expectation value\\footnote{It is worthy to note that $\\mcav{\\cdots}$ is different for the average $\\braket{\\cdots}$ defined in~\\refeq{average_o}. It is true that, for an equilibrated system, the Markov\\index{Markov chain} chain samples\\index{sample} the configurational\\index{configuration!space} space according to the Boltzmann-Gibbs\\index{Boltzmann!-Gibbs distribution} distribution and, therefore, $\\mcav{\\cdots}$ can estimate $\\braket{\\cdots}$, but it is not true in general for a non-equilibrated system. We stress the difference between them by adopting different notations.} of the quantity $f$. The required equilibrium condition implies that $\\mcav{f(t_1)} = \\mcav{f(t_2)}$. \n\nThe normalized autocorrelation function\\index{correlation function!time auto-} $\\hat{C}(f)$ is\n\\begin{equation}\n\\hat{C}_f(t) = \\dfrac{C_f(t)}{C_f(0)} \\, . \\labeq{normalized_autocorrelation_function}\n\\end{equation}\n\nThis quantity decays exponentially for large $t$ as $\\hat{C}_f(t) \\sim \\exp\\left(-\\abs{t}\/\\texpf\\right)$. For each quantity we can extract the value of $\\texpf$\n\\begin{equation}\n\\texpf = \\limsup_{t \\to \\infty} \\dfrac{t}{- \\log \\abs{\\hat{C}_f(t)}} \n\\end{equation}\nand the supreme of $\\texpf$ for every quantity $f$ \n\\begin{equation}\n\\texp = \\sup_f \\texpf \\, , \\labeq{exponential_autocorrelation_time}\n\\end{equation}\nis what we call \\textit{exponential autocorrelation time}\\index{autocorrelation time!exponential} which is the time scale ruling the transition of the system from an arbitrary configuration\\index{configuration} to the equilibrium, that is, this is the quantity we are looking for.\n\nNonetheless, there are two main difficulties in the computation of $\\texp$ from a practical point of view. The first one concerns the selection of the quantity $f$: we do not know which is the quantity $f$ that maximizes $\\texpf$. The second one is a technical problem: the fit of a numeric autocorrelation function\\index{correlation function!time auto-} to an arbitrary function from which we only know that its long-term behavior is an exponential decay is not an easy task. Procedures followed in previous works~\\cite{janus:10} suggest considering the autocorrelation function\\index{correlation function!time auto-} as a sum of two decaying exponentials from which we can extract the value of $\\texp$. This procedure is very delicate because a large number of simulated samples\\index{sample} requires an automatic way to obtain $\\texp$ and this automatic protocol can fail, requiring human intervention and, most of the time, demanding to extend the simulations. The full protocol is carefully detailed in Appendix A of \\cite{janus:10}.\n\nIt would be desirable to introduce a different time scale with better properties. At this point, we define the \\textit{integrated autocorrelation time}\\index{autocorrelation time!integrated} $\\tintf$\n\\begin{equation}\n\\tintf = \\dfrac{1}{2}+ \\sum_{t=1}^\\infty \\hat{C}_f(t) \\, . \\labeq{integrated_autocorrelation_time}\n\\end{equation}\n\nIt is not hard to prove~\\cite{sokal:97} that $\\tintf$ controls the statistical errors in measuring the quantity $f$. Specifically, if an equilibrated system generates one configuration\\index{configuration} at the Monte\\index{Monte Carlo} Carlo time $t$, we need to wait until $t + 2\\tintf$ to obtain a statistically independent configuration\\index{configuration}, but only as far as the quantity $f$ is concerned.\n\nThe normalized autocorrelation function\\index{correlation function!time auto-} can be expressed in terms of the eigenvalues $\\lambda_n$ of the transition probability matrix\\index{transition matrix} projected on the subspace orthogonal to its eigenvalue 1 ($1>|\\lambda_1|\\geq |\\lambda_2|\\geq\\ldots$), see Ref.~\\cite{sokal:97},\n\\begin{equation}\n\\hat C_f(t)=\\sum_n A_{n,f} \\lambda_n^{|t|}\\,,\\quad \\sum_n A_{n,f}=1\\,, \\labeq{autocorrelation_function_decomposition}\n\\end{equation}\nwhere the index $n$ runs from 1 to the size of the transition matrix\\index{transition matrix}, in our case $N_T!2^{N_TL^D} - 1$.\n\nThe amplitudes $A_{n,f}$  depend on $f$, while the $\\lambda_n$ are $f$-independent. We can plug~\\refeq{autocorrelation_function_decomposition} into \\refeq{integrated_autocorrelation_time} and, by computing the sum of a geometric series, we have\n\\begin{equation}\n\\tintf = \\dfrac{1}{2} + \\sum_n A_{n,f} \\dfrac{\\lambda_n}{1-\\lambda_n} \\, .\n\\end{equation}\n\nNow, in practical applications the (leading) $A_{n,f}$'s and $\\lambda_n$'s are real positive. Hence, $\\lambda_n=\\mathrm{e}^{-1\/\\tau_n}$ defines the characteristic time $\\tau_n$. The exponential autocorrelation time\\index{autocorrelation time!exponential} of the Markov\\index{Markov chain} chain $\\texp$ is just $\\tau_1$, the largest of the $\\tau_n$ (see \\cite{sokal:97}). Now, for $\\tau_n\\gg 1$ we can perform a Taylor expansion and we obtain $\\lambda_n\/(1-\\lambda_n)=\\tau_n - \\dfrac{1}{2} + \\mathcal{O}(1\/\\tau_n)$. \\refeq{normalized_autocorrelation_function} and \\refeq{integrated_autocorrelation_time} become\n\\begin{equation}\n\\hat{C}_f(t) = \\sum_n A_{n,f} e^{-\\abs{t}\/\\tau_n} \\>\\> , \\>\\> \\tintf = \\sum_n A_{n,f} \\tau_n \\, . \\labeq{autocorrelation_decomposition}\n\\end{equation}\nThe integrated autocorrelation\\index{autocorrelation time!integrated} time $\\tintf$ for the quantity $f$ is just an average of the decay modes of the correlation function, being the slower mode the exponential autocorrelation time\\index{autocorrelation time!exponential} $\\texp$ and the weights of that average the coefficients $A_{n,f}$. It is straightforward to prove that \n\\begin{equation}\n\\tintf \\leq \\texp \\, , \\labeq{inequality_autocorrelation_time}\n\\end{equation}\nand the equality is reached when $A_{1,f} = 1$.\n\n$\\tintf$ is the quantity we were looking for because, although it has the same disadvantage that $\\texpf$ with respect to the chosen $f$, it is much simpler to compute. In addition, in this work we overcome the problem of the quantity $f$ by using a variational method\\index{variational method} very similar to the Rayleigh-Ritz variational principle in Quantum Mechanics.\n\nThe last consideration has to be done before explaining our variational method\\index{variational method}. In this work, although we have proposed a new thermalization\\index{thermalization} protocol, we needed to compare our results with the reliable results provided by the computation of the exponential autocorrelation time\\index{autocorrelation time!exponential}. The thermal protocol followed here is the same described in appendix A of~\\cite{janus:10}.\n\n\\section{The variational method: dynamic Temperature Chaos} \\labsec{variational_method_eq_chaos}\nThis sections aims to describe the variational method\\index{variational method} used to compute our estimation of $\\texp$. The idea is very simple, from~\\refeq{autocorrelation_decomposition} and~\\refeq{inequality_autocorrelation_time} we can deduce that a very good estimation of $\\texp$ from $\\tintf$ would imply to choose a quantity $f$ with $A_{1,f} \\approx 1$ and $A_{n>1,f} \\approx 0$. As we have already introduce above, it has been suggested that we should focus on the temperature flow along the \\gls{PT} dynamics in order to compute the autocorrelation function\\index{correlation function!time auto-}~\\cite{janus:10,fernandez:13,martin-mayor:15}.\n\nThe computations of time autocorrelation functions\\index{correlation function!time auto-} are usually performed with spin-configuration\\index{configuration} functions $f$. In our protocol, we focus on the temperature random-walk\\index{random walk!temperature} that each copy of the system performs over the temperature mesh in the \\gls{PT} dynamics. At the first sight, it might be surprising that this temperature random-walk\\index{random walk!temperature} can provide information about the thermalization\\index{thermalization} of the system. The reader may find in \\refsec{thermalizing_PT} a detailed discussion about this fact.\n\nLet us consider, for a given sample\\index{sample}, the $N$ system copies in the \\gls{PT} dynamics. Our Markov\\index{Markov chain} chain will be the temperature random-walk\\index{random walk!temperature} through the index $i_t$ that indicates that, at time $t$, our system copy is at temperature $T_{i_t}$. At equilibrium, all the copies spend the same time in each temperature and, therefore, the probability for $i_t$ is just the uniform probability over the set $\\{1,2,\\dots,N \\}$. In consequence, the expectation value of the quantity $f$ is just the arithmetic mean\n\\begin{equation}\n\\mcav{f} = \\dfrac{1}{N} \\sum_{i=0}^N f(i) \\, . \\labeq{expectation_value_f}\n\\end{equation} \nWe shall consider, as well, functions that depends on pairs of system copies. For a given time $t$, these pairs will be described by two indices $i_t \\neq j_t$. The equilibrium value for an arbitrary function of a pair of system copies is\n\\begin{equation}\n\\mcav{f_{\\mathrm{pairs}}} = \\dfrac{1}{N(N-1)}\\sum_{i=0}^N \\sum_{j\\neq i}^N f_{\\mathrm{pairs}}(i,j) \\, .\n\\end{equation}\nBy looking at the definition of the time autocorrelation function\\index{correlation function!time auto-} in~\\refeq{unnormalized_autocorrelation_function} it is clear that, for computational purposes, it is convenient to define a function $f$ with $\\mcav{f}=0$. Therefore, for each function $f$ we define\n\\begin{equation}\n\\tilde{f} = f - \\mcav{f} \\, . \\labeq{f_expectation_0}\n\\end{equation}\nWe can define now our (not normalized) time autocorrelation function\\index{correlation function!time auto-} as\n\\begin{equation}\nC_f(t) = \\dfrac{1}{N_s-t_0-t} \\sum_{t'=t_0}^{N_s-t} \\tilde{f}(i_{t'})\\tilde{f}(i_{t'+t}) \\, ,\n\\end{equation}\nwhere $N_s$ is the total number of times we have \\textit{measured} the state of the \\gls{PT} indices $i_t$. Here, of course, we are considering the equilibrium autocorrelation function\\index{correlation function!time auto-} and, therefore, $t_0 \\gg \\tintf$\\footnote{The reader may find this statement a little bit contradictory since we are trying to estimate $\\tintf$, however a self-consistent procedure is followed, similarly to that one explained in~\\cite{janus:10}.}. Note that $C_f(t)$ is independent of the system copy as well as of the replica\\index{replica} and, hence, we can improve our statistics by averaging over the $N \\times \\Nrep$ numerical estimations of $C_f(t)$. All the statements for $f$ depending on a single copy are straightforwardly transferable to functions depending of a pair of system copies. \n\nOnce we have an estimation of $C_f(t)$ we can estimate the integrated autocorrelation\\index{autocorrelation time!integrated} time\n\\begin{equation}\n\\tintf \\approx \\nmet \\left( \\dfrac{1}{2} + \\sum_{t=0}^W \\hat{C}_f(t) \\right) \\, , \\labeq{window_integrated_autocorrelation_time}\n\\end{equation}\nwhere $\\nmet$ is the periodicity with which we record the time indices $i_t$ of our random walker\\index{random walk} and $\\hat{C}_f(t) = C_f(t)\/C_f(0)$ is the normalized autocorrelation function. In our simulations $\\nmet=25000$ Metropolis sweeps most of the times. The  original definition of $\\tintf$ [see \\refeq{integrated_autocorrelation_time}] involves an infinite sum, here we restrict the sum only to the first $W$ values, being $W$ a self-consistent window (see \\cite{sokal:97}) that avoids the divergence of the variance of $\\tintf$. We impose $\\tintf < 10W$.\n\nIn order to compute $\\tintf$ as closely as possible to the $\\texp$ value, we consider three different parameters to optimize: the type of function $f$, the temperature $T^*$ at which $f$ is zero, and a Wilson-Kadanoff\\index{Wilson-Kadanoff} renormalization\\index{renormalization group} block length $\\lblo$. We describe here the three parameters.\n\n\\begin{table}\n\\centering\n\\begin{tabular*}{0.6\\columnwidth}{@{\\extracolsep{\\fill}}cc}\n\\toprule\n\\toprule\n\\textbf{Identifier} & \\textbf{Function} \\\\\n\\toprule\n$0$ & piecewise constant \\\\\n$1$ & piecewise linear\\\\\n$2$ & piecewise quadratic\\\\\n$3$ & piecewise cubic\\\\\n$|$ & OR in couples\\\\\n$\\&$ & AND in couples\\\\\n$\\wedge$ &  XOR in couples\\\\\n$*$ &  Multiplication in couples \\\\\n\\bottomrule\n\\end{tabular*}\n\\caption[\\textbf{Functions of the variational method.}]{\\textbf{Functions of the variational method\\index{variational method}.} Different choices of the function $f$ used in the variational method\\index{variational method}.}\n\\labtab{functions_variational_method}\n\\end{table}\n\n\\begin{itemize}\n\\item \\textbf{The type of function \\boldmath $f$.} Similarly to the Rayleigh-Ritz variational principle in Quantum Mechanics we consider test-functions $f$, belonging to eight different classes (see~\\reftab{functions_variational_method}). The four first functions depend only on a single system copy. Specifically, the function labeled as $0$ is just the complementary of the Heaviside function $1-\\Theta(T^*)$. The function labeled as $1$ is a piecewise linear function that has already been used before in~\\cite{janus:10}. As is quite evident by the description, the functions labeled as $2$ and $3$ are quadratic and cubic piecewise functions respectively. We will specify their specific functional form below. \n\nThe last four functions depend on two system copies. For the functions labeled as $\\lvert$, $\\&$ and $\\wedge$, each system copy of the pair has associated the value of the function labeled as $0$. Then, the value of the function is the corresponding binary operation of the pair of values obtained for each system copy. Finally, the $*$ function is just the multiplication of the piecewise linear function for each system copy (with the corresponding normalization).\n\n\\item \\textbf{The temperature $\\mathbf{T^*}$.} We require for the temperature $T^* \\in \\{T_1,T_2, \\dots, T_{N\/2}\\}$ that $f(T^*) = 0$. The value of $T^*$ is our second variational parameter. In addition, the condition $\\mcav{f_{T^*}}=0$ together with $f(T^*) = 0$ define our test-functions. Specifically, the linear piecewise function is\n\\begin{equation}\n\\begin{aligned}\nT > T^*\\,&: \\quad &f_{T^*}(T) = a_+ (T-T^*) \\, ,\\\\\nT < T^*\\,&: \\quad &f_{T^*}(T) = a_- (T-T^*) \\, .\n\\end{aligned}\n\\end{equation}\nWe require $a_+$ and $a_-$ to be positive and their ratio is fixed by the condition $\\mcav{f_{T^*}}=0$. Indeed, we only need to fix the ratio, because the overall scale of the test function $f_{T^*}$ is irrelevant.\n\nWith an analogous procedure we can define the quadratic ($p=1$) and the cubic ($p=2$) functions\n\\begin{equation}\n\\begin{aligned}\nT>T^* \\, : \\quad f_{T^*}(T) = a_+ (T-T^*)^p (2T_N - T^* -T) \\, ,\\\\\nT<T^* \\, : \\quad f_{T^*}(T) = a_- (T-T^*)^p (2T_1 - T^* -T) \\, .\\\\\n\\end{aligned}\n\\end{equation}\nAgain, we choose $a_+>0$ and $a_->0$ and the ratio is fixed by $\\mcav{f_{T^*}}=0$. We try all the possible values of $T^*$ in the lower half part of the set of temperatures in our \\gls{PT} simulation.\n\n\\item \\textbf{The renormalization\\index{renormalization group} time-block \\boldmath $\\lblo$.} We can modify the value of the time autocorrelation function\\index{correlation function!time auto-} by changing the function $f$ itself (as it has been introduced in the two previous parameters of the variational method\\index{variational method}) but we can also modify it by changing the temporal series from which we compute that autocorrelation function\\index{correlation function!time auto-}. We build Wilson-Kadanoff\\index{Wilson-Kadanoff} blocks: the Monte\\index{Monte Carlo} Carlo sequence $f_{T^*}(i_1),f_{T^*}(i_2),\\dots,f_{T^*}(i_{N_s})$ is divided into blocks of $\\lblo$ consecutive data (see e.g. \\cite{amit:05}). For each block, we compute the average and we build a new sequence $f'_{T^*}(j_1),f'_{T^*}(j_2),\\dots,f'_{T^*}(j_{N_s\/\\lblo})$ from which we compute the integrated autocorrelation time\\index{autocorrelation time!integrated} just as we did for $\\lblo=1$. Of course, after computing the integrated autocorrelation time\\index{autocorrelation time!integrated}, we need to rescale it to recover the original time units. The idea of this parameter is to reduce the high-frequency fluctuations of the time autocorrelation function\\index{correlation function!time auto-}.\n\nHowever, the parameter $\\lblo$ needs to be controlled, otherwise, it can become larger than $\\tintfTlblo$ erasing all the information of the autocorrelation function\\index{correlation function!time auto-} and giving us spurious results. If that happens, each block would be just the expectation value (well, a finite estimation) of the function $f$, and the autocorrelation function\\index{correlation function!time auto-} would vanish for times $t\\neq 0$. From~\\refeq{window_integrated_autocorrelation_time} we deduce that, after converting $\\tintf$ to the correct time units, we would have $\\tintfTlblo = \\lblo \\nmet\/2$, which diverges with $\\lblo$. With the aim to control that spurious effect, we impose\n\\begin{equation}\n\\tintfTlblo > \\dfrac{5}{2} \\nmet \\lblo \\, . \\labeq{tintfTlblo_condition}\n\\end{equation}\nThe values of $\\lblo$ are taken from the list $\\lblo =\\{1,2,5,10,20,50,100,200,500,1000$\n$,2000\\}$.\n\\end{itemize}\n\nOur estimation of the integrated autocorrelation time\\index{autocorrelation time!integrated}, namely $\\tintvar$, is just the highest value of all the $\\tintfTlblo$. This estimation has great advantages from its predecessor (which corresponds with the linear piecewise function at the critical\\index{critical temperature} temperature and $\\lblo=1$). Firstly, our estimation is robust in the sense that it does not produce spurious values. Moreover, the process can be easily implemented in an automatic way which is a \\textit{sine qua non} condition given the huge number of possible combinations of parameters.\n\nThe worse scenario would be to found that our effort has been in vain and the automatic process always chooses the piecewise linear function with $T^* = \\ensuremath{T_\\mathrm{c}}\\xspace$ and $\\lblo=1$. Fortunately, this is not the case, in~\\reftab{frequency_functions_variational_method} we can see the numbers of times that our method chooses each function. Indeed, it is notorious that almost all of the time, the method chooses a single-copy function. The same happens with $T^*$, the chosen value is not always $\\ensuremath{T_\\mathrm{c}}\\xspace$, actually, we called the chosen temperature the \\textit{dynamic chaotic temperature} $T_d$ that will be useful in the following analysis. The effects of $\\lblo$ or, more accurately, the effects of the discretization of $\\lblo$ can also be observed in the results (for example \\reffig{log_tau_I}). Specifically, the low density in the $\\lblo$ mesh leads to small gaps in the determination of the autocorrelation time\\index{autocorrelation time!integrated} $\\tintvar$.\n\n\\begin{table}\n\\centering\n\\begin{tabular*}{0.8\\columnwidth}{@{\\extracolsep{\\fill}}cccccccccc}\n\\toprule\n\\toprule\n$L$ & $0$ & $1$ & $2$ & $3$ & $|$ & $\\&$ & $\\wedge$ & $*$ & Total  \\\\\n\\toprule\n$16$ & $2032$ & $5320$ &  $3875$ & $1374$  & $4$ & $115$ & $74$ & $6$ & $12800$ \\\\\n$24$ & $1556$ & $7196$ &  $3089$ & $820$  & $0$ & $127$ & $11$ & $1$ & $12800$ \\\\\n\\bottomrule\n\\end{tabular*}\n\\caption[\\textbf{Frequency of choice of the variational method.}]{\\textbf{Frequency of choice of the variational method\\index{variational method}.} Number of times the variational method\\index{variational method} has picked one of the eight choices among the functions $f$ described in the text. $L$ denotes the lattice size.}\n\\labtab{frequency_functions_variational_method}\n\\end{table}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{eq_chaos\/func_corr}\n\\caption[\\textbf{Improvement of the estimation of the time autocorrelation functions.}]{\\textbf{Improvement of the estimation of the time autocorrelation functions.}\\index{correlation function!time auto-} Auto-correlation function for the most chaotic sample\\index{sample} for $L=16$ (left) and $L=24$ (right): (Top) Auto-correlation function computed using the method of \\cite{janus:10} and (Bottom) using the variational method\\index{variational method} presented here. Note that the improvement of the new method is notorious is we focus on the y-intercept (further details can be found in the text). \\textbf{Inset:} Linear-log plot showing the small $t$ behavior of the autocorrelation function\\index{correlation function!time auto-}.}\n\\labfig{example_autocorrelation_function}\n\\end{figure}\n\nAnother question has to be answered. It is true that our method chooses a variety of parameters, to the detriment of the classical choice but, is there a significant improvement in the estimation of $\\texp$ or are all the estimations just small fluctuations of the previous estimation $\\tint$? An example of the improvement obtained in the computation of the autocorrelation function\\index{correlation function!time auto-} is shown in~\\reffig{example_autocorrelation_function}. The main problem of the previous estimation becomes obvious from the figure: the value of $A_{1,f}$ [see~\\refeq{autocorrelation_decomposition}] could be, indeed, fairly small $A_{1,f} \\approx 0.1$. In the figure, this amplitude roughly corresponds to the abscissa of the initial point of the linear decreasing that we are able to see in the log-log scale (i.e. the beginning of the domination of the large time-scale corresponding to $\\texp$). Our new estimations (bottom panels) are rather better.\n\nThis hand-waving argument can be made quantitative. Let us denominate $\\tintold$ to the methodology of estimation of $\\tint$ for previous works~\\cite{janus:10} and $\\tintvar$ to our variational-method\\index{variational method} estimation. In~\\reffig{histograma_taus_multiplot_g} we separate our samples\\index{sample} in deciles according to its $\\tintvar$ value so that the first decile corresponds to the $1280$ samples\\index{sample} with smaller $\\tintvar$. We have argued that the most chaotic samples\\index{sample} will have larger $\\texp$, so those deciles are our proposal for separating the \\textit{most chaotic} and \\textit{less chaotic} samples\\index{sample}.\n\nThen, we build up the histogram of the ratio $\\tintold\/\\tintvar$ for the samples\\index{sample} on a given decile. Top panels in~\\reffig{histograma_taus_multiplot_g} show that the gain of considering $\\tintvar$ is sizable but, if we focus on the most chaotic samples\\index{sample} (i.e. the tenth decile, in the bottom panels) the benefits of our variational method\\index{variational method} are more than evident with a significant fraction of the samples\\index{sample} with $\\tintold\/\\tintvar<0.1$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{eq_chaos\/histograma_taus_multiplot_g}\n\\caption[\\textbf{A quantitative argument for the variational method.}]{\\textbf{A quantitative argument for the variational method\\index{variational method}.} Conditional probability density function of the ratio $\\tintold\/\\tintvar$, given that $\\tintvar$ belongs to a given decile is plotted. We show the data for the first decile (left) and the tenth decile (right) for $L=16$ (top) and $L=24$ (bottom).}\n\\labfig{histograma_taus_multiplot_g}\n\\end{figure}\n\n\n\n\\section{Static Temperature Chaos}\nAt this point, we momentarily forget about previous disquisitions on \\gls{TC}\\index{temperature chaos} from the dynamical point of view and we come back to the classical static definition: \\gls{TC}\\index{temperature chaos} is the complete rearrangement of the equilibrium configurations\\index{configuration} upon any change of temperature. This phenomenon has been traditionally studied~\\cite{billoire:00,billoire:02,katzgraber:07} through the probability density function of the overlap\\index{overlap!distribution} between the spin configurations\\index{configuration} at temperatures $T_1$ and $T_2$,\n\\begin{equation}\nq_{T_1,T_2} = \\dfrac{1}{V} \\sum_i s_i^{T_1} s_i^{T_2} \\, . \\labeq{overlap_2T}\n\\end{equation}\nDue to the limitation of the maximum size that can be simulated, this quantity is strongly affected by finite-size effects\\index{finite-size effects}. We focus therefore on other quantity, introduced in~\\cite{ritort:94}: the \\textit{chaotic parameter}\n\\begin{equation}\n\\xchaos = \\dfrac{\\braket{q^2_{T_1,T_2}}_J}{\\sqrt{\\braket{q^2_{T_1,T_1}}_J\\braket{q^2_{T_2,T_2}}_J}} \\, , \\labeq{chaotic_parameter}\n\\end{equation}\nwhere $\\braket{\\cdots}_J$ stands for the usual thermal average but we stress the sample\\index{sample} dependency with the sub-index $J$. It has been proposed that the \\gls{TC}\\index{temperature chaos} phenomenon should be studied through a detailed analysis of the distribution of this sample-dependent chaotic parameter~\\cite{fernandez:13,billoire:14}.\n\nThe reader should notice that $0 < \\xchaos \\lesssim 1$. The extreme values are clear; $\\xchaos = 1$ means that both configurations\\index{configuration} at temperatures $T_1$ and $T_2$ are indistinguishable i.e. absence of chaos. On the contrary, $\\xchaos = 0$ means that both configurations\\index{configuration} are completely different which would indicate strong chaos.\n\nWe select the most chaotic samples\\index{sample} and the less chaotic ones accordingly to the estimation $\\tintvar$ and we plot in~\\reffig{X_TminT} their chaotic parameter as a function of temperature by keeping $T_1$ to the lower simulated temperature $T_{\\min}$ and varying $T_2$. It is clear that qualitative different behaviors on the quantity $X_{T_{\\min},T}^J$ are present in both sets. The less chaotic ones tend to decrease smoothly as $T_2$ increases while the more chaotic ones suffer sharp drops at well-defined temperatures, namely \\textit{chaotic events}. In addition, it was empirically observed~\\cite{fernandez:13} that chaotic events occurring at low temperatures are more harmful to the performance of \\gls{PT}. With this information in mind, we are looking for a single number that could quantify the \\textit{chaoticity} of a given sample\\index{sample}. The introduced observable~\\cite{fernandez:13} was the chaotic integral\n\\begin{equation}\nI = \\int_{T_{\\min}}^{T_{\\max}} \\xchaosmin dT_2 \\, . \\labeq{chaotic_integral}\n\\end{equation}\nThis quantity will be smaller if the sample\\index{sample} suffers a chaotic event that ``cuts'' the integral. Moreover, for the chaotic samples\\index{sample} is usual that, once the chaotic event takes place at temperature $T^*$, the chaotic signal for temperatures $T>T^*$ is low but the fluctuations of the value are still present. To minimize this effect, we propose the parameter $I_2$ that reduces the integration range to the first half of the simulated temperatures.\n\nFinally, looking at~\\reffig{X_TminT}, we noticed that some samples\\index{sample} presented strong decays but then, they maintain a relatively high value of the chaotic parameter for higher temperatures (for example, look at the purple curves in the top panels). To take into account that effect we define the quantity\n\\begin{equation}\nK_i = 1-X_{T_{i},T_{i+1}} \\, , \\labeq{finite_differences}\n\\end{equation} \nwhich is essentially the finite difference of two consecutive points in the curve. After some trials based on heuristic arguments and after seeing a lot of $\\xchaosmin$ vs $T$ curves, we define the quantity\n\\begin{equation}\nI_X = aI_2 - b \\min_i \\left( -\\log K_i^2\\right) - c \\sum_i \\left(-\\log K_i^2 \\right) \\, ,\n\\end{equation}\nwhere the coefficients $a$, $b$ and $c$, that depend on the lattice size $L$, are obtained through a minimization of the correlation Pearson coefficient $r$ between $I_X$ and $\\log(\\tintvar)$ (as we will discuss in the following section).\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{eq_chaos\/X_TminT}\n\\caption[\\textbf{Temperature dependence of the chaotic parameter.}]{\\textbf{Temperature dependence of the chaotic parameter.} Plot of $X_{T_{\\min},T}^J$ versus $T$ for the five most chaotic samples\\index{sample} (top) and the five less chaotic ones (bottom): $L=16$ case (left) and $L=24$ case (right).}\n\\labfig{X_TminT}\n\\end{figure}\n\n\\section{Correlation dynamics-statics} \\labsec{correlation_dynamics_static}\nWe study here the correlation between the static characterization of chaos through the previously defined observables and the dynamic one, through the autocorrelation time\\index{autocorrelation time!integrated} $\\tintvar$ that we will call from now on, simply $\\tint$. \n\nWe also find it useful to address the failures in the process of finding quantities relating to both perspectives. Usually, space requirements in publications or the clarity of the message make the rules of choosing the appropriate results to show, and it is perfectly reasonable. However, this privileged format allows us to extend a little bit more and show the path of the research which often contains failures. In addition, addressing those quantities not related to chaos would be also helpful for future works.\n\\subsection{The failures}\nHere, we address some \\textit{a priori} reasonable quantities that turned out to be not related with \\gls{TC}\\index{temperature chaos}.\n\nFirst, we recall the previously defined $T_d$ which is the temperature $T^*$ chosen by the variational method\\index{variational method}. We compute also, from the static characterization, the temperature $T_s$ at which the \\textit{bigger} chaotic event occurs\\footnote{Note that, for the less chaotic samples\\index{sample} with a smooth decay of the function $\\xchaosmin$ against $T_2$, this chaotic event can be fairly small.} i.e. the temperature for which $\\xchaosmin$ presents the maximum (negative) slope.\n\nThe correlation between both quantities is almost absent as can be seen in~\\reffig{Td_Ts}. We can see an over-density, however, the points out of the principal density are too dispersed. For $L=16$ (top) the number of points within the lines is $8017$ ($62.63 \\%$ of the total) while for $L=24$ (bottom) the number of points within the lines is $7539$ ($58.90 \\%$ of the total). If we compute the correlation coefficients we obtain~\\reftab{coef_correla_TdTs}. The errors in the determination of the correlation coefficients are computed from the Bootstrap\\index{Bootstrap} method (see \\refsec{estimating_errorbars}).\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.6\\columnwidth]{eq_chaos\/Td_Ts_g}\n\\caption[\\textbf{\\boldmath Scatter plot of $T_d$ versus $T_s$.}]{\\textbf{Scatter plot of $\\mathbf{T_d}$ versus $\\mathbf{T_s}$.} We present the $L=16$-data (top) and the $L=24$-ones (bottom). Points are calculated with a special procedure. First, samples\\index{sample} are classified on deciles according to $\\log(\\tau_{\\mathrm{int}})$. The points coordinates were obtained by computing the median $T_d$ and the median $T_s$ within each decile (errors from Bootstrap\\index{Bootstrap}). The golden parallel lines enclose the area of over-density that presents a higher correlation for later recount.}\n\\labfig{Td_Ts} \n\\end{figure}\n\n\n\\begin{table}\n\\centering\n\\begin{tabular*}{0.5\\columnwidth}{@{\\extracolsep{\\fill}} ccccc}\n\\toprule\n\\toprule\n&$L$  & & $r$ & \\\\\n\\toprule\n&$16$ & & $0.348 \\pm 0.008$ &\\\\\n&$24$ & & $ 0.342 \\pm 0.007$ & \\\\\n\\bottomrule\n\\end{tabular*}\n\\caption[\\textbf{\\boldmath $T_d$ vs $T_s$.}]{\\textbf{$\\mathbf{T_d}$ vs $\\mathbf{T_s}$.} Correlation coefficients of the scatter plot of $T_d$ against $T_s$ for the simulated two lattice sizes.}\n\\labtab{coef_correla_TdTs}\n\\end{table}\n\n\nWe can try to relate $T_s$ to other dynamic estimation of chaos, for example, the integrated autocorrelation time\\index{autocorrelation time!integrated} $\\tint$. Unfortunately, although slightly better than our previous attempt, we observe a weak correlation between both estimators, $\\tint$ and $T_s$, (see \\reffig{log_tau_Ts}) and we can check it quantitatively through \\reftab{coef_correla_Ts}.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.6\\columnwidth]{eq_chaos\/log_tau_Ts_g}\n\\caption[\\textbf{\\boldmath Scatter plot of $\\log(\\tint)$ against $T_s$.}]{\\textbf{Scatter plot of $\\mathbf{\\log(\\tint)}$ against $\\mathbf{T_s}$.} We show $L = 16$ (top) and $L=24$ (bottom).}\n\\labfig{log_tau_Ts} \n\\end{figure}\n\n\\begin{table}\n\\centering\n\\begin{tabular*}{0.5\\columnwidth}{@{\\extracolsep{\\fill}}ccccc}\n\\toprule\n\\toprule\n&$L$   && $r$ & \\\\\n\\toprule\n&$16$  && $-0.621 \\pm 0.006$ &\\\\\n&$24$  && $ -0.621 \\pm 0.006$ &\\\\\n\\bottomrule\n\\end{tabular*}\n\\caption[\\textbf{Correlation coefficients for the scatter plot of $\\mathbf{\\log(\\tau_{\\mathrm{int}})}$ versus $\\mathbf{T_s}$ for the two simulated lattice sizes.}]{\\textbf{Correlation coefficients for the scatter plot of $\\mathbf{\\log(\\tau_{\\mathrm{int}})}$ versus $\\mathbf{T_s}$ for the two simulated lattice sizes.}}\n\\labtab{coef_correla_Ts}\n\\end{table}\n\n\n\\subsection{The success}\nHere, we present our most successful attempts to relate both dynamic and static characterization of chaos. In \\reffig{log_tau_I}, we confront the most representative\nestimator for the dynamical chaos, namely the largest integrated autocorrelation time\\index{autocorrelation time!integrated} $\\tint$ found in our variational\\index{variational method} study, with the static chaotic integrals $I$, $I_2$ and $I_X$. We can observe how spurious values of the original parameter $I$ (i.e. large values of $I$ associated with large $\\tau_\\mathrm{int}$) are displaced towards lower values when we use the improved parameters $I_2$ and $I_X$.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{eq_chaos\/log_tau_I_g}\n\\caption[\\textbf{Scatter plot of $\\mathbf{\\log(\\tintvar)}$ versus integrated chaotic parameters.}]{\\textbf{Scatter plot of $\\mathbf{\\log(\\tintvar)}$ versus integrated chaotic parameters.} We present data for two lattice sizes and for the three definitions of the integrated chaotic parameter defined in the text ($I, I_2$ and $I_X$). The pattern of depleted horizontal bands is due to our choice of a few $l_\\mathrm{blo}$.}\n\\labfig{log_tau_I}\n\\end{figure}\n\nThe value of the correlation coefficients is reported in~\\reftab{coef_correla} (as before, the errors are computed by using a Bootstrap\\index{Bootstrap} method, see \\refsec{estimating_errorbars}). We observe a strong anti-correlation in  $I_X$, which improves over the previous indicator of correlation $I$. \\cite{fernandez:13} The improvement is less clear for $I_2$.\n\n\\begin{table}\n\\centering\n\\begin{tabular*}{0.6\\columnwidth}{@{\\extracolsep{\\fill}}ccccc}\n\\toprule\n\\toprule\n&$L$ & Integral & $r$  \\\\\n\\toprule\n&$16$ & $I$   & $-0.714 \\pm 0.005$ \\\\\n&$16$ & $I_2$ & $-0.751 \\pm 0.005$ \\\\\n&$16$ & $I_X$ & $-0.795 \\pm 0.004$ \\\\\n\\toprule\n&$24$ & $I$   & $-0.725 \\pm 0.005$ \\\\\n&$24$ & $I_2$ & $-0.746 \\pm 0.005$ \\\\\n&$24$ & $I_X$ & $-0.786 \\pm 0.004$ \\\\\n\\bottomrule\n\\end{tabular*}\n\\caption[\\textbf{Correlation coefficients for $\\mathbf{\\log(\\tint)}$ versus the integrated chaotic parameters.}]{\\textbf{Correlation coefficients for $\\mathbf{\\log(\\tint)}$ versus the integrated chaotic parameters.} Correlation coefficients are shown for each two lattice sizes and for the three definitions of the parameter ($I, I_2$ and $I_X$).}\n\\labtab{coef_correla}\n\\end{table}\n\n\\section{Finite size scaling} \\labsec{scaling_eq_chaos}\n\\index{finite size scaling}\nThis section is devoted to study the size-scaling behavior of the dynamic characterization of \\gls{TC}\\index{temperature chaos}. It has been observed \\cite{fernandez:13} that chaotic events are less common in small systems. This suggests a large $L$ limit for the chaotic behavior that we investigate here.\n\nAn implicit assumption of our study is that the scaling behavior of $\\tint$ is mostly decided by the value $\\Tmin$. Other details, such as the number of temperatures in the \\gls{PT} mesh, are expected to play a minor role (if kept in a reasonable range). Our choice of simulated parameters (see, \\reftab{parameters_simulation_MUSA_MUSI}) does not allow us to check the impact of the number of temperatures in the \\gls{PT} mesh, but we can justify that $\\Tmin$ has a deep impact on the determination of $\\tint$.\n\n\\subsection{Temperature chaos depends on \\boldmath $\\Tmin$}\nIn order to study how the range of temperatures in the \\gls{PT} affects the dynamics, we have confronted both simulations for $L=16$, one with $N=13$ and $\\Tmin=0.698$, the other one with $N=16$ and $\\Tmin=0.479$. We need to increase the number of temperatures $N$ in the mesh to keep the interval between adjacent temperatures fixed.\n\nSince the simulation with $N = 16$ reaches a lower minimum temperature than the simulation with $N = 13$, we expect to find chaos events (i.e a jam in the \\gls{PT} temperature flow) that the simulation with $N=13$ cannot ``see''. In~\\reffig{cociente_taus} we show a scatter plot of $\\log(\\tau_{\\mathrm{int},16}\/\\tau_{\\mathrm{int},13})$ versus $T_d$ for the 12800 samples\\index{sample} ($\\tau_{\\mathrm{int},16}$ and $\\tau_{\\mathrm{int},13}$ are the autocorrelation times for $N=16$ and $N=13$ respectively). \n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{eq_chaos\/cociente_taus_g}\n\\caption[\\textbf{Scatter plot of $\\mathbf{\\log({\\tau_{\\mathrm{int,}16}}\/{\\tau_{\\mathrm{int,}13}})}$ versus $\\mathbf{T_d}$.}]{\\textbf{Scatter plot of $\\mathbf{\\log({\\tau_{\\mathrm{int,}16}}\/{\\tau_{\\mathrm{int,}13}})}$ versus $\\mathbf{T_d}$.} The lattice size is $L=16$, $\\tau_{\\mathrm{int},16}$ is the relaxation\\index{relaxation} time for $N=16$ ($T_\\mathrm{min}=0.479$), $\\tau_{\\mathrm{int},13}$ is the relaxation\\index{relaxation} time for $N=13$ ($T_\\mathrm{min}=0.698$), $T_d$ is the temperature of chaos from a dynamical point of view (defined in the variational method\\index{variational method}) of the simulation with $N = 16$. Both simulations have the same number of disorder\\index{disorder} samples\\index{sample}. The vertical black line represents the minimum temperature simulated in the $N= 13$ simulation. (We added a small Gaussian\\index{Gaussian!noise} white noise to $T_d$, which is a discrete variable, to avoid the cluttering of data in vertical lines). }\n\\labfig{cociente_taus}\n\\end{figure}\n\nFor $T_d>0.698$ the ratio takes values of order one for most samples\\index{sample}, while for $T_d < 0.698$ there is a huge number of samples\\index{sample} with $\\tau_{\\mathrm{int},16} \\gg \\tau_{\\mathrm{int},13}$,\ni.e. there are a lot of samples\\index{sample} with a chaotic behavior in a temperature-range  below $T_\\mathrm{min}=0.698$.\n\nThe same idea can be analyzed from a different point of view. Imagine that we have studied with great care a given sample\\index{sample} down to some temperature $T_\\mathrm{min}$. Can we say something about possible chaotic effects at lower temperatures? The question is answered negatively in \\reffig{no-prediction-from-small-Tmin}: the probability that a sample\\index{sample} has a large $\\tau_{\\mathrm{int}}$ for the simulation with a lower $T_\\mathrm{min}$ is not correlated to the value of $\\tau_{\\mathrm{int}}$ for the first simulation.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{eq_chaos\/dist_tau_quintiles_g}\n\\caption[\\textbf{Conditional probability distribution of $\\mathbf{\\tint}$.}]{\\textbf{Conditional probability distribution of $\\mathbf{\\tint}$.} The empirical probability distribution as a function of $\\tint$ for the $N=16$ simulation, conditional to the $\\tint$ obtained from $N=13$ simulation belonging to a given quintile. The non-conditional probability distribution function is also shown ($L_{16}$ curve).  \\textbf{Inset.} Blowup of the top right part of the main figure. For the hard samples\\index{sample}, the simulation with $T_\\mathrm{min}=0.698$ conveys little or no information on the difficulty of the $T_\\mathrm{min}=0.479$ simulation.}\n\\labfig{no-prediction-from-small-Tmin}\n\\end{figure}\n\n\\subsection{The scaling}\nWe have discussed that $\\Tmin$ has a great impact on the $\\tint$ value, therefore, we need to fix the same $\\Tmin$ for all the simulations in order to establish fair comparisons. We study the \\gls{PT} dynamics for $L=8,12,16,24$ and $32$ with $\\Tmin\\approx 0.7$. An important advantage of $T_\\mathrm{min}\\approx 0.7$ is that \\gls{TC}\\index{temperature chaos} has been already characterized at such temperatures, in the equilibrium setting~\\cite{fernandez:13}. Lowering $T_\\mathrm{min}$ would increase chaos effects, which would have been good in principle, but it would have been also extremely difficult to reach thermal equilibrium. Instead, increasing $T_\\mathrm{min}$ to approach the critical point would make the results irrelevant, because samples\\index{sample} displaying \\gls{TC}\\index{temperature chaos} would be too scarce (besides, we want to study the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass}, rather than critical effects).\n\nThe $L=32$ data are from Ref.~\\cite{janus:10} and have been obtained with the dedicated Janus\\index{Janus} computer~\\cite{janus:09}. The Janus\\index{Janus} simulation used heat bath dynamics, rather than Metropolis, and the \\gls{PT} there had $N_T=34$ and $T_\\mathrm{min}=0.703$. In order to be sure that heat bath autocorrelation times are consistent with Metropolis times (as we would expect) we simulated with Janus\\index{Janus} ten randomly selected samples\\index{sample} with both algorithms, finding that $\\tau_\\mathrm{Metropolis}\\approx \\tau_\\mathrm{heat-bath}\/3$.\n\nWe show in~\\reffig{all_L_prob_tau} the cumulative distribution function of $\\tau = \\tint$, $F(\\tau)$. It can be seen qualitatively from the figure that the maximum slope of $F$ decreases with $L$ for the small systems, and it stabilizes between $L=24$ and $L=32$; indeed these two distributions can be approximately superposed by a simple translation. This is reminiscent of a critical slowing-down~\\cite{zinn-justin:05}\n\\begin{equation}\\label{eq:zPT-def}\n\\tau\\sim L^{z^\\mathrm{PT}(T_\\mathrm{min})}\\,.\n\\end{equation}\n\nIt is not obvious \\emph{a priori} that such a simple scaling should hold in the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass}. As a working, simplifying hypothesis we assume that the exponent $z^\\mathrm{PT}$ only depends on the value of the lowest temperature in the \\gls{PT} grid, $T_\\mathrm{min}$ (and not on the number of temperatures).\n\nThe reader may warn that in~\\reffig{all_L_prob_tau} the distribution functions are not drawn for small values of $F(\\tau)$ in the $L=8$ and $L=16$ cases. The reason is that we could not find with our variational method\\index{variational method} a $\\tint$ that fulfills the condition of~\\refeq{tintfTlblo_condition} and therefore we can not provide a safe computation of $\\tau$. As long as we are concerned about the top part of the curve, we ignore this problem that does not appear in the simulation of $L=16$ with $N=16$, which is the simulation used in the rest of the study.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{eq_chaos\/all_L_prob_tau_g}\n\\caption[\\textbf{Empirical probability distribution of $\\mathbf{\\tau}$ for $\\mathbf{L=8,12,16,24}$ and 32.}]{\\textbf{Empirical probability distribution of $\\mathbf{\\tau}$ for $\\mathbf{L=8,12,16,24}$ and 32.} For $L=8$ and $L=16$ some of the samples\\index{sample} have $\\tau$ smaller than our minimal resolution (if $\\tau<n_{\\mathrm{Met}}$ we cannot compute it safely). We show only the part of the distribution function that can be safely computed.}\n\\labfig{all_L_prob_tau}\n\\end{figure}\n\nIt is clear from~\\reffig{all_L_prob_tau} that a simple translation would not attains the collapse of all the curves. Therefore, we study the exponent $z^{\\mathrm{PT}}$ for different parts of the distribution function, i.e. for different \\textit{percentiles}. We use the different simulated sizes to compute an effective $z^{\\mathrm{PT}}$ exponent for each pair of lattices ($L_1,L_2$) by means of the definition\n\\begin{equation}\nz^\\mathrm{PT}(L_1,L_2,p)=\\frac{\\log(\\tint(L_1,p)\/\\tint(L_2,p))}{\\log(L_1\/L_2)}\\,,\n\\end{equation}\nwhere $\\tau(L_i,p)$ is determined by the implicit equation $F(\\tau(L_i,p))=p\/100$ where $p=1,\\ldots,100$ is the so called percentile rank (i.e. $\\tau(L_i,p)$ is the $p$-th percentile of the distribution for the size $L_i$).  \n\nWe have computed $z^\\mathrm{PT}$ for three pairs of lattice sizes, (12,24), (16,24), and (24,32) and in~\\reffig{z_percentile} we show the results as a function of the rank. As expected, we see that the scaling does not hold for the whole curve in general. However, we observe that for higher ranks, i.e. the most chaotic samples\\index{sample}, the scaling hold for all the pairs of lattices. In addition, for the largest pair of lattices (24,32), the exponent $z^{\\mathrm{PT}}$ is independent of the percentile, within the statistical errors. This result is consistent with the previously drawn picture in~\\cite{fernandez:13}: for small system sizes, it is almost impossible to find samples\\index{sample} displaying strong \\gls{TC}\\index{temperature chaos} and one needs to go to larger lattices, like $L=32$ to find chaotic samples\\index{sample} with a significant probability.\n\nAn interesting coincidence with the results of non-equilibrium simulations~\\cite{janus:08b,janus:09b,fernandez:15,janus:17} could have a deep meaning. Indeed in non-equilibrium conditions one finds that the \\gls{SG} coherence length\\index{coherence length} $\\xi$, in a lattice of size $L\\gg\\xi$, at temperature $T=0.7$ grows with the simulation time $t_\\mathrm{w}$ as~\\cite{janus:09b}\n\\begin{equation}\n\\xi(t_\\mathrm{w})\\propto t_w^{1\/z(T)}\\,,\\quad z(T=0.7)=11.64(15)\\,, \\labeq{zeta_janus}\n\\end{equation}\nwhere $z(T)$ is the so-called dynamic critical exponent\\index{dynamic critical exponent}\\index{dynamic critical exponent!zzzzz@\\Also{aging rate}|gobbleone}, that turns out to be strongly temperature-dependent in the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass} $z(T)\\propto T_\\mathrm{c}\/T$, see \\refch{aging_rate}.\n\nOur results for the lattice pair $(24,32)$ suggest that\n\\begin{equation}\nz(T=0.7)\\approx z^\\mathrm{PT}(T_\\mathrm{min}=0.7)\\,. \\labeq{z-coincide}\n\\end{equation}\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{eq_chaos\/z_percentil_g}\n\\caption[\\textbf{The effective exponent $\\mathbf{z^{\\mathrm{PT}}}$.}]{\\textbf{The effective exponente $\\mathbf{z^{\\mathrm{PT}}}$.} The effective exponent $z^\\mathrm{PT}(L_1,L_2,p)$ for three different pairs of lattice sizes (12,24), (16,24) and (24,32) as a function of the percentile rank $p$.  The two horizontal lines are the bounds for the off-equilibrium value $z=11.64(15)$ [see \\refeq{zeta_janus}]. The numerical values of $z^\\mathrm{PT}$ for the largest pair are compatible with the off-equilibrium value.}\n\\labfig{z_percentile}\n\\end{figure}\n\nThe surprising result of~\\refeq{z-coincide} suggests the rescaling of the whole probability distribution as a test. This is done in figure \\reffig{all_L_prob_tau_reescaled} (main) that shows $F(\\tau)$ as a function of $y = \\tau\/L^z$. \n\nAs expected, the data for $L=24$ and $L=32$ present a nice collapse. The curve corresponding to $L=16$ collapses into them for percentile ranks higher than $80$ only and the curve corresponding to $L=12$ collapses for percentile ranks higher than $90$. This is pointing to us that the chaotic behavior of the large $L$ limit is harder to find as we go to smaller $L$.\n\nIn figure~\\ref{fig:all_L_prob_tau_reescaled} (inset), we show a log-log plot of $1\\!-\\!F(\\tau)$ as a function of $\\tau\/L^z$, that emphasizes the large $\\tau$ tail of the distribution. The fit presented shows that the probability density function of $\\tau$ behaves, asymptotically for large $y$, like a fat tailed distribution:\n\\begin{equation}\n\\rho(y\\equiv \\tau\/L^z) \\sim y^{-1-a_1}\\,,\\quad a_1\\approx 1.38 \\,.\n\\end{equation}\n\nThe distribution seems to reach its asymptotic form for $L\\geq 24$. Perhaps unsurprisingly, the thermodynamic (i.e. equilibrium) effective potential that characterizes temperature chaos turns out to be also asymptotic for $L\\geq 24$~\\cite{fernandez:13}.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{eq_chaos\/all_L_prob_tau_reescaled_g}\n\\caption[\\textbf{Probability distribution function of the rescaled variable \\boldmath $y = \\tau\/L^{z}$}.]{\\textbf{Probability distribution function of the rescaled variable \\boldmath $y = \\tau\/L^{z}$}. $z$ is the dynamic exponent corresponding to $T_\\mathrm{min}=0.7$, namely $z(T=0.7)=11.64(15)$. \\textbf{(Inset)} Plot of $\\log(1-F(\\tau))$ versus $\\log\\left(\\tau\/L^{z}\\right)$; the straight black line is a fit to the form $a_0 - a_1 \\log(\\tau\/L^z)$ yielding $a_0 = -29.33$ and $a_1=-1.38$.}\n\\labfig{all_L_prob_tau_reescaled}\n\\end{figure}\n\n\n\n\n\\chapter{General Introduction}\n\\labch{general_introduction}\nAs usually happens in science, we have just used the light of our predecessors to open the door in front of us and we expect that our work provides enough light to those who are to come.\n\nThis thesis pretends to be another step in the development of numerical research in disordered\\index{disorder!systems} systems. Specifically, we will focus on spin glasses which have demonstrated to be a fertile field from both, experimental and theoretical approaches. Throughout this thesis, we will discuss a variety of interesting phenomenons and we will also open new avenues to previously unexplored effects in the context of spin glasses. However, without a doubt, the leitmotiv conducting this thesis is the role of numerical simulations as a valuable tool to explore spin-glass physics.\n\nIndeed, the development of this thesis has been possible due to the high-quality data obtained from a huge numerical effort. In particular, the role of the FPGA\\index{FPGA}-based supercomputer Janus\\index{Janus} II has been determinant in this thesis. Unprecedented large off-equilibrium simulations have allowed obtaining high-accuracy data that are the basis of several chapters in the present document. The Janus\\index{Janus} II success has been possible due to the Janus\\index{Janus} Collaboration, an international project involving five different universities from Italy and Spain that have been joining in their efforts to design and program the spin-glass dedicated computer which is at the forefront of numerical research. In addition to Janus\\index{Janus} II, thousands of hours of computational time in CPUs have been also fundamental in the analysis of data. Specifically I would like to mention Cierzo supercomputer in BIFI and the Madrid's cluster in the UCM.\n\nThis thesis is organized into five different parts. The first part, containing the~\\refch{def_spinglass}, is focused on introducing the spin glasses to the reader. It is impossible to account for all the work developed in the spin-glass field. Nonetheless, we try to properly introduce the reader into the spin-glass context, starting from the very definition of the studied system. We also discuss relevant experimental results, trying to focus on those that will come up throughout the thesis. The most relevant theoretical results are also provided in this introduction. Finally, numerical simulations, the main research-resource in this thesis, are introduced.\n\nThe second part, containing the~\\refch{metastate}, is dedicated to discussing the metastate\\index{metastate}. The theoretical development of the spin glasses in the thermodynamic limit\\index{thermodynamic limit} has historically suffered mathematical inconsistencies. The irruption of the physical-mathematics in the context of spin glasses brought a solution to this problem introducing the concept of metastate\\index{metastate}. However, for finite dimensions, there exist different metastate\\index{metastate} pictures with different predictions. Since the invention of the metastate\\index{metastate} in 1990, this concept has been in the theoretical \\textit{world}, separated from experiments and numerical simulations, nonetheless, the current state of the art in numerical simulations have allowed us to numerically construct the metastate\\index{metastate} and to (partially) elucidate between the competing metastate\\index{metastate} pictures.\n\nThe third part, shaped by the~\\refch{aging_rate} and \\refch{mpemba}, is devoted to studying the off-equilibrium dynamics in spin glasses. This topic is of glaring importance in the spin glasses since experiments are (almost) always conducted in off-equilibrium conditions. Moreover recently, quantitative relevant relations have been found between equilibrium and out-equilibrium spin glasses. Learning about off-equilibrium dynamics might be useful to unveil the equilibrium properties of spin glasses concerning their very nature. Specifically, \\refch{aging_rate} is focused on discussing the growth of the coherence length\\index{coherence length} in spin glasses, a key quantity that characterizes the off-equilibrium evolution of those systems. In that chapter, we will witness a \\textit{tour de force} of Janus\\index{Janus} II, and we will be able to solve a discrepancy between numerical simulations and experiments by extrapolating our results to the relevant experimental time-scales. In~\\refch{mpemba} we will discuss an interesting phenomenon: the Mpemba effect. It has been observed that, under some circumstances, if two beakers of water, one hotter than the other, are put in contact with a thermal reservoir at low temperatures, the hot water freezes faster than the cold water. We translate this off-equilibrium phenomenon to the spin-glass context for the first time and provide a satisfactory explanation.\n\nThe fourth part, containing the~\\refch{Introduction_chaos}, \\refch{equilibrium_chaos} and \\refch{out-eq_chaos} is devoted to study the Temperature Chaos phenomenon in spin glasses. In~\\refch{Introduction_chaos} we introduce the main historical results on Temperature Chaos, from its origins to the last steps. In~\\refch{equilibrium_chaos} we study equilibrated spin glasses and we characterize the Temperature Chaos from a static and a dynamical point of view. We also reproduce previous results by relating both approaches through an exploration of the correlations between them and proposing new observables to improve those correlations. Moreover, in this chapter, we develop a numerical method to improve the dynamical estimation of Temperature Chaos. In~\\refch{out-eq_chaos} we tackle the problem of characterizing Temperature Chaos in off-equilibrium dynamics. The very definition of Temperature Chaos refers to the reorganization of the \\textit{equilibrium} configurations\\index{configuration} of a spin glass upon small changes in the temperature. Then, the existence of the phenomenon in out-equilibrium systems needs to be established. In that chapter, we find a phenomenon that closely mimics the equilibrium Temperature Chaos. Indeed, we observe a strong relationship between the equilibrium and the off-equilibrium Temperature Chaos.\n\nThe fifth and last part of the main body of the thesis corresponds to the conclusions (\\refch{conclusions}). There, we recall the relevant results of the thesis and we also discuss possible future works.\n\nFinally, some appendixes can be found in the last part of the document. In~\\refch{AP_statistics} we explain the relevant statistical tools that we have used in the analysis of this thesis. In~\\refch{AP_technical_details_aging} and~\\refch{AP_technical_details_out-eq_chaos} we provide technical details on the~\\refch{aging_rate} and~\\refch{out-eq_chaos} respectively. \\refch{AP_PT} is devoted to explaining technical details on the use of the parallel\\index{parallel!tempering} tempering method to compute the characteristic time-scales in Markov\\index{Markov chain} chains. \\refch{AP_multispin_coding} provides technical details on general methods of parallelization\\index{parallel!computation} that are useful in our simulations and analysis.\n\nThe original results of this thesis are published in the following articles\n\\begin{itemize}\n\\item A. Billoire, L. A. Fernandez, A. Maiorano, E. Marinari, V. Martin-Mayor, J. Moreno-Gordo, G. Parisi, F. Ricci-Tersenghi, and J. J. Ruiz-Lorenzo. 'Numerical Construction of the Aizenman-Wehr Metastate'. In: Phys. Rev. Lett. 119 (3 July 2017), p. 037203. doi: 10.1103\/PhysRevLett.119.037203 \\cite{billoire:17}\n\n\\item A Billoire, L A Fernandez, A Maiorano, E Marinari, V Martin-Mayor, J Moreno-Gordo, G Parisi, F Ricci-Tersenghi, and J J Ruiz-Lorenzo. 'Dynamic variational study of chaos: spin glasses in three dimensions'. In: Journal of Statistical Mechanics: Theory and Experiment 2018.3 (2018), p. 033302. doi: 10.1088\/1742-5468\/aaa387 \\cite{billoire:18}\n\n\\item M. Baity-Jesi, E. Calore, A. Cruz, L. A. Fernandez, J. M. Gil-Narvion, A. Gordillo-Guerrero, D. I\u00f1iguez, A. Maiorano, E. Marinari, V. Martin-Mayor, J. Moreno-Gordo, A. Mu\u00f1oz-Sudupe, D. Navarro, G. Parisi, S. Perez-Gaviro, F. Ricci-Tersenghi, J. J. Ruiz-Lorenzo, S. F. Schifano, B. Seoane, A. Tarancon, R. Tripiccione, and D. Yllanes. 'Aging Rate of Spin Glasses from Simulations Matches Experiments'. In: Phys. Rev. Lett. 120 (26 June 2018), p. 267203. doi: 10.1103\/PhysRevLett.120.267203 \\cite{janus:18}\n\n\\item M. Baity-Jesi, E. Calore, A. Cruz, L. A. Fernandez, J. M. Gil-Narvi\u00f3n, A. Gordillo-Guerrero, D. I\u00f1iguez, A. Lasanta, A. Maiorano, E. Marinari, V. Martin-Mayor, J. Moreno-Gordo, A. Mu\u00f1oz-Sudupe, D. Navarro, G. Parisi, S. Perez-Gaviro, F. Ricci-Tersenghi, J. J. Ruiz-Lorenzo, S. F. Schifano, B. Seoane, A. Tarancon, R. Tripiccione, and D. Yllanes. 'The Mpemba effect in spin glasses is a persistent memory effect'. In: Proceedings of the National Academy of Sciences 116.31 (2019), pp. 15350\u201315355 \\cite{janus:19}\n\n\\item M. Baity-Jesi, E. Calore, A. Cruz, L. A. Fernandez, J. M. Gil-Narvi\u00f3n, I. Gonzalez-Adalid Pemartin, A. Gordillo-Guerrero, D. I\u00f1iguez, A. Maiorano, E. Marinari, V. Martin-Mayor, J. Moreno-Gordo, A. Mu\u00f1oz-Sudupe, D. Navarro, I. Paga, G. Parisi, S. Perez-Gaviro, F. Ricci-Tersenghi, J. J. Ruiz-Lorenzo, S. F. Schifano, B. Seoane, A. Tarancon, R. Tripiccione, and D. Yllanes. 'Temperature chaos is present in off-equilibrium spin-glass dynamics'. In: Communications Physics 4.1 (Apr. 2021), p. 74 \\cite{janus:21}\n\\end{itemize}\n\n\n\\setcounter{chapter}{-1}\n\\chapter{Introducci\u00f3n General}\n\\labch{introduccion_general}\n\nComo habitualmente ocurre en ciencias, nuestro cometido ha sido usar la luz de nuestros predecesores para poder abrir la puerta que se encontraba delante de nosotros. Esperamos que nuestro trabajo d\u00e9 la suficiente luz a aquellos que est\u00e1n por venir.\n\nEsta tesis pretende ser otro paso m\u00e1s en el desarrollo de la investigaci\u00f3n num\u00e9rica en los sistemas desordenados. En concreto, nos centraremos en los vidrios de esp\u00edn, que han demostrado ser un campo de conocimiento f\u00e9rtil tanto para los experimentos como para el desarrollo te\u00f3rico. A lo largo de esta tesis discutiremos una gran variedad de fen\u00f3menos interesantes y tambi\u00e9n abriremos nuevos caminos hacia efectos que no hab\u00edan sido explorados previamente en el contexto de los vidrios de esp\u00edn. Sin embargo, sin g\u00e9nero de duda, el leitmotiv de esta tesis es el papel de las simulaciones num\u00e9ricas como una valiosa herramienta para explorar la f\u00edsica de los vidrios de esp\u00edn.\n\nCiertamente, el desarrollo de esta tesis ha sido posible gracias a la alta calidad de los datos obtenidos con un inmenso esfuerzo num\u00e9rico. En concreto, el rol de la supercomputadora, basada en FPGA, Janus II ha sido determinante en esta tesis. Simulaciones masivas fuera del equilibrio sin precedentes han permitido obtener datos altamente precisos que son la base de varios cap\u00edtulos en este texto. El \u00e9xito de Janus II ha sido posible gracias a la colaboraci\u00f3n Janus, un projecto internacional que involucra cinco universidades distintas de Italia y Espa\u00f1a que han unido sus fuerzas para dise\u00f1ar y programar un hardware dedicado a la simulaci\u00f3n de vidrios de esp\u00edn en la vanguardia de la investigaci\u00f3n num\u00e9rica. Adem\u00e1s de Janus II, miles de hora de tiempo computacional en CPU han tenido tambi\u00e9n un papel fundamental en el an\u00e1lisis de datos. En concreto, me gustar\u00eda mencionar la supercomputadora Cierzo, en el BIFI y el cluster de Madrid, en la UCM.\n\n\nEsta tesis est\u00e1 organizada en cinco partes diferentes. La primera parte, formada por el Cap\u00edtulo 1, est\u00e1 centrada en introducir los vidrios de esp\u00edn al lector. Es imposible dar cuenta de todo el trabajo desarrollado en el campo de los vidrios de esp\u00edn. No obstante, intentamos introducir apropiadamente al lector al contexto de los vidrios de esp\u00edn, empezando por la definici\u00f3n del sistema en cuesti\u00f3n. En este cap\u00edtulo tambi\u00e9n discutimos los resultados experimentales m\u00e1s importantes, tratando de dar m\u00e1s relevancia a aquellos que guardan relaci\u00f3n directa con los temas tratados en esta tesis. Los resultados te\u00f3ricos m\u00e1s relevantes tambi\u00e9n se plasman en este cap\u00edtulo introductorio. Finalmente, las simulaciones num\u00e9ricas, el principal recurso para el desarrollo de la investigaci\u00f3n en esta tesis, son introducidas en este cap\u00edtulo.\n\nLa segunda parte, que incluye el Cap\u00edtulo 2, est\u00e1 dedicada a discutir el metaestado. El desarrollo te\u00f3rico de los vidrios de esp\u00edn en el l\u00edmite termodin\u00e1mico ha sufrido hist\u00f3ricamente de inconsistencias matem\u00e1ticas. La irrupci\u00f3n de la f\u00edsica-matem\u00e1tica en el contexto de los vidrios de esp\u00edn trajo la soluci\u00f3n a este problema introduciendo el concepto del metaestado. Sin embargo, para dimensi\u00f3n finita, existen diferentes teor\u00edas del metaestado con diferentes predicciones. Desde la invenci\u00f3n del metaestado en 1990, el concepto ha estado restringido al mundo te\u00f3rico, separado de experimentos y de simulaciones num\u00e9ricas, no obstante, el actual estado del arte en las simulaciones num\u00e9ricas ha permitido construir num\u00e9ricamente el metaestado y (al menos parcialmente) dilucidar entre las distintas teor\u00edas que lo describen.\n\nLa tercera parte, formada por los Cap\u00edtulos 3 y 4, est\u00e1 dedicada al estudio de la din\u00e1mica de no equilibrio de los vidrios de esp\u00edn. Este tema es de palmaria importancia en estos sistemas puesto que los experimentos (casi) siempre son llevados a cabo fuera del equilibrio. Adem\u00e1s, relaciones cuantitativas relevantes han sido halladas entre los vidrios de esp\u00edn en equilibrio y fuera del equilibrio. Aprender sobre la din\u00e1mica fuera del equilibrio puede ser \u00fatil para desvelas las propiedades de equilibrio de los vidrios de esp\u00edn, relacionadas con la misma naturaleza de estos sistemas. Espec\u00edficamente, el Cap\u00edtulo 3 est\u00e1 centrado en discutir el crecimiento de la longitud de coherencia en los vidrios de esp\u00edn, una cantidad clave que caracteriza la evoluci\u00f3n fuera del equilibrio de estos sistemas. En este cap\u00edtulo, presenciaremos un aut\u00e9ntico \\textit{tour de force} de Janus II y seremos capaces de resolver una discrepacia entre experimentos y simulaciones num\u00e9ricas extrapolando nuestros resultados hasta las escalas de tiempo relevantes para los experimentos. En el Cap\u00edtulo 4 discutiremos un fen\u00f3meno interesante: el efecto Mpemba. Se ha observado que, bajo ciertas circunstancias, si dos recipientes de agua, uno caliente y otro fr\u00edo, son puestos en contacto con un ba\u00f1o t\u00e9rmico a baja temperatura, el agua caliente se congela antes que el agua fr\u00eda. Nosotros traducimos este fen\u00f3meno que se da fuera del equilibrio al contexto de los vidrios de esp\u00edn por primera vez y damos una primera explicaci\u00f3n del mismo.\n\nLa cuarta parte, formada por los Cap\u00edtulos 5, 6 y 7, est\u00e1 dedicada a estudiar el fen\u00f3meno del Caos en Temperatura en vidrios de esp\u00edn. En el Cap\u00edtulo 5 introducimos los principales resultados hist\u00f3ricos sobre el Caos en Temperatura, desde sus or\u00edgenes hasta los \u00faltimos pasos. En el Cap\u00edtulo 6 estudiamos vidrios de esp\u00edn en equilibrio y caracterizamos el Caos en Temperatura desde un punto de vista est\u00e1tico y din\u00e1mico. Adem\u00e1s, relacionamos ambos enfoques explorando las correlaciones entre ambos. Adem\u00e1s, en este cap\u00edtulo desarrollamos un m\u00e9todo num\u00e9rico para mejorar la estimaci\u00f3n din\u00e1mica del Caos en Temperatura. En el Cap\u00edtulo 7, atacamos el problema de caracterizar el Caos en Temperatura en una din\u00e1mica fuera del equilibrio. La propia definici\u00f3n del Caos en Temperatura se refiere a las configuraciones de \\textit{equilibrio} del vidrio de esp\u00edn bajo un cambio arbitrariamente peque\u00f1o de temperatura. Por lo tanto, la existencia del fen\u00f3meno fuera del equilibrio debe ser establecida. En este cap\u00edtulo, encontramos un fen\u00f3meno que imita a la perfecci\u00f3n al fen\u00f3meno del Caos en Temperatura en equilibrio. De hecho, observamos una fuerte relaci\u00f3n entre el Caos en Temperatura en equilibrio y fuera del equilibrio.\n\nLa quinta y \u00faltima parte del cuerpo principal de la tesis corresponde a las conclusiones. En el Cap\u00edtulo 8 revisitamos los resultados m\u00e1s relevantes de esta tesis y tambi\u00e9n discutimos sobre posibles futuros trabajos.\n\nFinalmente, el lector puede encontrar algunos ap\u00e9ndices en la parte final de este texto. En el ap\u00e9ndice A explicamos cu\u00e1les son las herramientas estad\u00edsticas m\u00e1s relevantes que hemos usado a lo largo de esta tesis. En los ap\u00e9ndices B y D se dan detalles t\u00e9cnicos de los cap\u00edtulos 3 y 7 respectivamente. En el ap\u00e9ndice C se explican los detalles t\u00e9cnicos del uso del m\u00e9todo de \\textit{Parallel Tempering} para calcular las escala de tiempo caracter\u00edsticas en las cadenas de Markov. Por \u00faltimo, en el ap\u00e9ndice E se dan los detalles t\u00e9cnicos de m\u00e9todos generales de paralelizaci\u00f3n que han sido \u00fatiles en nuestras simulaciones y an\u00e1lisis.\n\nLos resultados originales de esta tesis est\u00e1n publicados en los siguientes art\u00edculos\n\\begin{itemize}\n\\item A. Billoire, L. A. Fernandez, A. Maiorano, E. Marinari, V. Martin-Mayor, J. Moreno-Gordo, G. Parisi, F. Ricci-Tersenghi, and J. J. Ruiz-Lorenzo. 'Numerical Construction of the Aizenman-Wehr Metastate'. In: Phys. Rev. Lett. 119 (3 July 2017), p. 037203. doi: 10.1103\/PhysRevLett.119.037203 \\cite{billoire:17}\n\n\\item A Billoire, L A Fernandez, A Maiorano, E Marinari, V Martin-Mayor, J Moreno-Gordo, G Parisi, F Ricci-Tersenghi, and J J Ruiz-Lorenzo. 'Dynamic variational study of chaos: spin glasses in three dimensions'. In: Journal of Statistical Mechanics: Theory and Experiment 2018.3 (2018), p. 033302. doi: 10.1088\/1742-5468\/aaa387 \\cite{billoire:18}\n\n\\item M. Baity-Jesi, E. Calore, A. Cruz, L. A. Fernandez, J. M. Gil-Narvion, A. Gordillo-Guerrero, D. I\u00f1iguez, A. Maiorano, E. Marinari, V. Martin-Mayor, J. Moreno-Gordo, A. Mu\u00f1oz-Sudupe, D. Navarro, G. Parisi, S. Perez-Gaviro, F. Ricci-Tersenghi, J. J. Ruiz-Lorenzo, S. F. Schifano, B. Seoane, A. Tarancon, R. Tripiccione, and D. Yllanes. 'Aging Rate of Spin Glasses from Simulations Matches Experiments'. In: Phys. Rev. Lett. 120 (26 June 2018), p. 267203. doi: 10.1103\/PhysRevLett.120.267203 \\cite{janus:18}\n\n\\item M. Baity-Jesi, E. Calore, A. Cruz, L. A. Fernandez, J. M. Gil-Narvi\u00f3n, A. Gordillo-Guerrero, D. I\u00f1iguez, A. Lasanta, A. Maiorano, E. Marinari, V. Martin-Mayor, J. Moreno-Gordo, A. Mu\u00f1oz-Sudupe, D. Navarro, G. Parisi, S. Perez-Gaviro, F. Ricci-Tersenghi, J. J. Ruiz-Lorenzo, S. F. Schifano, B. Seoane, A. Tarancon, R. Tripiccione, and D. Yllanes. 'The Mpemba effect in spin glasses is a persistent memory effect'. In: Proceedings of the National Academy of Sciences 116.31 (2019), pp. 15350\u201315355 \\cite{janus:19}\n\n\\item M. Baity-Jesi, E. Calore, A. Cruz, L. A. Fernandez, J. M. Gil-Narvi\u00f3n, I. Gonzalez-Adalid Pemartin, A. Gordillo-Guerrero, D. I\u00f1iguez, A. Maiorano, E. Marinari, V. Martin-Mayor, J. Moreno-Gordo, A. Mu\u00f1oz-Sudupe, D. Navarro, I. Paga, G. Parisi, S. Perez-Gaviro, F. Ricci-Tersenghi, J. J. Ruiz-Lorenzo, S. F. Schifano, B. Seoane, A. Tarancon, R. Tripiccione, and D. Yllanes. 'Temperature chaos is present in off-equilibrium spin-glass dynamics'. In: Communications Physics 4.1 (Apr. 2021), p. 74 \\cite{janus:21}\n\\end{itemize}\n\n\n\n\n\n\n\n\\chapter[A brief introduction to spin glasses]{A brief introduction \\\\ to spin glasses}\n\\labch{def_spinglass}\n\\setlength\\epigraphwidth{.5\\textwidth}\n\n\\epigraph{\\textit{Todo empez\u00f3 aquel fat\u00eddico d\u00eda \\dots}}{-- Jonathan Stroud, \\textit{El amuleto de Samarkanda} }\n\nThe first signs of unusual properties of alloys between noble metals and transition metals such as \\gls{AuFe} or \\gls{CuMn}, which later become the paradigmatic examples of spin glasses, were discovered in the second half of the 50's and the 60's through measurement of the specific heat\\index{specific heat} at low temperatures \\cite{denobel:59,zimmerman:60}, observation of time-dependent remanent magnetization\\index{magnetization} \\cite{kouvel:60,kouvel:61} and measurements of the ESR spectra \\cite{owen:56}. In parallel, the first interpretations of those experimental results from a theoretical point of view came out \\cite{marshall:60,klein:63} through \\gls{RKKY}\\index{RKKY interaction} interactions \\cite{ruderman:54,kasuya:56,yosida:57}.\n\nThe experiments and the first developed theories suggested that those systems present a low-temperature magnetic order\\index{magnetic order} which exhibits different properties from the known condensed-matter phases. Therefore, a natural question arose: there exists a sharp phase transition\\index{phase transition} from a high-temperature phase\\index{phase!high-temperature\/paramagnetic} to a low-temperature phase\\index{phase!low-temperature\/spin-glass} capable of describing the observed phenomena? \n\nThe very existence of this sharp phase transition\\index{phase transition} was discussed during the 1960s and the first half of the 1970s \\cite{violet:66,violet:67,beck:71,tholence:74}. However, the experiments of Canella, Mydosh, and Budnick \\cite{cannella:71,cannella:72} unveiled a sharp cusp in the ac susceptibility\\index{susceptibility!ac} of spin glasses such as \\gls{CuMn}. Those experiments were strong support for the sharp transition\\index{phase transition} hypothesis between a high-temperature paramagnetic phase\\index{phase!high-temperature\/paramagnetic} and a, at that moment new, low-temperature phase\\index{phase!low-temperature\/spin-glass}. Indeed, subsequent works in the following years brought evidence against this sharp transition\\index{phase transition}~\\cite{mulder:81,mulder:82} unveiling the rounded nature of the susceptibility\\index{susceptibility} curve. It was in 1991 when the community got convinced about the existence of the phase transition\\index{phase transition} in experimental spin-glasses \\cite{gunnarsson:91}. Nonetheless, at that time (the early 1970s), although still debated, the experimental results were supporting the phase transition\\index{phase transition} to occur.\n\nMoreover, previous experiments of neutron scattering\\index{neutron scattering}, indicated that there was not an order of the constitutive elements responsible for the magnetization\\index{magnetization}, the spins, in the low-temperature phase\\index{phase!low-temperature\/spin-glass} \\cite{arrott:65}. \n\nThe evidence of no order in the low-temperature phase\\index{phase!low-temperature\/spin-glass}, together with the results that suggested a sharp phase transition\\index{phase transition}, led Edwards and Anderson to propose their theory \\cite{edwards:75}. This theory was a breakthrough in the study of the spin glasses because, although it was a mean-field\\index{Mean-Field!theory} theory, proposed a model that put aside the \\gls{RKKY}\\index{RKKY interaction} interactions in favor of random interactions following probability distribution. This simple conceptual change opened the door to a large class of systems to be studied in the same terms that spin glasses did.\n\nAfter the Edwards-Anderson theory\\index{Edwards-Anderson!theory} not only the spin glasses became a hot topic in the condensed matter\\index{condensed matter} research, but also the divergence between the theoretical and the experimental researches was considerable. Some effort was indeed done to establish some interplay, for example, scaling theories of domain\\index{magnetic domain} growth \\cite{fisher:88,koper:88,ocio:90,bouchaud:01}, however, the mainstream of both, theory and experiments, were quite apart from each other, as we briefly discuss below.\n\nDue to the glassy nature of the spin glasses in low temperatures, experiments always were developed in off-equilibrium conditions while the theoretical work put its efforts on capture the nature of spin glasses at low temperatures, with results that require access to the spin configurations\\index{configuration} for equilibrated systems in the thermodynamic limit\\index{thermodynamic limit}.\n\nTraditionally, numerical work has been a powerful tool of theoretical research. Numerical studies were, almost always, developed in small systems at thermal equilibrium, very far from the experiments, most of them performed out of equilibrium in very large systems. Moreover, the numerical simulations performed in off-equilibrium conditions could only reach time-scales much smaller than the experimental ones. However, the continuous increase of the computational power and the development of special-purpose computers\\index{special-purpose computer} have allowed the study of the spin glasses at low temperatures out of equilibrium in the relevant experimental scales.\n\nAn exciting new venue for numerical research is bridging the experimental approach to spin glasses and the theoretical one. This thesis pretends to be a step forward in this sense.\n\nIn this chapter, the reader may find a general introduction to spin glasses. First of all, a first definition of the concept of spin glass will be introduced (see \\refsec{definition_spinglass}). Then, some experimental background will be provided in \\refsec{experimental_spinglass}. The main theoretical pictures, with their different quantitative predictions, will be exposed in \\refsec{theoretical_spinglass}. Finally, a brief introduction to the numerical work performed in this thesis together with some useful definitions can be found in \\refsec{numerical_spinglass}.\n\nAll the results presented in this chapter are not original from this thesis. In addition to the cited papers that will appear in this chapter, the reader may find useful general references about spin glasses in \\cite{binder:86,mezard:87,mydosh:93,fischer:93,young:98,dotsenko:01,dedominicis:06}.\n\n\\section{A first definition of spin glass} \\labsec{definition_spinglass}\nSpin glasses are magnetic systems whose low-temperature phase\\index{phase!low-temperature\/spin-glass} is frozen and disordered\\index{disorder}. From a practical point of view, we consider the spins (i.e. the magnetic moments) as the constitutive elements of the spin glass. We denote $\\vec{s}_{\\vec{r}}$ as the spin at the position $\\vec{r}$, and $\\braket{\\dots}_t$ as the mean over the experimental time.\n\nA direct consequence of the low-temperature phase\\index{phase!low-temperature\/spin-glass} to be frozen is $\\braket{\\vec{s}_{\\vec{r}}}_t \\neq 0$. Moreover, the direction in which the spins freeze are random, which has a direct implication in the magnetization\\index{magnetization} $\\sum_{\\vec{r}} \\braket{\\vec{s}_{\\vec{r}}}_t = 0$. Actually, no long-range magnetic order\\index{magnetic order} will appear in the low-temperature phase\\index{phase!low-temperature\/spin-glass}\n\\begin{equation}\n \\vec{M}_{\\vec{k}} = \\dfrac{1}{N} \\sum_i e^{-i\\vec{k}\\cdot\\vec{r}}\\braket{\\vec{s}_{\\vec{r}}}_t = 0 \\quad \\,\\, \\forall \\, \\vec{k} .  \\labeq{no_magnetic_order}\n\\end{equation} \nThis general expression includes the ferromagnetic order parameter\\index{order parameter} $\\vec{k} = \\vec{0}$ and the antiferromagnetic one $\\vec{k} = (\\pi,\\pi,\\pi)$.\n\nThese two characteristics, freezing and disorder\\index{disorder}, give us the key to understand the name \\gls{SG} which was proposed for the first time by Anderson \\cite{anderson:70}. The sluggish evolution of the system together with the disorder\\index{disorder} reminds the structural glass\\index{structural glass} main properties trading the magnetic moments by the position of the particles. In structural glasses\\index{structural glass}, particles occupy random positions with no time-evolution (disorder\\index{disorder} and freezing).\n\nThere exists a far well establish consensus about the principal ingredients that one model which aspires to exhibit the \\gls{SG}'s phenomenology must include: \\textit{randomness}\\index{randomness} in the interactions and \\textit{frustration}\\index{frustration} \\cite{toulouse:77,blandin:78}. \n\n\\subsection{A key property: frustration}\nSuppose that a pair of spins $\\vec{s}_{\\vec{r}_i}$ and $\\vec{s}_{\\vec{r}_j}$ interact with each other through the Hamiltonian\\index{Hamiltonian} $\\mathcal{H}_{ij} = -J_{ij}\\vec{s}_{\\vec{r}_i}\\cdot \\vec{s}_{\\vec{r}_j}$, being $J_{ij}$ the \\textit{coupling}\\index{couplings} interaction between the spins at the positions $\\vec{r}_i$ and $\\vec{r}_j$. For $J_{ij}>0$ the more energetically favorable values for the pair of spins $\\vec{s}$ i.e. the more energetically favorable \\textit{configurations},\\index{configuration} are those in which the spins are parallel to each other. In the same way, for $J_{ij}<0$ the spins tends to align in an anti-parallel way. For a collection of spins, the Hamiltonian\\index{Hamiltonian} can be easily generalized\n\\begin{equation}\n\\mathcal{H} = -\\sum_{i,j}J_{ij}\\vec{s}_{\\vec{r}_i}\\cdot \\vec{s}_{\\vec{r}_j} \\labeq{first_Hamiltonian} \\, .\n\\end{equation}\nWe will say that a system is frustrated\\index{frustration} when it is not possible to satisfy simultaneously all the pairwise interactions i.e. there is no way to maximize simultaneously all the summands of \\refeq{first_Hamiltonian}.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{intro\/frustration}\n\\caption[\\textbf{A graphical example of frustration.}]{\\textbf{A graphical example of frustration\\index{frustration}.} Plaquette\\index{plaquette} 1 is said to be frustrated\\index{frustration} because the spins, that tend to align in a parallel or anti-parallel way depending on the couplings\\index{couplings} (+ or - respectively), can not satisfy all the interactions simultaneously. However, in the unfrustrated plaquette\\index{plaquette} 2, we observe that neither the size of the plaquette\\index{plaquette} nor the mixture of positive and negative interactions is responsible for the frustration\\index{frustration}. The key is the number of negative couplings\\index{couplings} (respectively positive): if the number of, say, negative couplings\\index{couplings} is odd, then we will have a frustrated\\index{frustration} plaquette\\index{plaquette}, otherwise, we will have an unfrustrated plaquette\\index{plaquette}.}\n\\labfig{frustration}\n\\end{figure}\n\nThe canonical example of a frustrated\\index{frustration} system can be found in \\reffig{frustration} where the spins lie in the nodes of a square-regular\\index{regular lattice!square} lattice\\footnote{As far as we are just dealing with the sign of the couplings\\index{couplings}, that particular disposition of the spins only try to simplify the visualization without any generality loss.} in which the edges represent the coupling\\index{couplings} interactions between the spins. We firstly focus our attention in the closed-loop, also called \\textit{plaquette}\\index{plaquette}, labeled with the number 1. As long as we are discussing here just if the spins are parallel or anti-parallel, let us take for the sake of simplicity spins with values $\\uparrow$ (up) and $\\downarrow$ (down). Suppose that the spin (1) is $\\uparrow$, following to spin (2) through a positive coupling\\index{couplings} ($+$) which favors the parallel interactions, the value of spin (2) also should be $\\uparrow$. The same reasoning can be applied to the spin (3). Finally, the spin (4) is at the end of the loop and, therefore, has to satisfy two couplings\\index{couplings}: the positive ($+$) coupling\\index{couplings} with the spin (3) and the negative ($-$) coupling\\index{couplings} with the spin (1). Any value that the spin (4) takes will lead to one unsatisfied interaction. We say that the plaquette\\index{plaquette} 1 is frustrated\\index{frustration}. \n\nHowever, if we focus now on plaquette\\index{plaquette} 2 and we apply the same mental exercise, we can easily find a configuration\\index{configuration} of spins that satisfies all the interactions, for example, the sequence (1)$\\uparrow$, (2)$\\uparrow$, (3)$\\downarrow$, (4) $\\uparrow$, (5) $\\uparrow$, (6) $\\uparrow$, (7) $\\uparrow$, (8) $\\uparrow$, (9) $\\downarrow$ and (10) $\\downarrow$. The length of the plaquette\\index{plaquette} is irrelevant and one can infer from these examples that the key is the number of negatives (or equivalently positives) interactions; if the number of negative (positive) interactions that favor anti-parallel (respectively parallel) spin-alignment is odd, then the plaquette\\index{plaquette} will be frustrated\\index{frustration}, otherwise, we will say that the plaquette\\index{plaquette} is unfrustrated.\n\nIn frustrated\\index{frustration} systems there exist many configurations\\index{configuration} with several unsatisfied interactions in which any local change would lead to an increase of the energy\\index{energy}, thus, frustration\\index{frustration} draws a rugged free-energy\\index{free energy!landscape} landscape with many metastable\\index{metastability} states and high-energy barriers. Indeed, the rugged free-energy\\index{free energy!landscape} landscape is directly related to the frozen nature of \\gls{SG}s and other characteristic properties as the slow time-evolution.\n\nTwo clarifications should be made at this point. First, the mixture of positive and negative interactions do not guarantee the system to be frustrated\\index{frustration},\\index{frustration} see \\reffig{frustration}, plaquette\\index{plaquette} 2, or Mattis model\\index{Mattis model} \\cite{mattis:76} which can be mapped to a uniform ferromagnet\\index{ferromagnet}. Lastly, frustration\\index{frustration} without randomness\\index{randomness} (or vice versa) does not lead to \\gls{SG} behavior, the most simple example is the regular\\index{regular lattice!triangular} triangular lattice with antiferromagnetic\\index{antiferromagnetic} interactions\\footnote{Exactly solvable, see \\cite{wannier:50}}, which is a fully frustrated\\index{frustration} system with a large ground-state\\index{ground-state} degeneracy but without phase transition\\index{phase transition} at finite temperature\\footnote{More examples of frustrated\\index{frustration} systems without randomness\\index{randomness} can be found in \\cite{villain:77,villain:77b,wolff:82,wolff:83,wolff:83b,mackenzie:81}}.\n\nWe conclude that frustration\\index{frustration} and randomness\\index{frustration}\\index{randomness} are necessary conditions to have a \\gls{SG} but not sufficient ones. An illustrative example with this respect can be found in the Ising\\index{Ising} ferromagnet in a small random magnetic field, where frustration\\index{frustration} and randomness\\index{randomness} appear in a very weak way and it is possible to find long-range magnetic order\\index{magnetic order}.\n\n\\subsection{Why spin glasses?}\nA natural question is, if the \\gls{SG}s reproduce the glassy behavior at low temperatures, why should we focus on them instead on the structural\\index{structural glass} glasses? The main reason to study the glassy behavior through \\gls{SG}s is their simplicity. This simplicity allows the development of theoretical tools in \\gls{SG}s that can be later applied to other fields of the complex systems \\cite{mezard:85c,mezard:86b,amit:85,amit:85b,goldstein:92} (paradoxically, including the structural\\index{structural glass} glasses \\cite{charbonneau:14}).\n\nMoreover, \\gls{SG}s exhibit a wide set of characteristic phenomenons of glassy behavior with several advantages from the theoretical and experimental points of view. \n\nFirst of all, it is worthy to note that, unlike the structural\\index{structural glass} glasses, the phase transition\\index{phase transition} is well-known in \\gls{SG}s, both in experiments through the study of the susceptibility\\index{susceptibility} \\cite{gunnarsson:91} and in the theoretical models \\cite{palassini:99,ballesteros:00}. The physicists can greatly benefit from this fact by establishing quantitative criteria to determine whether or not they are studying the glassy phase\\index{phase!low-temperature\/spin-glass}.\n\nBesides, from the experimental point of view, \\gls{SQUID}\\index{Superconducting Quantum Interference Device (SQUID)} allows for very precise measures, much more difficult to perform in structural\\index{structural glass} glasses.\n\nFinally, a technical reason is that \\gls{SG}s are much simpler to simulate than structural\\index{structural glass} glasses. The lattice models are very easy to simulate numerically.\n\n\\subsubsection{Further motivations in the study of spin glasses}\nIn addition to the above-exposed advantages of studying \\gls{SG}s in order to explore the glassy behavior, there exist other fields in which the study of these systems is interesting and prolific. The study of complexity in optimization problems constitutes a paradigmatic example in this regard~\\cite{barahona:82b}. \n\nThe Turing machine\\index{Turing machine} is a theoretical machine widely used in computation theory introduced by Turing~\\cite{turing:37}. The deterministic version of the machine is able to give at most one result for every situation while the nondeterministic one is able to provide more than one result in each situation.\n\nThe set of problems that can be solved by a deterministic Turing machine\\index{Turing machine} in polynomial time belongs to the set P\\index{complexity!P} of problems. In the same way, the set of problems that can be solved by a non-deterministic Turing machine in polynomial time belong to the set of NP problems. It is trivial to see that P\\index{complexity!P} problems are a subset of NP\\index{complexity!NP} problems.\n\nSpecifically, there exists a subset of NP\\index{complexity!NP} which is called NP-complete\\index{complexity!NP-complete}\\footnote{The concept of NP-completeness was introduced in~\\cite{cook:71}.}, which is of special interest. We say that a problem is NP-complete if it belongs to the complexity class NP and all the NP problems are reducible (in polynomial time) to that problem. From a computational point of view, the NP-complete\\index{complexity!NP-complete} problems are the hardest in the set NP\\index{complexity!NP} and are equivalent to each other (see Cook-Levin theorem~\\cite[p.38]{garey:79}), in the sense that founding a polynomial-bounded algorithm for any one of them would effectively yield a polynomial-bounded algorithm for all.\n\nThere exist many problems in the NP-complete\\index{complexity!NP-complete} set (see, for instance ~\\cite{karp:72}) and, specifically, the problem of finding the ground-state\\index{ground-state} for a three-dimensional Ising \\gls{SG} \\footnote{See~\\refsubsec{source_randomness} for the concept of Ising \\gls{SG}.} is NP-complete\\index{complexity!NP-complete}~\\cite{barahona:82}. \n\nAlthough the three-dimensional case is of special interest in this thesis, there exist several models of \\gls{SG}s that have been studied with great detail from the complexity point of view. In particular, the question of the \\textit{planarity}\\index{planar graph} of the \\gls{SG} lattice has proven to be central in the complexity discussion~\\cite{istrail:00}.\n\nThe concept of planar graph\\index{planar graph} is rather intuitive. A graph is said to be planar\\index{planar graph} if it can be drawn in a plane without edge-crossing\\footnote{Although the concept is rather intuitive, in general, prove that a graph is planar\\index{planar graph} requires non-trivial criteria like Kuratowski's theorem.}. On the contrary, the fact that the problem of finding the ground-state\\index{ground-state} in a non-planar graph\\index{non-planar graph} is NP-complete\\index{complexity!NP-complete} is certainly not intuitive. The reader may find an interesting study of this problem for several graphs in~\\cite{istrail:00}.\n\nAn interesting fact in the case of \\gls{SG}s is the two-dimensional case. The typical case in the study of spin systems is to consider the spins placed in the vertex of a lattice and only take into account nearest-neighbors interactions. In this case, for the two-dimensional case, finding the ground-state is a P problem. However, if one considers next-nearest-neighbors interactions, the problem of finding the ground-state becomes NP-complete. This case might be surprising at first sight since the basic elements building the lattice are complete graphs of fourth order\\footnote{A complete graph of $n^{\\mathrm{th}}$ order, also knows as $K_n$ is a graph with $n$ nodes where every node is connected to the rest of them. In other contexts are also known as \\textit{fully-connected networks}.}, that fulfill Kuratowski's criteria and, therefore, are planar. The reader may find a deep discussion in this respect in~\\cite{istrail:00}.\n\nTherefore, the study of \\gls{SG}s is not only interesting from the statistical physics or the solid-state physics point of view but also from the complexity point of view.\n\n\\subsection{Beyond spin glasses. Weakly disordered versus strongly disordered systems}\nThroughout this first approach to \\gls{SG}s, we have set that the disorder is one of the essential characteristics that a system must have in order to exhibit the \\gls{SG} behavior. However, there exist a variety of systems exhibiting disorder that cannot be identified as \\gls{SG}s.\n\nWe have already introduced the Hamiltonian for spin systems in~\\refeq{first_Hamiltonian}. The simplest case, with $J_{ij}=J$ constant, corresponds to the Ising ferromagnet. Therefore, the introduction of disorder can be regarded as an additional random term\n\\begin{equation}\nJ_{ij} = J + \\delta J_{ij} \\, .\n\\end{equation}\n\nThe limiting cases $\\delta J_{ij} \\ll J$ and $\\delta J_{ij} \\gg J$ correspond to weak and strong disorder respectively. Specifically, in this thesis we will focus on systems with a strong disorder\\index{disorder!systems}, which is the case of \\gls{SG}s. However, there exist a variety of systems exhibiting weak disorder with a very different but rich phenomenology.\n\nThe study of weak-disorder systems\\index{disorder!systems} is a natural generalization of the study of pure systems\\footnote{In this context, pure systems refer to the absence of impurities in their composition.} by the introduction of impurities that are unavoidable in real systems. In these systems, the ground-state\\index{ground-state} and the equilibrium properties keep a close relationship with the pure system obtained by removing its impurities. However, the presence of these impurities may affect the behavior of the system in the neighborhood of the critical temperature\\index{critical temperature} \\cite{mccoy:70,balagurov:73,harris:74,harris:74b,khmelnitskii:75,grinstein:76}. \n\nMoreover, for systems undergoing a first-order phase transition, the effects of the weak disorder have been widely studied. Particularly, in two-dimensional systems, any arbitrarily small amount of disorder makes the first-order transition to become a second-order one~\\cite{hui:89,aizenman:90,cardy:97,jacobsen:98,chatelain:98}. The three-dimensional case is more difficult to study but there exist results suggesting that a change of the order of the transition from first- to second-order occurs when the disorder increases~\\cite{fernandez:12}. This is known as the Cardy-Jacobsen conjecture\\index{Cardy-Jacobsen conjecture}~\\cite{cardy:97,jacobsen:98}.\n\nOn the contrary, in strong-disorder systems, like \\gls{SG}s, the ground-state, the equilibrium properties of the system, and the phase transition completely differs from the pure system, as we will illustrate throughout this thesis.\n\nThe disorder can also be present in the form of an external random field $h_i$. This type of disorder, with the absence of the coupling disorder, leads to the well-known and widely-studied \\gls{RFIM}\\index{Random Field Ising Model}~\\cite{imry:75,nattermann:98,belanger:98}. This model is paradigmatic in the study of disordered systems and can be regarded as an intermediate case between weak and strong disorder.\n\nIn the two-dimensional case, the ordered phase is destroyed by the random field~\\cite{binder:84,aizenman:90}, i.e. the effects in the transition are strong and the properties of the pure system give no clue to understand the disordered system. However, in the three-dimensional case, the critical behavior of the system is changed by the presence of the random fields but still exhibits an ordered phase~\\cite{imbrie:84,bricmont:87}, similarly to what occur in the weak-disorder systems.\n\nThe importance of the \\gls{RFIM} in the statistical physics literature yields over several reasons. Standing as one of the simplest disordered systems with a rich phenomenology is one. Moreover, this model has numerous representatives in nature, for example, the diluted antiferromagnets in a homogeneous external field~\\cite{fishman:79} and it has been intensively studied, from both the experimental~\\cite{belanger:98} and the theoretical~\\cite{nattermann:98} point of view. Besides, contrary to the \\gls{SG} case (as we will discuss in subsequent sections), the theoretical and experimental \\gls{RFIM} have been developed in parallel for many years.\n\nThe number of weak-disorder systems and the results characterizing them is large and of central importance in the statistical physics field, as we have illustrated above. Nonetheless, the presence of weak disorder is not always affecting the physics of the pure system. We discuss this issue next.\n\n\\subsubsection{The Harris criterion}\nWe have stated above that the presence of impurities may affect the critical behavior of a system. There exists a quantitative criterion to know if the presence of the disorder is going to be relevant: the Harris criterion\\index{Harris criterion}~\\cite{harris:74}.\n\nThe argument behind the original formulation of the Harris criterion is rather simple and a useful discussion can be found in~\\cite{brooks:16}. The main idea is that the sign of the critical exponent associated with the specific heat\\index{specific heat} ($\\alpha$) for the pure system, is the stability condition itself.\n\nIf $\\alpha>0$, the disorder will change the critical behavior. On the contrary, the disorder will be irrelevant and the critical behavior of the system will not change.\n\n\n\\section{Experimental spin glasses} \\labsec{experimental_spinglass}\nAs we said in the introduction of the chapter, the paradigmatic examples of \\gls{SG}s are transition metal impurities in noble metal hosts like \\gls{CuMn}, \\gls{AuFe}, \\gls{AgMn}, \\gls{CuFe}, etc. However, there exist other types of materials that exhibit the \\gls{SG} phenomenology. Is the case of rare earth constituents in metallic host \\gls{YDy}, \\gls{YEr}, \\gls{ScTb}, etc. Also the same holds for ternary systems, e.g \\gls{LaGdAl}. We will discuss the interactions that are the source of the randomness\\index{randomness} and frustration\\index{frustration} necessary to have a \\gls{SG} low-temperature phase\\index{phase!low-temperature\/spin-glass} in \\refsubsec{source_randomness} and we will expose relevant experiments that had a historical impact in the development of the \\gls{SG}s field and that are somehow related to the results presented in this thesis in \\refsubsec{aging_memory_rejuvenation}\n\n\\subsection{Internal structure and magnetic interactions: the source of randomness}\n\\labsubsec{source_randomness}\nWe have set that one of the main ingredients to have a \\gls{SG} is the randomness\\index{randomness}. How the real systems achieve that randomness\\index{randomness} in the interactions? There exist two main ways to obtain it: \\textit{bond randomness}\\index{randomness!bond} and \\textit{site randomness}\\index{randomness!site}.\n\nBond randomness\\index{randomness!bond} is a type of disorder\\index{disorder} present in real systems like \\gls{RbCuCoF} and \\gls{FeMnTiO}. These systems present regular\\index{regular lattice} lattices where the dominant magnetic interactions are short-ranged, then the impurities (cobalt and manganese respectively) are introduced. This procedure mimics what is called \\textit{ideal spin-glass} i.e. a regular\\index{regular lattice} lattice of spins interacting with their nearest-neighbors in a ferromagnetic or antiferromagnetic random way.\n\nSite randomness\\index{randomness!site} is the type of disorder\\index{disorder} that is present in the most commonly studied \\gls{SG}s such as \\gls{CuMn}. In this system, the substitution of the magnetic solute for the non-magnetic solvent should occur completely randomly\\footnote{There are other procedures used to create this type of disorder\\index{disorder} by destroying the crystal structure of the materials, making them amorphous.}. However, this type of disorder\\index{disorder} needs something else to generate randomness\\index{randomness} in the interactions. We need a kind of magnetic interaction that depending on the distance between the magnetic impurities generates antiferromagnetic or ferromagnetic couplings\\index{couplings}.\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\nThe dominant interaction in those systems is the so-called \\gls{RKKY}\\index{RKKY interaction} interaction \\cite{ruderman:54,kasuya:56,yosida:57}. This interaction is long-ranged and the underlying mechanism is the conduction electrons of the host metal acting as intermediaries between the magnetic moments of the magnetic solute. \n\nA magnetic impurity placed at $\\vec{r}_i$ changes the susceptibility\\index{susceptibility} of the conduction electrons surrounding it through hyperfine interaction. A second magnetic impurity placed at $\\vec{r}_j$ will behave in the same way, thus the two \\gls{RKKY}\\index{RKKY interaction} polarization will overlap, establishing an effective interaction between the two spins of the magnetic impurities. This interaction is given by\n\\begin{equation}\nJ(r_{ij}) = 6\\pi Z J^2 N(E_F) \\left[ \\dfrac{\\sin (2k_F r_{ij})}{(2k_F r_{ij})^4} - \\dfrac{\\cos (2k_F r_{ij})}{(2k_Fr_{ij})^3}\\right] \\quad , \\labeq{RKKY}\n\\end{equation}\nwhere $Z$ is the number of conduction electrons per atom, $J$ is the s-d exchange constant, $N(E_F)$ is the density of states at the Fermi level, $k_F$ is the Fermi momentum and $r_{ij} = \\lvert \\vec{r}_i - \\vec{r}_j \\rvert$. At large distance the quartic term of the distance becomes irrelevant with respect to the cubic one, therefore\n\\begin{equation}\nJ(r_{ij}) \\approx \\dfrac{J_0 \\cos (2k_F r_{ij} + \\phi)}{(2k_F r_{ij})^3} \\quad ,\\labeq{RKKY_reduced}\n\\end{equation}\nbeing $J_0$ a constant which agglutinates all the constant terms of \\refeq{RKKY} and $\\phi$ a phase that takes into account the charge difference between impurity and host. The reader may wonder whether the decaying-sinusoidal behavior of \\refeq{RKKY} or \\refeq{RKKY_reduced} would be enough to generate random interactions. We would like to stress the fact that the Fermi moment $k_F$ is, actually, quite large (of the order of the inverse of the interatomic spacing). That makes the sinusoidal oscillations to be very sensitive to any change of the distance $r_{ij}$.\n\nOf course, there exist other types of interactions between spins capable to generate randomness\\index{randomness}\\footnote{For example, superexchange interaction is relevant in insulating and semiconducting materials due to the lack of conduction electrons. Moreover, there exist weaker interactions like dipolar\\index{dipolar} interaction that play an important role because they introduce anisotropies\\index{anisotropy}.}, however, we only stop to explain \\gls{RKKY}\\index{RKKY interaction} for historical and practical reasons. From the historical point of view, \\gls{RKKY}\\index{RKKY interaction} interactions are bounded to the birth of the \\gls{SG} research. Moreover, along this thesis experiments with \\gls{CuMn} will have an important role and the dominant interaction in \\gls{CuMn} turn out to be the \\gls{RKKY}\\index{RKKY interaction} interaction.\n\nLast, we have to keep in mind that the computation of \\refeq{RKKY} involves several approximations. The assumption of the free electron and the random impurities are the stronger ones. On the one hand, the consideration of an electronic band structure leads to considerable modifications of the \\gls{RKKY}\\index{RKKY interaction} interaction, see for example \\cite{narita:84} or, for computations in specific materials like graphene \\cite{annica:10,sherafati:11}. On the other hand, the positions of the impurities are not truly random as we assumed before. Experimentally it is possible to find significant correlations in the position of the impurities through neutron-scattering\\index{neutron scattering} techniques. Actually, the knowledge of those correlations allows the experimental computation for the couplings\\index{couplings} in different \\gls{SG}s, see \\reffig{RKKY} extracted from \\cite{morgownik:83}.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{intro\/RKKY.png}\n\\caption[\\textbf{RKKY interaction for real Spin Glasses.}]{\\textbf{RKKY interaction for real Spin Glasses.} Dotted line represents the original computation for the \\gls{RKKY}\\index{RKKY interaction} coupling\\index{couplings} as a function of the interatomic distance. Computations over real \\gls{SG}s shows significant differences with the theoretical results due to the correlations between the position of the impurities in the host metal. Figure from \\cite{morgownik:83}.}\n\\labfig{RKKY}\n\\end{figure}\n\nThe naive computation may not be quantitatively accurate but this, \\textit{a priori}, unfortunate fact turns out to be a hope for the study of \\gls{SG}s. The \\gls{RKKY}\\index{RKKY interaction} model still captures the fundamental requirements to find glassy behavior and thus, open the door to theoretical models which we will discuss in future sections that do not reproduce the couplings\\index{couplings} distribution of the real systems but which still contains the two main ingredients needed for finding \\gls{SG} behavior: randomness\\index{randomness} and frustration\\index{frustration}.\n\n\\subsubsection{Subtle but fundamental: anisotropies}\nUp to now, the magnetic interaction that we have attributed to \\gls{CuMn} real systems, the \\gls{RKKY}\\index{RKKY interaction} interaction, presents isotropic behavior, thus, there is no reason to restrict the value of the spin $\\vec{s}$ of the impurities in any dimension. This three-dimensional spin leads to the so-called \\textit{Heisenberg\\index{Heisenberg} spin glass}.\n\nHowever, even the purest real system presents some anisotropies\\index{anisotropy}. The role of those anisotropies\\index{anisotropy} is fundamental because they can restrict the degrees of freedom\\index{degree of freedom} of the spins to a plane (resulting in the $XY$ \\textit{spin glass}) or to a single dimension, leading to the known as \\textit{Ising\\index{Ising} spin glass}.\n\nThroughout this thesis, several results will be compared with real \\gls{CuMn} systems which are, essentially, Heisenberg\\index{Heisenberg}-like \\gls{SG}. A lot has been said about the effect of the anisotropies\\index{anisotropy} in the \\gls{CuMn} \\gls{SG}s \\cite{prejean:80,fert:80,levy:81,bray:82,mendels:87,chu:94,petit:02,bert:04,zhai-janus:21} where the main ones that we can found are the dipolar\\index{dipolar} anisotropies\\index{anisotropy}, weak but present in every spin system, and the \\gls{DM} anisotropies\\index{anisotropy}\\index{Dzyaloshinksii-Moriya}, whose origin is a large spin-orbit coupling\\index{couplings} of the conduction electron with the impurities, acting the conduction electron as an intermediary (similar to \\gls{RKKY}\\index{RKKY interaction}). \n\nFrom the computational point of view, Ising\\index{Ising} \\gls{SG}s are very convenient since they are much easier to simulate than Heisenberg\\index{Heisenberg} \\gls{SG}s and the research developed in the context of this thesis is focused on the former. Differences between Heisenberg\\index{Heisenberg} and Ising\\index{Ising} \\gls{SG}s are numerous\\footnote{For example, the very existence of a phase transition\\index{phase transition} in 3D systems is not clear in the Heisenberg\\index{Heisenberg} models while is commonly accepted in 3D Ising\\index{Ising} \\gls{SG}s.}, therefore, a natural question is whether or not the results obtained in this thesis are general.\n\nThis question is positively answered in \\cite{baityjesi:14}, actually, we know now that the ruling universality\\index{universality} class in presence of coupling\\index{couplings} anisotropies\\index{anisotropy} is Ising\\index{Ising} and even the purest real \\gls{SG} will present some anisotropies\\index{anisotropy}.\n\n\\subsection{Aging, memory and rejuvenation}\\labsubsec{experiments_introduction}\n\\labsubsec{aging_memory_rejuvenation}\nHere, we present emblematic experiments showing that \\gls{SG}s are out of equilibrium in the experimental time-scales. We also take the opportunity to expose those experiments that will be, at least conceptually, related in some way to the original results presented throughout this thesis.\n\n\\subsubsection{Aging}\\index{aging}\nWe have said that \\gls{SG}s, in absence of an external magnetic field, have null magnetization\\index{magnetization!zero}\\footnote{Not only, but also no magnetic order\\index{magnetic order} can be found, see \\refeq{no_magnetic_order}} $M=\\sum_{\\vec{r}} \\braket{\\vec{s}_{\\vec{r}}}=0$, however, when a magnetic field is applied, a magnetization\\index{magnetization} $M \\neq 0$ can be measured. When the external magnetic field is turned off, the system evolves from $M=M(t_0)\\neq 0 $ to $M(t_f)=0$. This process is called \\textit{relaxation}\\index{relaxation}.\n\n\\gls{SG}s exhibit in the low-temperature phase\\index{phase!low-temperature\/spin-glass} extremely-large relaxation\\index{relaxation} times, remaining out of equilibrium during the whole experiments. Still, the most striking feature is the emergence of a second time-scale: the relaxation\\index{relaxation} process strongly depends on the time that the system spent in the low-temperature phase\\index{phase!low-temperature\/spin-glass} before turning off the external field. We say that the system \\textit{ages}.\n\nThere exist two mirror experimental setups that are proven to be equivalent \\cite{nordblad:86}: the relaxation\\index{relaxation} of the \\gls{TRM} and the relaxation\\index{relaxation} of the \\gls{ZFC} magnetization\\index{magnetization!thermo-remanent}.\n\nThe typical protocol to study the \\gls{TRM}\\index{magnetization!thermo-remanent} is the following. First, we set a small external magnetic field and the system is cooled in its presence from a temperature $T_0$ above the critical\\index{critical temperature} temperature\\footnote{The temperature that separates the low-temperature phase\\index{phase!low-temperature\/spin-glass} and the high-temperature one\\index{phase!high-temperature\/paramagnetic}.} $\\ensuremath{T_\\mathrm{c}}\\xspace$ to a temperature $T_1<\\ensuremath{T_\\mathrm{c}}\\xspace$. We let the system age at temperature $T_1$ for a waiting time $t_w$ and then, the external magnetic field is switched off. At that moment, the decreasing magnetization\\index{magnetization} is recorded as a function of time $t$ where $t=0$ corresponds to the moment in which we turned off the field. One part of the total magnetization\\index{magnetization} falls immediately, the so-called reversible magnetization\\index{magnetization}. The other part is the so-called remanent magnetization\\index{magnetization!thermo-remanent} and falls slowly with the time $t$. The slow fall of the remanent magnetization\\index{magnetization!thermo-remanent} is shown in \\reffig{TRM_relaxation} from \\cite{vincent:97}. The reader may find similar experiments of \\gls{TRM}\\index{magnetization!thermo-remanent} relaxation\\index{relaxation} processes in \\cite{chamberlin:84,ocio:85,nordblad:86,alba:86}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{intro\/TRM_relaxation.png}\n\\caption[\\textbf{Thermo-remanent magnetization in spin glass.}]{\\textbf{Thermo-remanent magnetization in spin glass.} Thermo-remanent magnetization\\index{magnetization!thermo-remanent} $M$ normalized by the field-cooled value $M_{fc}$ is plotted against the time $t$ in a semi-log scale. \\gls{AgMn} with the $2.6 \\%$ of impurities is cooled from $T_0 >\\ensuremath{T_\\mathrm{c}}\\xspace=10.4$ K to $T_1=9$ K$ = 0.87 \\ensuremath{T_\\mathrm{c}}\\xspace$ in the presence of an external field of $0.1$ Oe. The system stays at $T_1$ with the field for a time $t_w$, then, the field is cut and the decaying magnetization\\index{magnetization} is recorded as a function of time $t$ where $t=0$ corresponds to the moment of turning off the field. Different curves corresponds to different $t_w$. Figure from \\cite{vincent:97}.}\n\\labfig{TRM_relaxation}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{intro\/TRM_relaxation_collapse.png}\n\\caption[\\textbf{The relevant time-scale of the thermo-remanent magnetization relaxation.}]{\\textbf{The relevant time-scale of the thermo-remanent magnetization relaxation\\index{relaxation}.} Thermo-remanent magnetization\\index{magnetization!thermo-remanent} $M$ normalized by the field-cooled value $M_{fc}$ is plotted against $t\/t_w$ in a semi-log scale in the same experimental conditions that in \\reffig{TRM_relaxation}. An almost perfect collapse is observed. Figure from \\cite{vincent:97}.}\n\\labfig{TRM_relaxation_collapse}\n\\end{figure}\n\nIn fact, one can observe in \\reffig{TRM_relaxation} for every curve an inflection point which roughly corresponds to $t=t_w$. The natural representation is, therefore, the one showed in \\reffig{TRM_relaxation_collapse} where the abscissa axis corresponds to the time normalized as $t\/t_w$. A perfect collapse of the curves under this representation is known as \\textit{full aging\\index{aging}}. However, the collapse is only approximated. The scaling variable\\footnote{The original symbol associated with this quantity, and the most used is $\\xi$, however, we use here $\\tau$ to avoid confusion with the coherence length\\index{coherence length} $\\xi$.} $\\tau$, firstly used in structural\\index{structural glass} glasses \\cite[p.~129]{struik:80} and lately introduced in \\gls{SG} by \\cite{ocio:85} as quoted by \\cite{rodriguez:03}, solves the problem and achieves a much better collapse\n\\begin{equation}\n\\tau=\\dfrac{t_w^{1-\\mu}}{1-\\mu}\\left[ \\left( 1 + \\dfrac{t}{t_w} \\right)^{1-\\mu} -1 \\right] \\quad \\mu < 1 \\,\\, . \\labeq{not_full_aging}\n\\end{equation}\n\nPutting the subtleties aside, we now know that the relevant time-scale of the aging\\index{aging} processes is $t_w$ and this is fundamental, as we will discuss in the following chapters because it has a deep relation with the coherence length\\index{coherence length} $\\xi$ acting as the key quantity governing the non-equilibrium phenomena, see~\\refch{aging_rate}.\n\nThe mirror protocol is the \\gls{ZFC}. In that protocol, the sample\\index{sample} is cooled from a temperature $T_0 > \\ensuremath{T_\\mathrm{c}}\\xspace$ to a temperature $T_1<\\ensuremath{T_\\mathrm{c}}\\xspace$ in zero-field. After a time $t_w$ a small field is turned on and the magnetization\\index{magnetization} is recorded as a function of time $t$, analogously to the previous experiment, $t=0$ corresponds to the moment in which the field is applied. This protocol is equivalent to the \\gls{TRM}\\index{magnetization!thermo-remanent} but with an increasing magnetization\\index{magnetization}. Furthermore, the sum of the \\gls{ZFC}-magnetization\\index{magnetization!zero-field-cooled} plus the \\gls{TRM}\\index{magnetization!thermo-remanent} is equal to the field-cooled magnetization\\index{magnetization}\\footnote{This is not true in general, the reader should note that we are in the small field limit where the only relevant term is the linear one. The equality is not guaranteed for larger fields where the non-linear responses are sizable, see for example recent relevant works~\\cite{zhai-janus:20,zhai-janus:21}}. For experimental results of this protocol, see \\cite{lundgren:83,nordblad:86}.\n\nAn experiment that can be regarded as a generalization (if that word can be used in the context of experiments) was performed, actually, earlier by Nagata \\textit{et al.}, see \\cite{nagata:79}. The main results of their research can be summarized in \\reffig{nagata_dcsuscept}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{intro\/nagata_dcsuscept.png}\n\\caption[\\textbf{DC-Susceptibility in CuMn spin glass.}]{\\textbf{DC-Susceptibility\\index{susceptibility!dc} in CuMn spin glass.}  Susceptibility\\index{susceptibility!dc} of a DC applied-field $h=5.9$ G is measure in a $1.08 \\%$ and a $2.02 \\%$ CuMn spin-glass and plotted against the temperature. Curves (a) and (c) corresponds to a field cooled protocol while curves (b) and (d) corresponds to a zero-field cooled protocol. Arrows indicate the reversibility ($\\leftrightarrow$) or irreversibility ($\\rightarrow$) of the protocol. Figure from \\cite{nagata:79}.}\n\\labfig{nagata_dcsuscept}\n\\end{figure}\n\nIn this experiment, the authors measure the magnetic response to an applied magnetic field i.e. \\textit{the susceptibility}\\index{susceptibility} $\\chi = \\lvert \\vec{M} \\lvert \/ \\lvert \\vec{H} \\lvert$.\n\nTwo different protocols are performed in this experiment. In the protocol corresponding to the curves (a) and (c), the sample\\index{sample} is cooled in the presence of a constant field $h = 5.9$ G. Below the critical\\index{critical temperature} temperature $\\ensuremath{T_\\mathrm{c}}\\xspace$ the susceptibility\\index{susceptibility} remains almost constant and the process is reversible.\n\nOn the contrary, in the protocol corresponding to the curves (b) and (d), the sample\\index{sample} is cooled in absence of any field. Then, the sample\\index{sample} is heated and the susceptibility\\index{susceptibility} increases monotonically until it reaches the critical\\index{critical temperature} temperature. Moreover, cooling again the system in presence of the field leads to an irreversible behavior. Above $\\ensuremath{T_\\mathrm{c}}\\xspace$ both protocols are identical.\n\nMoreover, if we switch on the field after \\gls{ZFC} and let the system evolve at a fixed temperature, we observe that the susceptibility\\index{susceptibility} grows towards the field-cooled value but without reaching it in the experimental time-scales. This is the connection to the aging\\index{aging}-experiments showed above, where the temperature cycle\\index{temperature cycle} was performed only between two temperatures, but the aging\\index{aging} and the non-equilibrium behavior were captured the same.\n\nThe experiments exposed above are stressing us two main things:\n\\begin{enumerate}\n\\item Experimental \\gls{SG}s are out of equilibrium in the experimental time-scales.\n\\item The system ages, i.e. the time that the system expends below $\\ensuremath{T_\\mathrm{c}}\\xspace$ is a key quantity to understand its non-equilibrium behavior in the low-temperature phase\\index{phase!low-temperature\/spin-glass}.\n\\end{enumerate}\n\n\\subsubsection{Memory and rejuvenation}\n\n\\begin{figure}[h]\n\\centering\n\\hspace{-3cm}\n\\includegraphics[width=0.8\\textwidth]{intro\/memory_rejuvenation.png}\n\\caption[\\textbf{Memory and rejuvenation experiment in spin glasses.}]{\\textbf{Memory\\index{memory effects} and rejuvenation\\index{rejuvenation} experiment in spin glasses.} Out-of-phase susceptibility\\index{susceptibility!ac} is recorded when temperature vary in the presence of a sinusoidal field $h_\\mathrm{ac}$ of frequency $f=0.04$ Hz and peak amplitude of $0.3$ Oe. Main plot corresponds to CdCr$_{1.7}$In$_{0.3}$S$_4$ and the inset corresponds to the same plot for CuMn. Figure from \\cite{jonason:98}.}\n\\labfig{memory_rejuvenation}\n\\end{figure}\n \n \nIn previous experiments, when a constant field was applied, we defined the so-called \\textit{dc-susceptibility}\\index{susceptibility!dc}, therefore, a straightforward generalization is the \\textit{ac-susceptibility}\\index{susceptibility!ac}, which is none but the magnetic response of the system when a sinusoidal field is applied to it. \n\nIn \\gls{SG}s, the susceptibility\\index{susceptibility!ac} $\\chi_\\mathrm{ac}$ measured from a sinusoidal applied field $h_{\\mathrm{ac}}$ is also a sinusoidal quantity that presents a phase-delay $\\varphi$ with respect to the field $h_{\\mathrm{ac}}$. This delay lead to the definition of two quantities: the in-phase susceptibility\\index{susceptibility!ac} $\\chi'$ and the out-of-phase susceptibility\\index{susceptibility!ac} $\\chi''$\n\\begin{equation}\n\\begin{split}\n\\chi' & = \\chi \\cos \\varphi \\\\\n\\chi'' & = \\chi \\sin \\varphi\n\\end{split}\n\\end{equation}\n\nIn \\cite{jonason:98} \\footnote{Experiments of memory\\index{memory effects} and rejuvenation\\index{rejuvenation} in temperature cycles\\index{temperature cycle} of two temperatures were performed earlier, see for example \\cite{lefloch:92} however, the richer phenomenology of \\cite{jonason:98} make us to focus on this experiment for the sake of clarity.}, the authors measured the out-of-phase susceptibility\\index{susceptibility!ac} when a $h_\\mathrm{ac}$ field of frequency $f=0.04$ Hz and peak amplitude of $0.3$ Oe was applied to a \\gls{CdCrInS} \\gls{SG}. The main result of this research is summarized in \\reffig{memory_rejuvenation}\n\nThe system was cooled from $T>\\ensuremath{T_\\mathrm{c}}\\xspace=16.7$ K down to $T_f=5$ K at a constant cooling-rate of $0.1$ K\/min. Then, the system was heated back continuously at the same rate.\n\n$\\chi''$ was recorded during the cooling and heating protocol, resulting in two close curves (still slightly different) where the heated one was used as a reference curve (continuous line in \\reffig{memory_rejuvenation}). Next, the experiment was repeated but the system was let age at temperatures $T_1=12$ K and $T_2=9$ K for a waiting time $t_w=7$ h for $T_1$ and $t_w=40$ h for $T_2$. After the age, the cooling was restarted at the same rate and the out-of-phase susceptibility\\index{susceptibility!ac} merged back with the reference curve, it is said that the system \\textit{rejuvenates}. The measurements of this protocol correspond to the white squares in \\reffig{memory_rejuvenation}. \n\nLast, the system was reheated at a constant heating rate. When the temperature approached the age temperatures $T_1$ and $T_2$, the system, somehow, ``remembered'' the age and followed the susceptibility\\index{susceptibility} curve, reproducing the data of the cooling protocol. In the inset, the authors included the same experiment appearing in \\cite{djurberg:99} in a \\gls{CuMn} \\gls{SG}, where the behavior was completely similar.\n\nThis experiment sends a very clear message: the aging\\index{aging} at a temperature $T$ does not affect the susceptibility\\index{susceptibility} value at lower temperatures. This temperature independence has been commonly related to the \\textit{temperature chaos}\\index{temperature chaos} phenomenon that we extensively treat in \\refch{Introduction_chaos}, \\refch{equilibrium_chaos} and \\refch{out-eq_chaos} and, in particular, the \\refch{out-eq_chaos} sums up the spirit of this thesis: separated researches from the experimental and the theoretical point of view that can be related by numerical simulations.\n\n\\section{Theoretical spin glasses} \\labsec{theoretical_spinglass}\nIn this section, we will briefly expose the theoretical development of the \\gls{SG}'s theory. The aim of this section is not to deeply review all the important results on \\gls{SG} but to provide some context of its history and to highlight the main theoretical predictions which we will deal with by means of numerical simulations.\n\nThe tools of statistical mechanics that we use in this section will be introduced in \\refsubsec{tools_statistical_mechanics}. Then, the most popular theoretical models will be shown in \\refsubsec{theoretical_models}. Finally, we will present the main theoretical pictures which predict different results in the \\gls{SG} low-temperature phase\\index{phase!low-temperature\/spin-glass} in \\refsubsec{theoretical_pictures}. One of the tasks of our numerical simulations will be partially to discriminate between them.\n\n\\subsection{The tools of statistical mechanics}\n\\labsubsec{tools_statistical_mechanics}\nWe will present some basic results on statistical mechanics that will be useful in \\gls{SG} theory. The reader may find general references in \\cite{landau:80,lebellac:91,amit:05,greiner:12}.\n\nAlthough most of the things we are saying in this part are general for statistical mechanics, the notation and the language will be always focused on spin systems.\n\nWe consider a system described by $N$ microscopic variables $\\{s_i\\}$. The most basic quantity describing the system is the \\textit{energy}\\index{energy}\n\\begin{equation}\nE(\\{s_i\\}) \\equiv \\mathcal{H}(s_0,s_1,\\dots,s_n) \\, , \\labeq{Hamiltonian}\n\\end{equation}\na function (usually called \\textit{Hamiltonian}\\index{Hamiltonian}) of the microscopic variables $\\{s_i\\}$ that tends to be minimized when the system approach the thermal equilibrium.\n\nHowever, the energy\\index{energy} is not the only quantity describing the system, nor the only quantity we are interested in. A general quantity $\\mathcal{O}$ depending on the concrete values of the microscopic variables is called \\textit{observable}: $\\mathcal{O}(\\{s_i\\})$. \n\nIn general, the fluctuation of the observables due to the fluctuation of the microscopic variables makes desirable to consider averaged quantities\n\\begin{equation}\n\\braket{\\mathcal{O}} = \\lim_{t\\to \\infty} \\dfrac{1}{t} \\int_0^t \\mathcal{O}(\\{s_i\\}(\\tau)) d\\tau \\, . \\labeq{time_average}\n\\end{equation}\n\nThe problem now is clear. To compute these averages we need to know the time-evolution of a macroscopic number of variables, usually interacting with each other. Here emerges the fundamental principle of the statistical mechanics: the system, when it is at equilibrium, will eventually reach every possible set of microscopical variables, namely \\textit{configuration}\\index{configuration}. If we could know which is the probability of each configuration\\index{configuration} to appear, we could trade the integral over infinitely large periods with the weighted sum over all the possible configurations\\index{configuration} of the system.\n\nIn the canonical ensemble\\index{canonical ensemble}, the probability distribution of the configurations\\index{configuration} $\\{s_i\\}$ is\n\\begin{equation}\nP(\\{s_i\\}) = \\dfrac{e^{-\\beta \\mathcal{H}(\\{s_i\\})}}{Z} \\, , \\labeq{prob_configuration}\n\\end{equation}\nwhere $\\beta$ is the inverse of the temperature $T$ of the heat bath\\footnote{Which coincides with the temperature of the system in thermal equilibrium.} in units such that the \\textit{Boltzman constant} $k_B=1$ and $Z$ is a normalization factor called \\textit{Zustandssumme} or \\textit{partition function}\\index{partition function}\\footnote{For convenience, we are assuming a discrete number of possible configurations\\index{configuration} in the phase-space\\index{phase space}. In general, the expression of the partition function\\index{partition function} involves an integral instead of a sum.}\n\\begin{equation}\nZ = \\sum_{\\{s_i\\}} e^{-\\beta \\mathcal{H}(\\{s_i\\})} \\, . \\labeq{partition_function}\n\\end{equation}\n\nTherefore, the computation of the averages is\n\\begin{equation}\n\\braket{\\mathcal{O}} = \\dfrac{ \\sum_{\\{s_i\\}} \\mathcal{O}(\\{s_i\\}) e^{-\\beta \\mathcal{H}(\\{si\\})}}{Z} \\, . \\labeq{average_o}\n\\end{equation}\n\nIt is worthy to note that we have fully characterized the system at equilibrium through the partition function\\index{partition function} however, the very related quantity $F$, the \\textit{free energy\\index{free energy}}\\footnote{Note that we use $\\log$ as the symbol for the natural logarithm.}\n\\begin{equation}\nF = - \\dfrac{1}{\\beta} \\log Z(\\{s_i\\}) \\, ,\n\\end{equation}\nturns out to be much more practical because it is directly related to measurable quantities, fundamental in the thermodynamics of the system e.g. the \\textit{energy}\\index{energy} $U$ or the \\textit{magnetization}\\index{magnetization} $M$\n\\begin{equation}\nU = - \\left. \\dfrac{\\partial \\left( \\beta F\\right)}{\\partial T}\\right|_H \\quad , \\quad M = - \\left. \\dfrac{\\partial \\left( \\beta F\\right)}{\\partial H}\\right|_T \\labeq{thermodynamic_quantities} \n\\end{equation}\n\n\\subsubsection{Disorder and self-averaging}\nThe systems appearing along this thesis present a particularity: the interaction between any pair of particles is random. This randomness\\index{randomness}, extended in the whole system is what we call \\textit{disorder}\\index{disorder} and needs to be characterized by additional variables, taking into account the interaction between particles. We denote those variables $\\{J\\}$.\n\nThe disorder\\index{disorder} variables $\\{J\\}$ can exhibit a dynamical evolution and the time-scale of its evolution will determine if we are treating with \\textit{annealed} disorder or \\textit{quenched} disorder.\n\nOn the one side, the annealed disorder\\index{disorder!annealed} occurs when the time-scale of the dynamical evolution of $\\{J\\}$ is shorter than the observation time. Therefore, the interactions can be regarded as a sort of dynamic variables and we can average over them the same that we average over the configuration\\index{configuration} of the system\n\\begin{equation}\n\\overline{\\mathcal{O}_J} = \\int \\mathcal{O}(J) P(J) dJ \\, , \\labeq{average_disorder}\n\\end{equation}\nwhere $P(J)$ is the probability distribution that follows the disorder variables $J$. Therefore, in this situation, the free energy\\index{free energy} of the system is\n\\begin{equation}\nF = - \\dfrac{1}{\\beta} \\log \\overline{Z_J} \\, . \\labeq{free_energy_annealed}\n\\end{equation}\n\nThe computations associated with the annealed disorder have no additional difficulty, we have just added a fashion hat to our quantities, but the treatment is essentially the same. \n\nOn the other side, we have the quenched disorder\\index{disorder!quenched}, in which the time-scale of the dynamical evolution of $\\{J\\}$ is much larger than the observation time. In this case, each system is different due to the disorder\\index{disorder!quenched} \n\\begin{equation}\nF_J = - \\dfrac{1}{\\beta} \\log Z_J \\, .\n\\end{equation}\n\n\\textit{A priori}, there is no hope of universality\\index{universality} in those systems and we are forced to study them individually. However, the \\textit{self-averaging}\\index{self-averaging} property emerges to rescue us. In the thermodynamic limit\\index{thermodynamic limit}, for a system with $N$ degrees of freedom\\index{degree of freedom}, we have\n\\begin{equation} \n\\lim_{N \\to \\infty}\\dfrac{F_J}{N} = f_\\infty \\, , \\labeq{self_average}\n\\end{equation}\nbeing $f_\\infty$ the free-energy\\index{free energy!density} density in the thermodynamic limit\\index{thermodynamic limit}, which is independent of the disorder\\index{disorder} variables $\\{J\\}$.\n\nAn argument supporting this property can be found in \\cite{brout:59}. The reasoning is the following: any macroscopic system can be divided into a statistically large number $n$ of macroscopic systems. Due to the quenched disorder\\index{disorder!quenched}, every subsystem will have a different free energy\\index{free energy} $F_{J_j}$ with $j=(0,1,\\dots,n)$. Can we relate those free energies\\index{free energy} with the free energy\\index{free energy} of the whole system? \n\nThe free energy\\index{free energy} is just the logarithm of the partition function\\index{partition function} whose dependence of the interactions is codified in the Hamiltonian\\index{Hamiltonian} $\\mathcal{H}_J(\\{s_i\\})$. If we assume short-range interactions, which is a physically reasonable assumption, the total free energy\\index{free energy} will be the sum of the free energies\\index{free energy} of the subsystem plus the contribution to the free energy\\index{free energy} of the interface between the subsystems. As long as $N \\to \\infty$, the interface between subsystems will be negligible against the volume of those subsystems and, therefore \\refeq{self_average} holds. The implicit corollary is that computing the free energy\\index{free energy} of a large enough system is equivalent to compute the sum of the free energy\\index{free energy} for smaller systems.\n\nIf the distribution probability of the couplings\\index{couplings} $J$ is not pathological, we expect\n\\begin{equation}\n\\overline{F_J^2} - \\overline{F_J}^2 \\propto \\dfrac{1}{N} \\, .\n\\end{equation}\n\nThe computation of the disorder\\index{disorder!average} average of the logarithm \n\\begin{equation}\nF = \\overline{F_J} = -\\dfrac{1}{\\beta} \\overline{\\log Z_J} \\, ,\n\\end{equation}\nis one of the biggest difficulties on studying statistical mechanics on disordered\\index{disorder!systems} systems and we deal with it by using the so-called \\textit{replica method}\\index{replica!method\/trick}.\n\nIt is worthy to note that, the crucial hypothesis for Brout's argument is that the contribution of the boundaries of those subsystems is negligible. There exist some situations in which this assumption is no longer valid (see for instance \\cite{binder:86} for a more detailed explanation). For example, when a phase transition\\index{phase transition} occurs, the boundary conditions\\index{boundary conditions} become crucial and the system is no longer self-averaging\\index{self-averaging}. \n\nIt is well known that, for a spin glass below the critical\\index{critical temperature} temperature, some quantities are non-self-averaging\\index{self-averaging!non-}. This feature is closely related to the concept of \\textit{dispersed metastate\\index{metastate!dispersed}} (see \\refch{metastate}).\n\n\\subsubsection{The replica method}\nIn order to compute $\\overline{F_J}$ we use the so-called \\textit{replica method}\\index{replica!method\/trick} or \\textit{replica trick}, that was firstly introduced in the context of \\gls{SG}s in \\cite{edwards:75}.\n\nThe method is based on the expression\n\\begin{equation}\n\\log Z = \\lim_{n\\to 0} \\dfrac{Z^n -1}{n} \\, , \\labeq{replica_expression}\n\\end{equation}\nwhich is direct if we use the Taylor expansion of the right-side expression\n\\begin{equation}\n\\dfrac{Z^n -1}{n} = \\dfrac{e^{n \\log Z}-1}{n} = \\dfrac{1 + n \\log Z + O(n^2)-1}{n} = \\log Z + O(n) \\, . \\labeq{replica_expression_explained}\n\\end{equation}\n\nAt this point, we consider $n$ identical systems $Z_J^{(a)}$ $a=(0,1,\\dots,n)$ also called \\textit{replicas}\\index{replica} i.e. systems with the same realization of the disorder\\index{disorder} $J$, and we define\n\\begin{equation}\nZ_n \\equiv \\overline{Z_J^n} = \\overline{\\prod_{a=1}^n Z_J^{(a)}} \\, ,\n\\end{equation}\nand \n\\begin{equation}\nF_n = -\\lim_{n\\to 0} \\dfrac{1}{n \\beta}  \\log Z_n \\, .\n\\end{equation}\n\nBy using \\refeq{replica_expression}, is easy to see now that $F_n = \\overline{F_J}$\n\\begin{equation}\nF_n = -\\lim_{n\\to 0} \\dfrac{1}{n \\beta}  \\log \\overline{Z_J^n} = -\\dfrac{1}{\\beta} \\lim_{n \\to 0} \\dfrac{\\log \\left(1+n\\overline{\\log Z_J}+ O(n^2) \\right)}{n} = - \\dfrac{1}{\\beta} \\overline{\\log Z_J} = \\overline{F_J} \\, . \\labeq{replica_trick}\n\\end{equation}\n\nThe computation of $F_n$ is easier than the computation of $\\overline{F_J}$ if we first compute $\\log \\overline{Z_J^n}$ with $n$ integer and then we take the limit $n \\to 0$. This is, indeed, a very doubtful step in the mathematical sense. We consider $n$ to be an integer in order to define $Z_n$, then, we take the analytical extension of $Z_n$ for $n \\in \\mathbb{R}$ and finally we take the limit of $n\\to 0$. In \\refsubsec{theoretical_models} we will introduce a practical use for the replica method\\index{replica!method\/trick}.\n\nIn the cases in which the free energy\\index{free energy} is an analytic function of the temperature\\footnote{which usually happens in the high-temperature phase\\index{phase!high-temperature\/paramagnetic} of magnetic systems.} $\\beta$, the replica method\\index{replica!method\/trick} is exact. Moreover, when other methods are available to compute the free-energy\\index{free energy}, the results coincide with the predictions of the replica method\\index{replica!method\/trick}.\n\nThere exist also an alternative approximation to the replica method\\index{replica!method\/trick} which not requires the \\textit{trick} of the duality nature of $n$ integer-real, see \\cite{dotsenko:93,coolen:93,dotsenko:94}.\n\n\\subsection{Theoretical models} \\labsubsec{theoretical_models}\nHere, we discuss the main models that capture the \\gls{SG} physics and that are relevant in the development of the field. The trade-game is clear, on the one hand, the model should be detailed enough to exhibit the main \\gls{SG} phenomenology. We have discussed which are the basic ingredients for that: randomness\\index{randomness} and frustration\\index{frustration}. On the other hand, the model should be also simple enough to be analytically tractable. The \\gls{EA}\\index{Edwards-Anderson!model} model is the first we are going to introduce here, for historical reasons, but also due to its relevance in the actual context of \\gls{SG} and the present thesis. We are also going to define the \\gls{SK}\\index{Sherrington-Kirkpatrick} model, which represents the mean-field\\index{Mean-Field!model} approximation, which is analytically solvable, and which characterizes the \\gls{SG}s at infinite dimension.\n\nNonetheless, the aim of this part is not to deeply review all the theoretical results in \\gls{SG}s, that can be found in several places \\cite{mezard:87,dotsenko:01,dedominicis:06}, but to give some historical context and present results affecting the very nature of the \\gls{SG}s, an issue still debated nowadays at finite dimensions.\n\n\\subsubsection{Edwards-Anderson model}\nThis model was proposed by Edwards and Anderson\\index{Edwards-Anderson!model} in \\cite{edwards:75}. Now, the general degrees of freedom\\index{degree of freedom} we defined in \\refsubsec{tools_statistical_mechanics} are, indeed, spins which lie in a regular lattice\\index{regular lattice} and that interact with each other through the couplings\\index{couplings} $\\{J\\}$. The Hamiltonian\\index{Hamiltonian} of this model is\n\\begin{equation}\n\\mathcal{H} = - \\sum_{\\braket{i,j}} J_{ij} \\vec{s}_i \\vec{s}_j - \\sum_i \\vec{h}_i \\vec{s}_i \\, , \\labeq{ea_Hamiltonian_theory}\n\\end{equation}\nwhere $\\vec{s}_i$ is a unitary vector, $\\vec{h}$ is an external magnetic field and the sum over $\\braket{i,j}$ denotes the sum over the pairs of spins $s_i$, $s_j$ that are bounded by a coupling\\index{couplings} $J_{i,j}$ and that actually depends on the concrete form of the considered lattice. Along this thesis we will focus on the case $\\vec{h}_i = \\vec{0}$, therefore, from now on we address the particular case of non external magnetic field for the sake of simplicity.\n\nIf the spin vector is 3-dimensional, the system is called \\textit{Heisenberg\\index{Heisenberg} spin glass}, if the vector is 2-dimensional it is called $XY$ \\textit{spin glass} and, if the spin only can take values $s_i = \\pm 1$ we say that it is an \\textit{Ising\\index{Ising} spin glass}. From now on, we will focus on the Ising\\index{Ising} \\gls{SG}, which is actually a reasonable assumption in many systems (see \\refsubsec{source_randomness}) and is the particular case of our numerical simulations in the research developed throughout this thesis. \n\nMoreover, the most popular choice is to consider only nearest-neighbor interactions between the spins\\footnote{The rationale of this approach is the short-ranged nature of the interactions between spins.}, with the quenched variables $J_{ij}$ following a Gaussian\\index{Gaussian!distribution} probability distribution, however, the particular shape of the distribution is not very important and a very popular choice, apart from the Gaussian\\index{Gaussian!distribution}, is the bimodal one ($J_{ij} = \\pm J$) that we will use in the numerical simulations.\n\nThe \\gls{EA}\\index{Edwards-Anderson!model} model also brought a proposal for the order parameter\\index{order parameter!Edwards-Anderson} controlling the phase transition\\index{phase transition}: the \\textit{overlap\\index{overlap}}. The traditional order-parameters displayed in \\refeq{no_magnetic_order} are not valid in \\gls{SG} because, by definition, \\gls{SG} exhibit no long-range order. Nonetheless, the frozen and disorder\\index{disorder} nature of the glassy phase\\index{phase!low-temperature\/spin-glass} suggests a different order parameter\\index{order parameter!Edwards-Anderson} based on time correlations on the same site.\n\\begin{equation}\nq_{EA} = \\dfrac{1}{N} \\lim_{t \\to \\infty}  \\sum_i \\braket{s_i(t=0)s_i(t)}_t \\, . \\labeq{order_parameter_SG}\n\\end{equation}\n\nThe question that is answered by \\refeq{order_parameter_SG} is, therefore, how similar is the configuration\\index{configuration} of the system at a time $t$ compared to the configuration\\index{configuration} at time $t=0$? This time average does not seem very useful, but fortunately, by the same reasoning made in \\refsubsec{tools_statistical_mechanics} we can trade the time average by the weighted sum over all the possible configurations\\index{configuration}\n\\begin{equation}\nq_{EA} = \\dfrac{1}{N} \\sum_i \\braket{s_i}^2 \\, . \\labeq{order_parameter_SG_2}\n\\end{equation}\n\nWe expect $q_{EA} = 0$ for $T > \\ensuremath{T_\\mathrm{c}}\\xspace$ i.e. in the paramagnetic phase\\index{phase!high-temperature\/paramagnetic} $\\braket{s_i}=0$. The expectation for $T \\to 0$ is $q_{EA} \\to 1$.\n\nAs a final remark, let us note that the very existence of a phase transition\\index{phase transition} in the Ising\\index{Ising} \\gls{EA}\\index{Edwards-Anderson!model} model was not completely accepted (even with the existence of an earlier consensus~\\cite{kawashima:96,iniguez:96,iniguez:97,berg:98,janke:98,marinari:98}) until the beginning of the XXI century~\\cite{palassini:99,ballesteros:00}.\n\n\\subsubsection{Mean Field: the Sherrington-Kirkpatrick model}\n\nEven with the aim of simplicity in mind, the \\gls{EA}\\index{Edwards-Anderson!model} model is not simple to solve, nor its mean-field\\index{Mean-Field!model} version, the \\gls{SK}\\index{Sherrington-Kirkpatrick} model:\n\\begin{equation}\n\\mathcal{H} = - \\sum_{i<j} J_{ij} s_i s_j \\, .\n\\end{equation}\n\nThe reader may have noticed one main difference between this Hamiltonian\\index{Hamiltonian} and that appearing in \\refeq{ea_Hamiltonian_theory}, in addition to the absence of an external field $\\vec{h}$. Now, the sum is performed over every pair of sites with the restriction of $i<j$ to avoid double counting. In consequence, no useful concept of distance between spins exists, the interactions are infinite-ranged.\n\nMoreover, for convenience, the couplings\\index{couplings} $J_{ij}$ follow a Gaussian\\index{Gaussian!distribution} distribution probability\n\\begin{equation}\nP(J_{ij}) = \\dfrac{1}{\\sqrt{2\\pi \\sigma^2}} \\exp \\left[-\\dfrac{(J_{ij}-\\mu)^2}{2\\sigma^2} \\right]  \\, ,\n\\end{equation}\nwhere $\\mu$ and $\\sigma^2$ stand for the mean and the variance of the Gaussian\\index{Gaussian!distribution} distribution and take the values\n\\begin{equation}\n\\mu = \\overline{J_{ij}} = 0 \\quad , \\quad \\sigma^2 = \\overline{J_{ij}^2} = \\dfrac{1}{N} \\, .\n\\end{equation}\n\nThis particular choice of $\\sigma^2$ makes the free-energy\\index{free energy!density} density $f=F\/N$ independent of the total number of spins $N$.\n\n\\subsubsection{The replica symmetric solution}\n\nWith this information, we are able to compute the key quantity, the free energy\\index{free energy}, or equivalently, $Z_n$\n\\begin{align}\n Z_n & = \\int_{\\mydots} \\int P(J) \\prod_{a=1}^n \\sum_{\\{s^a\\}} \\exp \\left( \\beta \\sum_{i<j}^N J_{ij} s^a_i s^a_j \\right) {\\mathrm{d}} \\{J\\} = \\nonumber \\\\ \n & = \\left(\\dfrac{N}{2\\pi}\\right)^{N(N-1)\/4} \\sum_{\\{s^a\\}} \\int_{\\mydots} \\int \\exp \\left(\\beta \\sum_{a=1}^n \\sum_{i<j}^N J_{ij} s_i^a s_j^a - \\dfrac{1}{2}N \\sum_{i<j}^N J^2_{ij} \\right) {\\mathrm{d}} \\{J\\} \\, ,\n\\end{align} \nwhere the notation $\\int_{\\mydots} \\int$ and ${\\mathrm{d}} \\{J\\} = \\prod_{i<j} {\\mathrm{d}} J_{ij}$ denotes that we are dealing with multiple integrals, one for each pair $i<j$ and the sum over the configurations\\index{configuration} now runs over all the possible configurations\\index{configuration} for each replica\\index{replica} with superindex $a$.\n\nAfter some computations for the exchange in the sum order, we get the expression\n\\begin{equation}\nZ_n = \\sum_{\\{s^a\\}} \\exp \\left[ \\dfrac{\\beta^2 N n}{4} + \\dfrac{\\beta^2 N}{2} \\sum_{a<b}^n \\left( \\dfrac{1}{N}\\sum_i s_i^a s_i^b \\right)^2 \\right] \\, .\n\\end{equation}\n\nWe can now step back by introducing an integral variable depending on the pair of replicas\\index{replica} $a,b$ instead of the pair of sites $i,j$ with the help of the Hubbard-Stratonovich\\index{Hubbard-Stratonovich} transformation\n\\begin{equation}\nZ_n = \\beta \\sqrt{\\dfrac{N}{2\\pi}} \\int_{\\mydots}\\int \\sum_{\\{s^a\\}} \\exp \\left( \\dfrac{\\beta^2 N n}{4}-\\dfrac{\\beta^2 N}{2} \\sum_{a<b}^n Q_{ab}^2 + \\beta^2 \\sum_{a<b}^n \\sum_i^N Q_{ab}s_i^a s_i^b \\right) {\\mathrm{d}} \\{Q\\} \\, , \\labeq{partition_function_mean_field_1}\n\\end{equation}\nwhere now, $\\int_{\\mydots}\\int$ and $d\\{Q\\} = \\prod_{a<b} {\\mathrm{d}} Q_{ab}$ represent multiple integrals, one for each pair $a<b$.\n\nAfter some basic algebra, we can write \\refeq{partition_function_mean_field_1} as\n\\begin{equation}\nZ_n = \\int_{\\mydots}\\int e^{-\\beta N n f_n(\\{Q\\})} {\\mathrm{d}} \\{Q\\} \\, , \\labeq{partition_function_mean_field_2}\n\\end{equation}\nwhere we can see that $f_n(\\{Q\\})$ has a free-energy-like\\index{free energy} structure\n\\begin{equation}\nf_n(\\{Q\\})= -\\dfrac{\\beta}{4} + \\dfrac{\\beta}{2n}\\sum_{a<b}^n Q_{ab}^2 - \\dfrac{1}{\\beta N n} \\log \\left[ \\sum_{\\{s^a\\}} \\exp \\left( \\beta^2 \\sum_{i}^N \\sum_{a<b}^n Q_{ab} s^a_i s^b_i \\right) \\right] \\, ,\n\\end{equation}\nwith the effective Hamiltonian\\index{Hamiltonian}\n\\begin{equation}\n\\mathcal{H}_{\\mathrm{eff}} (\\{Q\\}) = -\\beta \\sum_i^N \\sum_{a<b}^n Q_{ab}s_i^a s_i^b \\, . \\labeq{effective_Hamiltonian_mean_field}\n\\end{equation}\n\nIn the thermodynamic limit\\index{thermodynamic limit} we can compute the integral of \\refeq{partition_function_mean_field_2} through the saddle-point\\index{saddle point} approximation. The minimization of $Z_n$ in \\refeq{partition_function_mean_field_2} is given by the equations $\\partial f_n(\\{Q\\})\/ \\partial Q_{ab}=0$ for all the pairs $a<b$\n\\begin{equation}\n\\dfrac{\\partial f(\\{Q\\})}{\\partial Q_{ab}} = \\dfrac{\\beta}{n} Q_{ab} - \\dfrac{\\beta}{nN} \\dfrac{\\sum_{\\{s^a\\}} \\sum_i^N s_i^a s_i^b e^{-\\beta \\mathcal{H}_{\\mathrm{eff}}(\\{Q\\})}}{\\sum_{\\{s^a\\}} e^{-\\beta \\mathcal{H}_{\\mathrm{eff}}(\\{Q\\})}} = 0 \\, ,\n\\end{equation}\nwhich leads to the solution\n\\begin{equation}\nQ_{ab} = \\dfrac{1}{N} \\sum_i^N \\braket{s_i^a s_i^b}_Q \\, , \\labeq{mean_field_solution_general}\n\\end{equation}\nwhere $\\braket{\\cdot}_Q$ represents the usual average defined in \\refeq{average_o} but with the effective Hamiltonian\\index{Hamiltonian} $\\mathcal{H}_{\\mathrm{eff}}$.\n\nUnfortunately, this is all that we can do with an arbitrary matrix $Q$. The first and most natural ansatz for the form of $Q$, as all the replicas\\index{replica!equivalence} were supposed to be equivalent, is the \\textit{replica symmetric} ansatz\\index{replica!symmetric ansatz}. In that case, we have the symmetric form\n\\begin{equation}\nQ_{ab} = (1-\\delta_{ab})q \\, , \\labeq{replica_ansatz}\n\\end{equation}\nwhere $\\delta_{ab}$ is the Kronecker delta.\n\nThis particular form of $Q$ leads, with standard methods of Gaussian\\index{Gaussian!integration} integration\\footnote{see for example~\\cite{dotsenko:01} for a detailed computation.}, to the saddle-point\\index{saddle point} equation\n\\begin{equation}\nq = \\int_{-\\infty}^\\infty \\dfrac{1}{\\sqrt{2\\pi}} e^{-z^2\/2} \\tanh^2 \\left( \\beta z \\sqrt{q} \\right) {\\mathrm{d}} z \\, .\n\\end{equation}\n\nThis equation can be numerically solved and it is straightforward to prove that, for $\\beta<1$ i.e. $T>\\ensuremath{T_\\mathrm{c}}\\xspace=1$, the only solution is the trivial one $q=0$ corresponding to the paramagnetic phase\\index{phase!high-temperature\/paramagnetic}. On the contrary, for $T<\\ensuremath{T_\\mathrm{c}}\\xspace$ not only $q \\neq 0$, but also $\\lim_{T \\to 0} q(T) = 1$.\n\nA deeper connection can be established by computing the disorder\\index{disorder!average} average of the \\gls{EA} order parameter\\index{order parameter!Edwards-Anderson} introduced in \\refeq{order_parameter_SG_2}\n\\begin{equation}\n\\overline{q_{EA}} = \\dfrac{1}{N} \\sum_i^N \\overline{\\braket{s_i}^2} = \\dfrac{1}{N} \\sum_i \\overline{\\left(\\dfrac{\\sum_{\\{s\\}} s_i \\exp \\left[-\\beta \\mathcal{H}_J(\\{s\\})\\right] }{Z_J}\\right)^2}\n\\end{equation}\nNow, we multiply the numerator and the denominator by $Z_J^{n-2}$ in order to use the replica\\index{replica!method\/trick} trick\n\\begin{equation}\n\\overline{q_{EA}}  =  \\dfrac{1}{N} \\sum_i^N \\overline{\\left( \\dfrac{\\sum_{\\{s^a\\}}s^r_i \\exp \\left[ -\\beta \\mathcal{H}_J^a(\\{s^a\\}) \\right]}{Z_J^n}\\right)^2} \\, ,\n\\end{equation}\nwhere $s^r$ represents an arbitrary replica\\index{replica} $r$. When the limit $n\\to 0$ is taken, the denominator $Z_J^n$ tends to $1$ and we finally get\n\\begin{equation}\n\\overline{q_{EA}} = \\dfrac{1}{N} \\sum_i^N \\overline{\\braket{s_i}^2} = \\dfrac{1}{N} \\sum_i^N \\lim_{n \\to 0} \\overline{\\braket{s_i^\\alpha s_i^\\beta}} \\, ,\n\\end{equation}\nhowever, performing the disorder\\index{disorder!average} average in the \\gls{SK}\\index{Sherrington-Kirkpatrick} formalism, as we have just done above, leads to\n\\begin{equation}\n\\overline{q_{EA}} = \\dfrac{1}{N} \\sum_i^N \\lim_{n \\to 0} \\braket{s_i^\\alpha s_i^\\beta}_Q \\, ,\n\\end{equation}\nand finally, from \\refeq{mean_field_solution_general} and \\refeq{replica_ansatz} we have that $q = \\overline{q_{EA}}$. $q$ is nothing but the order parameter\\index{order parameter!Edwards-Anderson} in the \\gls{EA}\\index{Edwards-Anderson!model} model.\n\nThis solution is, unfortunately, wrong. The first sign of a deep error in the replica symmetric\\index{replica!symmetric solution} solution was the computation of low-temperature entropy\\index{entropy} that turned out to be negative $S(T=0) = -1\/2\\pi <0$. Moreover, a detailed analysis of the solution \\cite{dealmeida:78} showed that it is unstable in the low-temperature phase\\index{phase!low-temperature\/spin-glass}.\n\n\\subsubsection{Parisi's solution: the Replica Symmetry Breaking}\n\n\\captionsetup{justification=centering}\n\nThe unsatisfactory previous results suggested that the replica symmetry \\index{replica!symmetric ansatz}should be broken and some attempts can be found in \\cite{bray:78} and \\cite{blandin:78} who actually proposed the first step of the general iterative solution. That solution came from Parisi \\cite{parisi:79,parisi:80,parisi:80b} and it is known as the \\gls{RSB}\\index{replica!symmetry breaking (RSB)} solution. The starting point is the replica symmetry\\index{replica!symmetric matrix} matrix $Q_{ab}^{\\text{RS}}$, see \\reffig{rs_qab}\n\\begin{figure}[h]\n\\begin{equation*}\nQ_{ab}^{\\text{RS}} = \\left(\\begin{array}{cccccccc}\n0 & & & & \\multicolumn{4}{c}{\\multirow{4}{*}{\\Huge $q_0$}}\\\\\n & 0 & &  & \\\\\n &  & 0 &  & \\\\\n & &  & 0 & \\\\\n\\multicolumn{4}{c}{\\multirow{4}{*}{\\Huge $q_0$}} & 0 &  &  &  \\\\\n & & & &   & 0 &  &   \\\\\n& & &  &  &  & 0 &   \\\\\n & & & &  &  &  & 0\n\\end{array}\\right) \n\\end{equation*}\n\\caption[\\textbf{Replica symmetry ansatz for the matrix $Q_{ab}$.}]{\\textbf{Replica symmetry ansatz\\index{replica!symmetric ansatz} for the matrix $Q_{ab}$.}}\n\\labfig{rs_qab}\n\\end{figure}\n\nFrom here the proposed matrix $Q_{ab}^{\\text{RSB}}$ is constructed through successive iterations. The first step, called the one-step \\gls{RSB}\\index{replica!symmetry breaking (RSB)} consists of dividing the $n$ replicas\\index{replica} into $n\/m_1$ groups, where $n$ and $n\/m_1$ are supposed to be integers at this point. The $n\/m_1$ groups are $m_1 \\times m_1$ squares placed in the diagonal of the matrix $Q_{ab}^{\\text{1-step}}$. All the elements of the matrix where $a=b$ remain equal to $0$, the elements in the $m_1 \\times m_1$ squares with $a \\neq b$ are equal to $q_1$ and the rest of the elements are equal to $q_0$ . The compact form to represent the values of $Q_{ab}^{\\text{1-step}}$ is\n\\begin{equation}\nQ^{\\text{1-step}}_{ab} =\n\\begin{cases}\n0 & \\text{if $a=b$}\\\\\nq_0 & \\text{if $\\Ceil{a\/m_1} \\neq \\Ceil{b\/m_1}$} \\\\\nq_1 & \\text{if $\\Ceil{a\/m_1} = \\Ceil{b\/m_1}$}\n\\end{cases}\n\\, , \\labeq{1step_RSB_compact} \n\\end{equation}\nbeing $\\Ceil{x}$ the ceiling function. The schematic representation of \\refeq{1step_RSB_compact} is in \\reffig{1step_qab}.\n\\begin{figure}[h]\n\\begin{equation*}\nQ_{ab}^{\\text{1-step}} = \\left(\\begin{array}{cccc|cccc}\n0 &  &\\multicolumn{2}{r|}{\\multirow{2}{*}{\\Large $\\ \\ q_1$}} & \\multicolumn{4}{c}{\\multirow{4}{*}{\\Huge $q_0$}}\\\\\n & 0 & &  & \\\\\n\\multicolumn{2}{c}{\\multirow{2}{*}{\\Large $q_1$}}  & 0 & & \\\\\n &  &  & 0 & \\\\\n\\hline\n\\multicolumn{4}{c|}{\\multirow{4}{*}{\\Huge $q_0$}} & 0 & & \\multicolumn{2}{c}{\\multirow{2}{*}{\\Large $q_1$}}\\\\\n & & & &  & 0 &   &  \\\\\n& & & & \\multicolumn{2}{c}{\\multirow{2}{*}{\\Large $q_1$}}  & 0 &   \\\\\n & & & &  & &  & 0\n\\end{array}\\right)\n\\end{equation*}\n\\caption[\\textbf{First step of Replica Symmetry Breaking.}]{\\textbf{First step of Replica Symmetry Breaking.\\index{replica!symmetry breaking (RSB)}}}\n\\labfig{1step_qab}\n\\end{figure}\n\n\\begin{figure}[h]\n\\begin{equation*}\nQ_{ab}^{\\text{2-steps}} = \\left(\\begin{array}{cc|cc|cc|cc}\n0 & q_2 &\\multicolumn{2}{c|}{\\multirow{2}{*}{\\Large $\\ \\ q_1$}} & \\multicolumn{4}{c}{\\multirow{4}{*}{\\Huge $q_0$}}\\\\\nq_2 & 0 & &  & \\\\\n\\cline{1-4}\n\\multicolumn{2}{c|}{\\multirow{2}{*}{\\Large $q_1$}}  & 0 &q_2 & \\\\\n &  & q_2 & 0 & \\\\\n\\hline\n\\multicolumn{4}{c|}{\\multirow{4}{*}{\\Huge $q_0$}} & 0 &q_2 & \\multicolumn{2}{c}{\\multirow{2}{*}{\\Large $q_1$}}\\\\\n\\multicolumn{4}{c|}{} &q_2  & 0 &   &  \\\\\n\\cline{5-8}\n\\multicolumn{4}{c|}{} & \\multicolumn{2}{c|}{\\multirow{2}{*}{\\Large $q_1$}}  & 0 & q_2  \\\\\n\\multicolumn{4}{c|}{} &  & &q_2  & 0\n\\end{array}\\right) \n\\end{equation*}\n\\caption[\\textbf{Second step of Replica Symmetry Breaking.}]{\\textbf{Second step of Replica Symmetry Breaking.\\index{replica!symmetry breaking (RSB)}}}\n\\labfig{2step_qab}\n\\end{figure}\n\\captionsetup{justification=raggedright}\n\nWe can repeat the computation of the free energy\\index{free energy}\\footnote{It is worthy to note that, contrary as usual, in the framework of the \\gls{RSB}\\index{replica!symmetry breaking (RSB)} formalism, the free-energy\\index{free energy} should be \\textit{maximized}. The formal reason is the number of components of the matrix $Q_{ab}$ becomes negative in $n \\to 0$ limit, see \\cite{mezard:87,dotsenko:01}.} with this matrix (see e.g. \\cite{dotsenko:01}) and the related thermodynamic quantities. The zero-temperature entropy\\index{entropy} is $S^{\\text{1-step}}(T=0) \\approx -0.01$ i.e. $|S^{\\text{1-step}}(T=0)| < |S^{RS}(T=0)|$ and the instability of the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass} is also reduced\\footnote{Actually, what is reduced is the most negative eigenvalue of the free-energy\\index{free energy} Hessian matrix near the critical\\index{critical temperature} temperature.}.\n\nThe \\gls{RSB}\\index{replica!symmetry breaking (RSB)} procedure can be generalized to an infinite number of steps. To obtain the two-steps \\gls{RSB}\\index{replica!symmetry breaking (RSB)}, we proceed in the same way as we did in the one-step \\gls{RSB}\\index{replica!symmetry breaking (RSB)} for each of the diagonal blocks of size $m_1 \\times m_1$, now, dividing them in blocks of size $m_2 \\times m_2$. An schematic view of the $Q_{ab}^{\\text{2-steps}}$ is in \\reffig{2step_qab}.\n\nSuccessive steps of \\gls{RSB}\\index{replica!symmetry breaking (RSB)} lead to $S(T=0) \\to 0$ and a less unstable solution in the low-temperature phase\\index{phase!low-temperature\/spin-glass}. It took a while, but finally, it was rigorously proved that the infinite-steps \\gls{RSB}\\index{replica!symmetry breaking (RSB)} produces the correct solution for the free energy\\index{free energy} in the \\gls{SK}\\index{Sherrington-Kirkpatrick} model \\cite{guerra:02,guerra:03,talagrand:06}.\n\nThe infinite-step solution, therefore, depends on an infinite number of parameters $q_i$. Each of those $q_i$ appear with a different weight in the \\gls{pdf} of the overlap\\index{overlap} $q$, that takes the form\n\\begin{align}\np(q) & = \\dfrac{1}{n(n-1)} \\sum_{a \\neq b} \\delta(Q^{\\infty\\text{-steps}}_{ab}-q) = \\nonumber \\\\\n& = \\dfrac{n}{n(n-1)}\\left[ (n-m_1)\\delta(q-q_0) + (m_1-m_2)\\delta(q-q_1) + \\dots \\right] \\, . \\labeq{pdf_q}\n\\end{align}\nThe $n \\to 0$ limit here is a delicate procedure in which we move the $n\\times n$ matrix $Q^{\\text{RSB}}_{ab}=Q^{\\infty\\text{-steps}}_{ab}$ to a $0 \\times 0$ matrix-space. Moreover, the construction of the matrix suggests the assimilation of $m_0 = n$ and, with the restriction of $n$ to be an integer, $m_\\infty=1$. Obviously, $m_k > m_{k+1}$, so $n=m_0 > m_1 > \\dots > m_{\\infty} = 1$. The analytical extension of $n$ and the limit $n \\to 0$ implies that there is no reason to still considering $m_k$ with $k=0,1,\\dots$ to be integers and, therefore, $m_k \\in [0,1]$. The direct implication is the reversing of the order of the coefficients $m_k$, the \\gls{pdf} now is\n\\begin{equation}\np(q) = m_1 \\delta(q-q_0) + (m_2-m_1)\\delta(q-q_1) + \\dots \\, ,\n\\end{equation} \nand the \\gls{SG} order parameter\\index{order parameter} is no longer a discrete set of parameters but a function $q(x)$ with $x \\in [0,1]$ defined as\n\\begin{equation}\nq(x) = q_k \\quad, \\quad 0 \\leq m_k < x < m_{k+1} \\leq 1 \\, .\n\\end{equation}\n\nWe stop here our brief recall of the \\gls{RSB}\\index{replica!symmetry breaking (RSB)} results in the \\gls{SK}\\index{Sherrington-Kirkpatrick} model, nonetheless, there exists a huge number of interesting results, for example in the rich physical interpretation (see e.g. \\cite{parisi:83,rammal:86}) or numerical results that agree with the \\gls{RSB}\\index{replica!symmetry breaking (RSB)} predictions \\cite{vannimenus:81,sommers:84,crisanti:02,aspelmeier:08}.\n\n\\subsection{Theoretical pictures in finite-dimension spin glasses} \\labsubsec{theoretical_pictures}\nThe \\gls{RSB}\\index{replica!symmetry breaking (RSB)} computation gives us the solution to the mean-field\\index{Mean-Field!model} model, but the behavior of the \\gls{SG}s in the low-temperature phase\\index{phase!low-temperature\/spin-glass} at finite dimensions is still a widely debated issue. Here, we briefly review the differences between the diverse pictures explaining the equilibrium \\gls{SG}-phase\\index{phase!low-temperature\/spin-glass} and we show the main predictions that will be crucial to elucidate their validity through experiments, analytical results, or in the case of this thesis, numerical simulations.\n\n\\subsubsection{The Droplet picture}\nAfter Parisi's solution for the mean-field\\index{Mean-Field!model} model, numerical studies of domain\\index{magnetic domain!walls} walls in \\gls{SG}s and their scaling properties \\cite{mcmillan:84,mcmillan:85,bray:84,bray:87}, based on Migdal-Kadanoff\\index{Midgal-Kadanoff!zzzzz@\\Also{Wilson-Kadanoff}|gobbleone} renormalization\\index{renormalization group} computations (that are exact for dimension $d=1$), were the seed for a completely different approach to explain the low-temperature phase\\index{phase!low-temperature\/spin-glass} in short-ranged Ising\\index{Ising} \\gls{SG}s. This picture, introduced along seminal works by Fisher, Huse, Bray and Moore \\cite{fisher:86,bray:87,fisher:88} is known as the \\textit{droplets}\\index{droplet!picture} picture.\n\nThe droplet\\index{droplet!picture} picture understands the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass} from its ground-state\\index{ground-state}. The basic object, the \\textit{droplet}\\index{droplet}, consists of a compact domain\\index{magnetic domain!compact} of linear size $L$ of coherently flipped spins with respect to the ground-state\\index{ground-state}, which constitutes the lowest-energy\\index{energy} excitation at this length-scale $L$. Actually, the droplets\\index{droplet} are expected to have fractal boundaries with a surface area of order $L^{d_s}$, $d-1 \\leq d_s < d$ \\cite{fisher:86}.\n\nThe droplets\\index{droplet} with zero energy\\index{energy} occurs with a probability $P \\propto L^{-\\theta}$ being $\\theta< (d-1)\/2$ the so-called \\textit{stiffness exponent\\index{stiffness!exponent}} and the free-energy\\index{free energy} cost of generating a droplet\\index{droplet} of linear size $L$ is $F_L \\sim \\varUpsilon L^\\theta$ where $\\varUpsilon$ is the \\textit{stiffness modulus\\index{stiffness!modulus}}. The computation of $\\theta$ have been performed numerically for $d=3$ resulting in $\\theta=0.27$ \\cite{boettcher:04,boettcher:05}, $\\theta=0.26$ \\cite{monthus:14}. For $d=2$ the exponent $\\theta$ is negative ($\\theta \\sim -0.28$ \\cite{boettcher:04}), thus, in the thermodynamic limit\\index{thermodynamic limit} the free-energy\\index{free energy} cost for generating a droplet\\index{droplet} tends to zero and the \\gls{SG} transition\\index{phase transition} disappears.\n\nThe most relevant results of the droplet\\index{droplet!picture} pictures are the following:\n\\begin{itemize}\n\\setlength\\itemsep{0.3cm}\n\n\\item The spatial correlation decays with the exponent $\\theta$ as \n\\begin{equation}\nC(r_{ij}) = \\overline{\\braket{s_i s_j}^2} - \\overline{\\braket{s_i}^2} \\overline{\\braket{s_j}^2} \\propto r_{ij}^{-\\theta} \\, .\n\\end{equation}\n\n\\item As a direct consequence of the long-distance vanishing limit of the correlation\\index{correlation function!four point} functions \\cite{fisher:86} the overlap\\index{overlap!distribution} distribution is trivial i.e. corresponds to a delta function $p(q) = \\delta(q^2 - q^2_{EA})$. The many-states nature of the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass} displayed by the \\gls{RSB}\\index{replica!symmetry breaking (RSB)} is no longer valid in the droplet\\index{droplet!picture} picture where only a pair of equilibrium states, related by spin-flip, appears\\footnote{It is commonly said that, according to the droplet\\index{droplet!picture} picture, the \\gls{SG} is a ``ferromagnet in disguise``, that is, a system with a complicated spin configuration\\index{configuration} for the ground-state\\index{ground-state} due to the randomness\\index{randomness!bond} of the couplings\\index{couplings} but that can be mapped to a ferromagnet by performing gauge transformations, similar to the Mattis model\\index{Mattis model} \\cite{mattis:76}.}. \n\n\\item Related to the dynamics, the aging\\index{aging} in the droplet\\index{droplet!picture} picture is explained through the growth of coherent domains\\index{magnetic domain} of spins. Moreover, the coarsening\\index{coarsening} domains\\index{magnetic domain!compact} would be compact objects where the overlap takes one of the two possible values associated with the two pure states allowed $q = \\pm q_{EA}$ \\cite{fisher:88}.\n\n\\item The presence of an external magnetic field suppresses the transition\\index{phase transition} to the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass}. The argument underlying this prediction is a generalization of the Imry-Ma \\cite{imry:75} argument. The energy cost of reversing the spins inside a droplet\\index{droplet} is, by an assumption of the droplets\\index{droplet!picture} model, proportional to $L^\\theta$, and the magnetization\\index{magnetization} of the droplet\\index{droplet} scales as $L^{d\/2}$. By introducing the Zeeman\\index{energy!Zeeman} energy, we can write the free-energy\\index{free energy} cost for flipping the droplet\\index{droplet} in the presence of a small external magnetic field $h$ as $L^\\theta - hL^{d\/2}$. Since $\\theta < (d-1)\/2 < d\/2$, the \\gls{SG} becomes unstable under the presence of any field $h$.\n\n\\item The \\gls{SG} phase\\index{phase!low-temperature\/spin-glass} exhibits a chaotic behavior under small changes of the temperature \\cite{banavar:87,bray:87}. This feature is a direct consequence of free-energy\\index{free energy} scaling ansatz $F_L \\propto L^\\theta$. The \\textit{naive} expectation for the free-energy\\index{free energy} of the droplet\\index{droplet} with surface $L^{d_s}$ would be $F_L \\propto L^{d_s}$, and since $d_s \\geq d - 1 > (d-1)\/2 > \\theta$, the difference between the \\textit{naive} expectation and the scaling ansatz is the presence of large cancellations of the contribution to the free-energy\\index{free energy} in different parts of the boundaries. This precarious equilibrium would be sensitive to small changes in the temperature due to changes in the sign of the free-energy\\index{free energy} at large scales (see e.g. \\cite{katzgraber:07} for further details). Thus, one would expect a complete reorganization of the spin equilibrium configurations\\index{configuration} upon small changes of the temperature. This sensitivity of the system is known as \\textit{temperature chaos}\\index{temperature chaos}. In~\\refch{Introduction_chaos} the reader may find a deeper discussion about this issue.\n\\end{itemize}\n\n\\subsubsection{The RSB picture}\nThe \\gls{RSB}\\index{replica!symmetry breaking (RSB)} solution for the mean-field\\index{Mean-Field!model} model is expected to be correct in short-ranged models like the \\gls{EA}\\index{Edwards-Anderson!model} model for dimensions higher than the \\textit{upper critical dimension}\\index{critical dimension!upper} $d>d_u = 6$. However, its validity in lower dimensions (in particular we are interested in dimension $d=3$) is still a debated issue.\n\nThe \\gls{RSB}\\index{replica!symmetry breaking (RSB)} theory in short-ranged finite dimensions is obtained from perturbative computations from the original mean-field\\index{Mean-Field!solution} solution but the physical behavior drawn is very similar to the mean-field\\index{Mean-Field} predictions. The most remarkable results are:\n\\begin{itemize}\n\\setlength\\itemsep{0.3cm}\n\\item In the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass}, the order parameter\\index{order parameter} is a function $q(x): [0,1] \\longrightarrow [-q_{EA},+q_{EA}]$. In particular, in the low-temperature phase\\index{phase!low-temperature\/spin-glass}, each pair of states will have an overlap\\index{overlap} $q \\in [-q_{EA},+q_{EA}]$ which follows the \\gls{pdf} of \\refeq{pdf_q} and that can be written as\n\\begin{equation}\np(q) = \\dfrac{{\\mathrm{d}} x(q)}{{\\mathrm{d}} q} \\, ,\n\\end{equation}\nby the introduction of the inverse function of $q(x)$\n\\begin{equation}\nx(q) = \\int_0^q P(q') {\\mathrm{d}} q' \\, .\n\\end{equation}\nThis non-trivial \\gls{pdf} is the sign of one of the most distinctive features of \\gls{RSB}\\index{replica!symmetry breaking (RSB)}: in the low-temperature phase\\index{phase!low-temperature\/spin-glass} there exist infinitely many states.\n\\item The organization of those infinitely many states is studied through the \\gls{pdf} of three pure states, see e.g. \\cite{rammal:86,dotsenko:01}. The main result is that, for any arbitrary tern of states $\\alpha$, $\\beta$ and $\\gamma$, the overlap\\index{overlap} between them must fulfill the condition\n\\begin{equation}\nq_{ab} = q_{bc} \\leq q_{ac} \\quad \\forall \\,\\, (a,b,c) \\in \\mathrm{Sym}(\\{\\alpha,\\beta,\\gamma\\}) \\, \\labeq{ultrametric}\n\\end{equation}\nbeing Sym$(\\{\\alpha,\\beta,\\gamma\\})$ the set of all permutations of the three states. Equivalently, \n\\begin{equation}\nq_{ab} \\geq \\min(q_{bc},q_{ac}) \\quad \\forall \\,\\, (a,b,c) \\in \\mathrm{Sym}(\\{\\alpha,\\beta,\\gamma\\}) \\, .\n\\end{equation}\nThis property defines a measure over the space of states and those spaces that present this particular metric are known as \\textit{ultrametric} spaces. Therefore, in the space of \\gls{SG} states, there exist no triangles with all three sides being different.\n\nThe usual way to visualize the ultrametricity\\index{ultrametricity} in \\gls{SG} is displayed in \\reffig{ultrametric} from \\cite{mydosh:93}. For each pair of states $\\alpha$ and $\\beta$, the overlap\\index{overlap} $q_{\\alpha \\beta}$ is obtained by going back in the tree until reaching the first common level. The ultrametric property of \\refeq{ultrametric} can be easily checked if one picks any three states (labeled with a number) in the referred figure.\n\\begin{figure}[h]\n\\includegraphics[width=0.5\\textwidth]{intro\/ultrametric.png}\n\\caption[\\textbf{Ultrametric organization of Replica Symmetry Breaking states.}]{\\textbf{Ultrametric organization of Replica Symmetry Breaking\\index{replica!symmetry breaking (RSB)} states.} The tree representation of the Replica Symmetry Breaking states\\index{replica!symmetry breaking (RSB)}. For any pair of states $\\alpha$ and $\\beta$ their corresponding overlap\\index{overlap} is obtained by downing the tree until reaching the encounter point. Figure from \\cite{mydosh:93}.}\n\\labfig{ultrametric}\n\\end{figure}\n\n\\item The ultrametricity\\index{ultrametricity} is argued to be related to the origin of the temperature chaos\\index{temperature chaos} phenomenon in the \\gls{RSB}\\index{replica!symmetry breaking (RSB)} picture, see for instance \\cite{vincent:97}. The ultrametric hierarchical structure of states is temperature-dependent, that is, the free-energy\\index{free energy!landscape} landscape changes with the temperature as sketched in \\reffig{ultrametric_temperature}. \n\nIn the thermodynamic limit\\index{thermodynamic limit}, any small change of the temperature will relocate the state to a different local minimum, leading to a complete reorganization of its equilibrium configuration\\index{configuration}. Furthermore, this explanation of the temperature chaos\\index{temperature chaos} phenomenon would also explain the experiments of memory\\index{memory effects} and rejuvenation\\index{rejuvenation}, commonly associated with it \\cite{picco:01,takayama:02,maiorano:05,jimenez:05} but not unanimously \\cite{berthier:02}.\n\nIn this picture, the system at a temperature $T$ would explore the metastable\\index{metastability} states\\footnote{In the thermodynamic limit\\index{thermodynamic limit}, the system would need infinite time to ``jump'' from one state to another.}. When the temperature is lowered by an amount $\\delta T$, the system would move in the branch corresponding to its actual state and restart the aging\\index{aging} process, this is the \\textit{rejuvenation}\\index{rejuvenation} effect. When the temperature is back to its previous value $T$, the system comes back to the initial state by moving in the same branch of the tree, this is the \\textit{memory}\\index{memory effects} effect.\n\n\\begin{figure}[h]\n\\includegraphics[width=0.7\\textwidth]{intro\/ultrametric_temperature.png}\n\\caption[\\textbf{Sketch of ultrametric structure as a function of the temperature.}]{\\textbf{Sketch of ultrametric structure as a function of the temperature.} The hierarchical structure of states as a function of temperature is commonly argued to be related to the temperature chaos\\index{temperature chaos} phenomenon in the Replica Symmetry Breaking\\index{replica!symmetry breaking (RSB)} picture. Figure from \\cite{vincent:97}.}\n\\labfig{ultrametric_temperature}\n\\end{figure}\n\n\\item In the \\gls{RSB}\\index{replica!symmetry breaking (RSB)} picture, contrarily to the expected in the droplet\\index{droplet!picture} picture, the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass} is not destroyed by a small magnetic field. The temperature-magnetic field plane is separated by the so-called \\textit{de Almeida-Thouless} line \\cite{dealmeida:78}. The part of the plane with a large magnetic field $h$ and a large temperature is paramagnetic-like while the opposite limit presents a \\gls{SG} behavior.\n\n\\item Similarly to the droplet\\index{droplet!picture} picture, the aging\\index{aging} in the \\gls{RSB}\\index{replica!symmetry breaking (RSB)} picture is explained through the growth of coherent domains\\index{magnetic domain} of spins. However, the predictions of both pictures split when trying to explain the nature of those domains\\index{magnetic domain}. The \\gls{RSB}\\index{replica!symmetry breaking (RSB)} theory expects space-filling domains\\index{magnetic domain!space-filling} i.e. the fractal dimension of the surface is $d_s=d$. \n\\end{itemize}\n\n\n\\subsubsection{Problems with early interpretation: the concept of metastate}\nHowever, the classic interpretation of \\gls{RSB}\\index{replica!symmetry breaking (RSB)} described above presents some issues. The properties of the theory were thought to be present in infinite systems but the procedure to obtain them was to average over the disorder\\index{disorder!average} and, only after that, the infinite-size limit was taken. The problem in disordered\\index{disorder!systems} systems is that, even if that limit exists for averaged quantities or distributions of quantities, it does not imply that an infinite system from which we obtain these quantities or distributions exists.\n\nIn order to solve this problem, mathematical approaches irrupted the physical debate through the concept of metastate\\index{metastate}, firstly proposed in a general context of disordered\\index{disorder!systems} systems by Aizenman and Wehr~\\cite{aizenman:90} and later introduced in the specific context of \\gls{SG}s to deal with this problem, by Newman and Stein~\\cite{newman:92,newman:96,newman:98,newman:03}.\n\nBy using the metastate\\index{metastate} formalism, two main pictures were introduced that rigorously solve the problem of taking the infinite-size limit: the metastate\\index{metastate} interpretation of \\gls{RSB}\\index{replica!symmetry breaking (RSB)}, and the chaotic-pairs picture. The reader may find a detailed discussion in~\\refch{metastate}.\n\n\n\\section{Numerical simulations in spin glasses} \\labsec{numerical_spinglass}\nThe previous sections showed us that, in general, the main theoretical results were far apart from the main experimental results. One role of numerical simulations is to fill that gap. On the one side, experiments are restricted to off-equilibrium conditions, and access to microscopical configurations\\index{configuration} is prohibited. On the other side, the theoretical works have been focused to understand the nature of the low-temperature phase\\index{phase!low-temperature\/spin-glass} in \\gls{SG}. Furthermore, the analytical results are only exact in unrealistic models: droplets\\index{droplet!picture}, exact in one dimensional \\gls{SG}s, and \\gls{RSB}\\index{replica!symmetry breaking (RSB)}, exact in the \\gls{SK}\\index{Sherrington-Kirkpatrick} model and, with almost total consensus, in \\gls{EA}\\index{Edwards-Anderson!model} model for dimensions $d>6$.\n\nNumerical simulations, mostly focused on Monte\\index{Monte Carlo} Carlo simulations \\cite{landau:05}, allow us to study off-equilibrium and equilibrium \\gls{SG}s. Moreover, from numerical data we can access the microscopical configurations\\index{configuration} and we have total control of the system. However, there are also obstacles in the path of numerical work. The equilibrium simulations are restricted to small system-sizes $L$ and temperatures $T$ not very far from the critical\\index{critical temperature} temperature due to the sluggish dynamics exhibited in the low-temperature phase\\index{phase!low-temperature\/spin-glass}, thus, the suspect of finite-size and critical effects hovers over the results. In the off-equilibrium case, again due to the extremely slow dynamics of \\gls{SG}s, the time-scale of the numerical work was traditionally very far from the time-scale of experiments.\n\nFortunately, this situation has improved significantly during the last years. The year-to-year increase of the computational power, the emergence of special-purpose\\index{special-purpose computer} computer like Janus\\index{Janus} \\cite{janus:06,janus:09} and Janus\\index{Janus} II \\cite{janus:14}, and the implementation of algorithms like Parallel\\index{parallel!tempering} Tempering have allowed to simulate larger systems with unprecedented precision and to achieve time-scales comparable with the experimental ones \\cite{janus:08b,janus:09b,janus:18}.\n\nThis thesis aims to be a step forward in the conversion of the numerical simulations from an extremely useful tool for theoretical studies to a bridge between theory and experiments. Along this document we will present several original works with at least one of the following tasks on mind:\n\\begin{enumerate}\n\\item \\textbf{Simulated systems must capture the observed phenomenology of real \\gls{SG}.} Physics is an experimental science and, therefore, every model that we take into consideration to explain the \\gls{SG}s physics must exhibit the same phenomenology observed in real systems. The state of the art for numerical simulations allows extrapolations to experimental scales (see~\\refch{aging_rate}) and the comparison between numerical and experimental results. \n\\item \\textbf{Relate theoretical results with experiments.} A few years ago, the idea of the statics-dynamics equivalence\\index{statics-dynamics equivalence} was proposed in numerical simulations \\cite{janus:08b,janus:10,janus:17}. Numerical evidence is suggesting that off-equilibrium systems of coherence length\\index{coherence length} $\\xi$ could be regarded as a set of equilibrated systems of linear size $L \\sim \\xi$. This concept allows us to relate theoretical predictions in equilibrated \\gls{SG}s with off-equilibrium measures that can be compared with experimental results (see \\refch{aging_rate}).\n\\item \\textbf{Discern between proposed theoretical pictures.} Although the new avenues open by the improvement of the computational power are exciting, we still can take advantage of the traditional purposes of the numerical works. We have discussed how different pictures provide different predictions in the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass} (\\refsubsec{theoretical_pictures}). Throughout this thesis we will face those predictions and we will compare them with numerical results (see~\\refch{metastate}, \\refch{aging_rate}).\n\\item \\textbf{Open new paths for experimental work.} The level of control that we have over simulated systems makes the numerical work an ideal field to find new phenomena that can be later addressed by experiments (see~\\refch{mpemba} and \\refch{out-eq_chaos}).\n\\item \\textbf{Develop new tools for numerical simulations.} Of course, the numerical simulations are not perfect, and not only the development of the hardware is capable of improving their performance. The numerical research to find new methods have been fundamental, from a historical point of view. Here we also focus on improve the numerical simulations, for example, in the study of the Temperature Chaos\\index{temperature chaos} (see~\\refch{equilibrium_chaos}), but also by implementing well-established methods in our works (\\refch{AP_statistics}, \\refch{AP_technical_details_aging}, \\refch{AP_PT}, \\refch{AP_technical_details_out-eq_chaos}, \\refch{AP_multispin_coding}).\n\\end{enumerate}\n\nThroughout this thesis we will be focus on the numerical study of the 3D \\gls{EA}\\index{Edwards-Anderson!model} model by using Monte\\index{Monte Carlo} Carlo methods, in the rest of this section, we will introduce the model and the observables computed with the goal to avoid repetitions in the subsequent chapters.\n\n\\subsection{3D Edwards-Anderson model} \\labsubsec{3D_EA_model}\nAll the numerical simulations carried out in the original works of this thesis are performed in the three-dimensional Ising\\index{Ising} \\gls{EA}\\index{Edwards-Anderson!model} model. In our simulations, the spins are disposed in a cubic lattice $\\Lambda_L$  with \\gls{PBC}\\index{boundary conditions!periodic} where the vertices corresponds to the location of the Ising\\index{Ising} spins $s_i = \\pm 1$. The edges of the cubic lattice correspond to the quenched couplings\\index{couplings} $J_{ij}$ and the energy\\index{energy} of the system is defined by the Hamiltonian\\index{Hamiltonian}\n\\begin{equation}\n\\mathcal{H}_{\\{J\\}} (\\{s\\})= - \\sum_{\\braket{i,j}} J_{ij}s_is_j \\, . \\labeq{EA_Hamiltonian}\n\\end{equation}\nIn our simulations, the couplings\\index{couplings} $J_{ij}$ are independent and identically distributed random variables drawn from a bimodal distribution ($J_{ij}= \\pm 1$ with a $50 \\%$ probability). This model exhibits a spin-glass transition\\index{phase transition} at temperature $\\ensuremath{T_\\mathrm{c}}\\xspace = 1.1019(29)$ \\cite{janus:13}.\n\nEach realization of the couplings\\index{couplings} $\\{J\\}$ is called a \\textit{sample}\\index{sample} and allows us to estimate the average over the disorder\\index{disorder!average}. For each sample\\index{sample}, we simulate statistically independent system copies, each of them evolving under the same couplings\\index{couplings} $\\{J\\}$ but with different thermal noise. Each of these copies is called a \\textit{replica}\\index{replica}. The need of simulating different replicas\\index{replica} will be exposed below.\n\nThe parameters of the simulation (number of samples\\index{sample}, number of replicas\\index{replica}, simulated temperatures, the size of the lattice, \\dots) and the corresponding Monte\\index{Monte Carlo} Carlo method used will be specified in the following chapters, depending on the simulation carried out. \n\n\\subsection{Monte Carlo simulations} \\labsubsec{Monte_Carlo}\nAs we have already anticipated, the studies carried out along this thesis are performed through Monte\\index{Monte Carlo} Carlo simulations \\cite{landau:05,amit:05}. Here, we briefly introduce this method for the reader unfamiliar with the Monte\\index{Monte Carlo} Carlo methods. We will quickly explain the basics of Markov\\index{Markov chain} chains, we will introduce the Metropolis\\index{Metropolis-Hastings} algorithm and the Parallel\\index{parallel!tempering} Tempering. Nonetheless, advanced applications of Markov\\index{Markov chain} chains and, specifically Parallel\\index{parallel!tempering} Tempering, related to the thermalization\\index{thermalization} process will be described in~\\refch{equilibrium_chaos}.\n\\subsubsection{Markov chains}\nWe are interested in the study of a very specific spin system in a lattice, as we have just discussed. The problem is that even for fairly small systems, the integrals involved in the computation of typical quantities have a very high dimensionality, which makes the numerical methods of integration very inefficient. We use, instead, a well-established method to obtain a sample\\index{sample} of configurations\\index{configuration} with the appropriate \\gls{pdf}: a dynamic Monte\\index{Monte Carlo} Carlo method.\n \nTo that purpose, we consider a random walk\\index{random walk} in the configuration\\index{configuration!space} space which hopefully allows us, for each temperature $T=1\/\\beta$, to \\textit{move} from an arbitrary point in the configuration\\index{configuration!space} space\\footnote{In general, we have no \\textit{a priori} information about how the typical configuration\\index{configuration} should be at the desired temperature.} to the relevant configurations\\index{configuration} at that temperature (i.e. the \\textit{thermalization}\\index{thermalization}) and to sample\\index{sample} configurations\\index{configuration} according to the Boltzmann-Gibbs\\index{Boltzmann!-Gibbs distribution} distribution [see \\refeq{prob_configuration}].\n\nBesides, our random walk\\index{random walk} should be Markovian\\index{Markovian}\\footnote{The next state depends only on the actual state and not on the history of the random walker\\index{random walk}.} and will be represented by a transition matrix\\index{transition matrix} $\\pi$. The transition matrix\\index{transition matrix} is 2-dimensional and the rows (equivalently the columns) correspond to every possible configuration\\index{configuration} of the system. The elements $\\pi_{XY}$ denote the probability of change from a configuration\\index{configuration} $X$ to another configuration\\index{configuration} $Y$ in the next step of the random walk\\index{random walk}. Therefore $\\pi_{XY} \\geq 0$ $\\forall X,Y$ and $\\sum_{Y} \\pi_{XY} = 1$.\n\nAs long as the system is Markovian\\index{Markovian}, the probability for going from one configuration\\index{configuration} $X_0$ to one configuration\\index{configuration} $X_n$ in $n>1$ steps is just the sum of the probabilities of the system running over all the possible paths from $X_0$ to $X_n$ in $n$ steps. In those paths, due to the \\textit{no-memory} characteristic of the Markovian\\index{Markovian}\\index{Markov chain} chains, the probability is just the product of the probabilities $\\pi^{(n)}_{X_0X_n} = \\sum_{ \\{ X_1,X_2,\\dots,X_{n-1} \\} } \\pi_{X_0X_1} \\pi_{X_1X_2} \\cdots \\pi_{X_{n-1}X_n}$. \n\nIn addition to those properties, which are inherent to all the Markov\\index{Markov chain} chains, we require other properties that are fundamental for the thermalization\\index{thermalization} process.\n\nFirst, the so-called \\textit{balance\\index{balance condition}} condition\n\\begin{equation}\n\\dfrac{\\exp\\left(-\\beta \\mathcal{H}(Y) \\right)}{Z} = \\sum_{X} \\pi_{XY} \\dfrac{\\exp\\left((-\\beta \\mathcal{H}(X) \\right)}{Z} \\, . \\labeq{balance_condition}\n\\end{equation}\nThis condition express the fact that, if we have a set of configurations\\index{configuration} distributed according to the Boltzmann-Gibbs\\index{Boltzmann!-Gibbs distribution} probability distribution at one step $t$, the set of configurations\\index{configuration} after one step ($t+1$) of the random walk\\index{random walk} for each element of the set will be also distributed according to the Boltzmann-Gibbs\\index{Boltzmann!-Gibbs distribution} probability distribution.\n\nMoreover, the \\textit{irreducibility}\\index{irreducibility}\\index{irreducibility} condition assures that all the configurations\\index{configuration} $Y$ are accessible from any configuration\\index{configuration} $X$, mathematically this is expressed as \n\\begin{equation}\n\\forall \\, X,Y \\, \\exists \\, n>0 \\, : \\, \\sum_{ \\{ X_1,X_2,\\dots,X_{n-1} \\} } \\pi_{X_0X_1} \\cdots \\pi_{X_{n-1}X_n} > 0 \\quad \\mathrm{with} \\quad X_n=Y \\, . \\labeq{irreducibility}\n\\end{equation}\nThere exists a more restricting condition, the \\textit{aperiodicity}\\index{aperiodicity} which needs the introduction of the concept of \\textit{period}. A period $d_X$ (being $X$ a configuration\\index{configuration}), is the greatest common divisor of the length of all the paths starting at configuration\\index{configuration} $X$ and finishing at the same configuration\\index{configuration} $X$. If that period is $d_X=1$ for all the states, we say that the Markov\\index{Markov chain} chain is aperiodic\\index{aperiodicity}.\n\nThe above properties allow us to introduce a specially useful theorem~\\cite{sokal:97}: if one Markov\\index{Markov chain} chain satisfies both, the aperiodic\\index{aperiodicity} and the balance\\index{balance condition} condition, then\n\\begin{equation}\n\\lim_{n\\to \\infty} \\pi^{(n)}_{XY} = \\dfrac{\\exp\\left( -\\beta \\mathcal{H}(Y)\\right)}{Z} \\, . \\labeq{thermalization_condition}\n\\end{equation}\nThis theorem implies that the starting configuration\\index{configuration} is immaterial, the random walk\\index{random walk} will eventually sample the desired Boltzmann-Gibbs\\index{Boltzmann!-Gibbs distribution} distribution. This theorem is not but a mathematical warranty of the fact that our system will thermalize.\n\n\\subsubsection{The Metropolis-Hastings algorithm}\nIn our numerical simulations, most of the time we are using the Metropolis-Hastings\\index{Metropolis-Hastings} algorithm\\footnote{Often called just Metropolis\\index{Metropolis-Hastings}, for short.}, which is nothing but a dynamic Monte\\index{Monte Carlo} Carlo method that fulfills the previously described conditions. This method is generally applicable to a multitude of contexts, but it is not our goal to provide general results on Monte\\index{Monte Carlo} Carlo methods\\footnote{To that purpose, the reader may consult \\cite{sokal:97,amit:05,landau:05}.}. Hence, we focus here on our particular context and identify a configuration\\index{configuration} $X$ with the set of all the spins in our lattice. \n\nThere exist several possible ways to propose a change from the configuration\\index{configuration} $X$ to the configuration\\index{configuration} $Y$. One very common choice is to attempt the change of individual spins in the lattice in a sequential way. As far as we are focus on the Ising\\index{Ising} model, for each spin there is only a possible change to perform: to flip the spin.\n\nIn the Metropolis-Hastings\\index{Metropolis-Hastings} algorithm, for one temperature $T=1\/\\beta$, we proceed in the following way:\n\\begin{enumerate}\n\\item Compute the energy\\index{energy} of the configuration\\index{configuration} $X$, namely $\\mathcal{H}(X)$\n\\item Flip the spin $i$ and compute the energy\\index{energy} of the new configuration\\index{configuration} $Y$: $\\mathcal{H}(Y)$.\n\\item Draw a random number $r \\in [0,1)$. If $r<\\exp\\left[ -\\beta(\\mathcal{H}(Y)-\\mathcal{H}(X))\\right]$ then we accept the change $X \\to Y$, otherwise, we remain in the configuration\\index{configuration} $X$.\n\\item Repeat the previous steps for all the spins $i$ in the lattice. We denote the realization of the Metropolis-Hastings\\index{Metropolis-Hastings} algorithm to all the lattice a \\textit{lattice sweep} or simply, a \\textit{sweep}.\n\\end{enumerate}\nThis algorithm makes the system evolve whenever the energy\\index{energy} is diminished, but also allows, with probability $e^{-\\beta \\Delta\\mathcal{H}}$, local moves which increase the energy\\index{energy} by an amount $\\Delta \\mathcal{H}$.\n\n\\subsubsection{Parallel Tempering}\nThe sluggish dynamics exhibited by the \\gls{SG}s is a major obstacle in the classical dynamic Monte\\index{Monte Carlo} Carlo methods, such as the Metropolis-Hastings\\index{Metropolis-Hastings} algorithm, because the necessary time to thermalize, even small systems, would be prohibitive. Several algorithms try to palliate this problem, here we introduce the \\gls{PT} algorithm \\cite{hukushima:96,marinari:98b}.\n\nThe general idea of the \\gls{PT} is to thermalize at the same time a set of $N$ identical copies which are at different temperatures $T_1 < T_2 < \\cdots < T_N$ (or equivalently $\\beta_N > \\cdots > \\beta_2 > \\beta_1$). For those samples\\index{sample} above the critical\\index{critical temperature} temperature $T>\\ensuremath{T_\\mathrm{c}}\\xspace$, the evolution of the system will be fast. On the contrary, for those systems that lie at temperatures $T<\\ensuremath{T_\\mathrm{c}}\\xspace$, the configurations\\index{configuration} will be almost frozen. The \\gls{PT} algorithm consists of two alternating sets of steps.\n\nFirst, each system copy independently undergoes standard Monte\\index{Monte Carlo} Carlo dynamics (for example Metropolis\\index{Metropolis-Hastings}) at its own temperature; one can use one or more Monte\\index{Monte Carlo} Carlo steps each time. Second, pairs of spin configurations\\index{configuration} attempt to exchange their temperatures by permuting the $N$ copies of configurations\\index{configuration} in the temperature mesh. The exchange rule between two copies labeled as $\\alpha$ and $\\alpha'$ with configurations\\index{configuration} $\\{s^{(\\alpha)}\\}$ and $\\{s^{(\\alpha')}\\}$ of the system follows the Metropolis\\index{Metropolis-Hastings} scheme but we trade the accepting probability to\n\\begin{equation}\nr < \\exp \\left\\lbrace \\left[\\beta_{\\alpha} - \\beta_{\\alpha'}\\right] \\left[\\mathcal{H}(\\{s^{(\\alpha)}\\}) - \\mathcal{H}(\\{s^{(\\alpha')}\\}) \\right] \\right\\rbrace \\, . \\labeq{PT_exchange_rule}\n\\end{equation}\n\nThe goal of the \\gls{PT} algorithm is to sample sets of $N$ configurations\\index{configuration} (one for each copy of the system $\\{s^{(\\alpha)}\\}_{\\alpha=1}^N$), each one at its given temperature. Of course, those copies are not ordered in the temperature mesh with the $\\alpha$ index because of the permutations introduced by the algorithm. In order to fully characterize the state of the system at a given time, we need to introduce $\\pi(\\alpha)$: the permutation of the $\\alpha=1,2,\\dots,N$ copies of systems in the temperature mesh. Now, the state of the \\gls{PT} can be described by $X=\\{ \\pi, \\{s^{(\\alpha)}\\}_{\\alpha=1}^N\\}$ and the stationary distribution of the \\gls{PT} algorithm would be\n\\begin{equation}\nP_{\\mathrm{eq}}(X) = \\dfrac{1}{N!}\\prod_{\\alpha=1}^N \\dfrac{\\exp \\left[-\\beta_{\\alpha} \\mathcal{H}(\\{s^{\\pi^{-1}(\\alpha)}\\})\\right]}{Z_{\\beta_{\\alpha}}} \\, , \\labeq{PT_eq_distribution}\n\\end{equation}\nbeing $Z_{\\beta_{\\alpha}}$ the partition function\\index{partition function} at the temperature $\\beta_{\\alpha}$ and $\\pi^{-1}(\\alpha)$ the inverse permutation of $\\pi$ that fulfills the condition $\\pi^{-1}\\left( \\pi(\\alpha)\\right) = \\alpha$ ; $\\forall$ $\\alpha$.\n\nThe rationale behind the \\gls{PT} method is simple. Each system copy undergoes a random walk\\index{random walk} in temperature space. When a system copy is at a low temperature, it only explores the nearby free-energy\\index{free energy} local minima. When its temperature is high, however, free-energy\\index{free energy!barrier} barriers disappear: the copy can freely wander in phase space\\index{phase space}, and when it cools again it will typically fall in a different free-energy\\index{free energy!valley} valley, with different local minima. For \\gls{PT} to effectively thermalize, it is crucial that any copy of the system spends its time roughly evenly at every temperature: high temperatures are needed to ensure visiting all the phase space\\index{phase space}; low temperatures are needed to visit its low free-energy\\index{free energy} regions. \n\nIn fact, \\gls{PT} is currently used in a very large number of very different applications (for example in physics, biology, chemistry, engineering, statistics), and considerable efforts have been devoted to improving it from various communities. Various temperature-exchange rules have been developed and tested \\cite{sugita:99,calvo:05,earl:05,brenner:07,bittner:08,malakis:13}. Furthermore, it has been suggested that a significant gain can be achieved by optimizing the choice of the $N$ temperatures \\cite{katzgraber:06,sabo:08}. Further details on the \\gls{PT} method can be found in~\\refch{equilibrium_chaos} and \\refsec{thermalizing_PT}.\n\n\n\\subsection{Observables}\\labsubsec{observables_introduction}\nHere, we present the observables that we will measure in most of the performed simulations. Nonetheless, in the subsequent chapters, we will present some observables that have to be announced in their context in order to be fully understood.\n\nOne more consideration needs to be done before defining the observables. The Hamiltonian\\index{Hamiltonian} \\refeq{EA_Hamiltonian} presents a gauge symmetry, specifically, the transformation \n\\begin{equation}\ns_i \\longrightarrow \\epsilon_i s_i \\quad \\quad J_{ij} \\longrightarrow \\epsilon_i \\epsilon_j J_{ij} \\, ,\n\\end{equation}\nleaves it unchanged, being $\\epsilon_i$ a random sign $\\pm 1$ for each site of the lattice $i$. When several samples\\index{sample} are simulated and averages over the disorder\\index{disorder!average} are taken, this gauge symmetry is just the expression of redundant degrees of freedom\\index{degree of freedom} of the system. If the measured observables are insensible to this symmetry, it would be possible to find pairs of $\\{ \\{J\\}, \\{s\\}\\}$ related by a gauge transformation and providing different \\textit{weights} to the disorder\\index{disorder!average} average i.e. totally equivalent configurations\\index{configuration} from the energetic point of view, provide different results for some observables in different samples\\index{sample}. Therefore, it is desirable to define gauge-invariant observables. The usual way to do that is by introducing \\textit{replicas}\\index{replica} (see \\refsubsec{3D_EA_model}). \n\nThe overlap\\index{overlap!field} field between two replicas\\index{replica} is defined as\n\\begin{equation}\nq^{\\sigma,\\tau}_{\\vec{x}}(t) = s_{\\vec{x}}^\\sigma(t) s_{\\vec{x}}^\\tau(t) \\, , \\labeq{def_overlap}\n\\end{equation}\nwhere $\\sigma$ and $\\tau$ are labels to denote two different replicas\\index{replica} and subscripts $\\vec{x}$ represent the position in the lattice. The four-point spatial correlation\\index{correlation function!four point} function is\n\\begin{equation}\nC_4(T,\\vec{r},t) = \\overline{\\braket{q^{\\sigma,\\tau}_{\\vec{x}}(t)q^{\\sigma,\\tau}_{\\vec{x}+\\vec{r}}(t)}} \\, , \\labeq{def_C4}\n\\end{equation}\nwhere $\\overline{\\left( \\cdots \\right)}$ is the disorder\\index{disorder!average} average defined in \\refeq{average_disorder} and $\\braket{\\cdots}$ the average over the thermal noise\\footnote{This correlation\\index{correlation function!four point} function will be measured in off-equilibrium systems along this thesis, therefore, the mean given by \\refeq{average_o} that holds for equilibrium systems do not apply. We estimate the thermal noise by averaging over all the pairs of replicas\\index{replica} $\\sigma$ and $\\tau$.}. In numerical simulations we can only estimate these means, since we only have finite number of samples\\index{sample} (much smaller than the possible set of couplings\\index{couplings} $\\{J\\}$) and a finite number of replicas\\index{replica}. The long distance decay of $C_4(T,\\vec{r},t)$ defines the coherence length\\index{coherence length} $\\xi(t)$, an observable of central importance as we will discuss below\n\\begin{equation}\nC_4(T,\\vec{r},t) \\sim r^{-\\vartheta} f(r\/\\xi(t)) \\, . \\labeq{long_distance_C4}\n\\end{equation}\nThe function $f(x)$ decreases faster than exponentially for large $x$, $f(x) \\sim e^{-x^{\\beta}}$ with $\\beta \\approx 1.7$ (see \\cite{jimenez:05}). The exponent $\\vartheta$ at the critical\\index{critical temperature} temperature $T=\\ensuremath{T_\\mathrm{c}}\\xspace$ is related to the anomalous dimension $\\eta$ (see for example \\cite{amit:05} for a definition), being $\\vartheta(\\ensuremath{T_\\mathrm{c}}\\xspace) = 1 + \\eta$ with $\\eta = -0.390(4)$ \\cite{janus:13}. For $T<\\ensuremath{T_\\mathrm{c}}\\xspace$, droplets\\index{droplet!picture} picture predicts compact domains\\index{magnetic domain!compact} and, therefore, $\\vartheta=0$. On the contrary, \\gls{RSB}\\index{replica!symmetry breaking (RSB)} picture expects space-filling domains\\index{magnetic domain!space-filling} where $C_4(T,\\vec{r},t)$ vanishes at fixed $r\/\\xi$ as $t$ grows. The value of $\\vartheta$ in this picture is given by the replicon\\index{replicon}, a critical mode analogous to magnons in Heisenberg\\index{Heisenberg} ferromagnets (see \\cite{janus:10b} for a detailed discussion and \\cite{dedominicis:06} for theoretical introduction of the replicon\\index{replicon}). Previous numerical studies give us the value $\\vartheta=0.38(2)$ \\cite{janus:09} with a small dependence on the temperature $T$ that was vaguely attributed to the effect of the critical point. We will discuss widely about this exponent in~\\refch{aging_rate}.\n\n\n\\subsubsection{Coherence length}\\labsubsubsec{coherence_length}\nThe coherence length\\index{coherence length} $\\xi(t)$ is a fundamental observable in the dynamics of the \\gls{SG}s because it rules the off-equilibrium phenomena a multitude of times as we will discuss along this thesis (see~\\refch{aging_rate},~\\refch{mpemba}, and \\refch{out-eq_chaos}). Conceptually, the coherence length\\index{coherence length} is a characteristic length-scale of the off-equilibrium systems that measure the linear size of domains\\index{magnetic domain} of correlated spins. Although, the name ``coherence length\\index{coherence length}''\\footnote{This coherence length\\index{coherence length} should not be confused with the so-called coherence length\\index{coherence length} in optics.} is not universal and some authors denominate it \\textit{dynamical correlation length} (see for example \\cite{parisi:99c}) or, by abuse of language, simply \\textit{correlation length} (see \\cite{joh:99}).\n\nThe relation of the coherence length\\index{coherence length} with the length of correlated spins deserves a real example which will (hopefully) help us to visualize the concept. Consider one three dimensional lattice of linear size $L=160$ in which we have simulated the \\gls{EA}\\index{Edwards-Anderson!model} model (see~\\refsubsec{3D_EA_model}) with a Metropolis-Hastings\\index{Metropolis-Hastings} algorithm for $t=2^{36}$ number of Monte\\index{Monte Carlo} Carlo steps at temperature $T=0.7$. In the left panels of~\\reffig{coherence_length_snapshot} two projections of configurations\\index{configuration} at this time over the $xy$ plane are showed. The \\textit{up} spins are plotted in yellow and the \\textit{down} spins are plotted in blue. We can no see any pattern in those configurations\\index{configuration} that appear to be random.\n\nHowever, if we consider the overlap\\index{overlap} between them (right panel of~\\reffig{coherence_length_snapshot}) by using the same \\textit{color code}, a pattern emerges. Islands of correlated spins appear in the plot. This, of course, is only a visual sketch of the concept but it encodes the idea that the four-point correlation\\index{correlation function!four point} function (which is just a correlation\\index{correlation function!four point} function of the overlaps\\index{overlap}) has encrypted the correlation length. Yet, the quantitative obtaining of such observable have required a long time in both, numerical simulations and experiments.\n\nThe problem of finding characteristic lengths in off-equilibrium systems have been widely discussed. The integral estimators have been used since 1982 \\cite{cooper:82}. Detailed numerical studies have concluded that a well-behaved estimator of $\\xi(t)$, that is very convenient from the numerical point of view (see \\cite{janus:09b}), should be computed through the integrals\\footnote{The alert reader may notice that in this expression, the correlation\\index{correlation function!four point} function $C_4(T,\\vec{r},t)$ has changed its vectorial dependence of the distance $\\vec{r}$ to a simple scalar dependence $r$. This rotational invariance is assumed and justified numerically with a careful study in \\cite{janus:09b}.}\n\\begin{equation}\nI_k(t) = \\int_0^{\\infty} r^k C_4(T,r,t) {\\mathrm{d}} r \\, . \\labeq{integral_estimator_xi}\n\\end{equation}\nLet us identify the correlation\\index{correlation function!four point} function $C_4(T,r,t)$ with its long range behavior displayed in \\refeq{long_distance_C4}. In that case, taking $x=r\/\\xi$ would lead to\n\\begin{equation}\nI_k(t) = \\int_0^{\\infty} \\xi^{k-\\vartheta} \\left( r\/\\xi \\right)^{k-\\vartheta} f(r\/\\xi) \\xi \\dfrac{{\\mathrm{d}} r}{\\xi} = \\xi^{k+1-\\vartheta} \\int_0^{\\infty} x^{k-\\vartheta} f(x) {\\mathrm{d}} x \\, .\n\\end{equation}\n\nTherefore, the knowledge about the concrete form of the function $f(x)$ is no longer needed and one finds an estimator of $\\xi$\n\\begin{equation}\n\\xi_{k,k+1}(t) = \\dfrac{I_{k+1}(t)}{I_k(t)} \\propto \\xi(t) \\, . \\labeq{def_xi}\n\\end{equation}\n\nHowever, we should not forget the previous assumption made concerning the correlation\\index{correlation function!four point} function. The estimation of \\refeq{def_xi} involves systematic errors and that result would be only valid in the $r \\to \\infty$ limit. The larger the value of $k$, the smaller is the deviation from the asymptotic behavior due to the term $r^k$, which rules the short distances. However,  increasing the exponent $k$ moves the peak of $r^k C_4(T,r,k)$ to larger values of $r$ where the relative error of the correlation\\index{correlation function!four point} function is larger. A compromise solution of the value $k$ is needed, here, we use $k=1$. This decision is numerically justified in \\cite{janus:09b} and used in several works \\cite{janus:10b,janus:14b,fernandez:15,manssen:15,janus:17,fernandez:18b,fernandez:19}. Further details of computation in numerical simulations are provided in~\\refsec{finite_size_effects}.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{intro\/coherence_length}\n\\caption[\\textbf{Spin-glass coherence length.}]{\\textbf{Spin-glass coherence length\\index{coherence length}.} \\textbf{Top left:} A snapshot of a configuration\\index{configuration} $\\{s_{\\boldsymbol{x}}^{(a)}\\}$, which has evolved for $t=2^{36}$ Monte\\index{Monte Carlo} Carlo steps at $T=0.7\\approx 0.64T_\\text{c}$.  We show the average magnetization\\index{magnetization} on the $xy$ plane, averaging over $z$. \\textbf{Bottom left:} Another configuration\\index{configuration} $\\{s_{\\boldsymbol{x}}^{(b)}\\}$ of the same sample\\index{sample}, prepared in the same way as $\\{s_{\\boldsymbol{x}}^{(a)}\\}$.  No visible ordering is present in either configuration\\index{configuration} because the preferred pattern of the magnetic domains\\index{magnetic domain} cannot be seen by eye ($s=1$ is plotted in yellow, and $-1$ in blue). \\textbf{Right:} If one measures the overlap\\index{overlap} between the two configurations\\index{configuration}, and with the same color code used for the spins, the preferred pattern of the magnetic domains\\index{magnetic domain}, of size $\\xi$, becomes visible. Figure from~\\cite{janus:19}.}\n\\labfig{coherence_length_snapshot}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\\chapter[Sweet introduction to Temperature Chaos]{Sweet introduction to \\\\ Temperature Chaos} \\labch{Introduction_chaos}\n\n\\setlength\\epigraphwidth{.5\\textwidth}\n\\epigraph{\\textit{Una racha de viento nos visit\u00f3 \\\\\nPero nuestra veleta ni se inmut\u00f3.\\\\\nLa canci\u00f3n de que el viento se parara \\\\\nDonde nunca pasa nada.}}{-- Extremoduro, \\textit{Dulce introducci\u00f3n al caos}}\n\nBefore the second half of the XIX century, it was commonly accepted that the \\textit{predictability} of a physical system was only constrained for technical reasons such as the limited knowledge of the position and speed of the particles. In the last part of the XIX century, however, Henri Poincar\u00e9, in his geometrical study of the stability of the Solar System, introduced the idea of the extreme sensibility of a system to small changes on its initial conditions.\n\nThat idea was relatively forgotten in the mainstream physics literature until $1963$ when Lorenz \\cite{lorenz:63} realized that this extreme sensitivity was exhibited by a system of coupled differential equations. He was simulating a simplified model of convection rolls and he noticed that starting his numerical simulations from two slightly different initial conditions led to completely different results, even in relatively short times. This evidence about the impossibility of long-term predictions in certain systems was, indeed, very attractive for the physics community and, the interest in that research topic notoriously increased.\n\nAlthough the concept of \\textit{chaos} have considerably evolved through the years and it is well-defined in the mathematical context~\\cite{hasselblatt:03}, it has been historically associated with the extreme sensitivity to small perturbations~\\cite{strogatz:18}. \\gls{SG}s borrow the term to describe the fragility of the glassy phase\\index{phase!low-temperature\/spin-glass} in response to perturbations.\n\nThe sensitivity of the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass} upon changes in the couplings\\index{couplings}, namely \\textit{disorder chaos}\\index{disorder!chaos}~\\cite{ney-nifle:97,ney-nifle:98,sasaki:05,katzgraber:07}, or in the external magnetic field~\\cite{kondor:89,ritort:94,billoire:03}, have been widely studied and satisfactorily described. \n\nThe temperature counterpart of this fragility is known as \\gls{TC}\\index{temperature chaos}, which means that the spin configurations\\index{configuration} which are typical from the Boltzmann\\index{Boltzmann!weight} weight at temperature $T_1$ are very atypical at temperature $T_2$ (no matter how close the two temperatures $T_1$ and $T_2$ are). This phenomenon has proved to be very elusive~\\cite{bray:87b,banavar:87,kondor:89,kondor:93,ney-nifle:97,ney-nifle:98,billoire:00,mulet:01,billoire:02,krzakala:02,rizzo:03,sasaki:05,katzgraber:07,parisi:10,fernandez:13,billoire:14,wang:15,billoire:18,janus:21} and remains to be fully understood.\n\nThis chapter\\footnote{The name of chapter is a small tribute to rock band Extremoduro and their song \\textit{Dulce introducci\u00f3n al caos} (Sweet introduction to chaos).} pretends to be a very brief introduction to the \\gls{TC}\\index{temperature chaos} phenomenon by wandering through the main historical results in this field. The aim is to understand the starting points of the two following chapters (\\refch{equilibrium_chaos} and \\refch{out-eq_chaos}) that will be devoted to exposing the original results of this thesis in \\gls{TC}\\index{temperature chaos}.\n\n\\section{The origin of Temperature Chaos} \\labsec{origin_tc}\nThe \\gls{TC}\\index{temperature chaos} phenomenon was originally predicted in finite-dimension \\gls{SG}s by Bray, Moore, and Banavar\\footnote{Although it was originally predicted in \\gls{SG}s, other systems like polymers~\\cite{sales:02,dasilveira:04} also exhibit it.}~\\cite{bray:87,bray:87b,banavar:87} in the context of renormalization\\index{renormalization group} studies and scaling arguments (the germ of the later-called \\textit{droplets picture}\\index{droplet!picture}, see~\\refsubsec{theoretical_pictures}). For the sake of clarity, we briefly recall the main characteristics of the droplets\\index{droplet!picture} to understand the emerging chaos feature in this theory (see for example \\cite{katzgraber:07} for further details).\n\nIn the droplet\\index{droplet!picture} picture, the low-temperature phase\\index{phase!low-temperature\/spin-glass} is understood in terms of excitation of the ground-state\\index{ground-state}. The energy\\index{energy} excitation occur through the so-called \\textit{droplets}\\index{droplet!picture}, which are compact domains\\index{magnetic domain!compact} of spins with linear size $L$ that have been coherently flipped and whose boundaries are expected to be fractal, with a surface area of the order $L ^{d_s}$, $d-1 \\leq d_s < d$. The free-energy\\index{free energy} cost of generating such a droplet\\index{droplet} is $F_L(T) \\sim \\Upsilon(T) L^{\\theta}$ with $0<\\theta<(d-1)\/2$, being $\\Upsilon$ and $\\theta$ the stiffness modulus\\index{stiffness!modulus} and the stiffness exponent\\index{stiffness!exponent} respectively. The entropy\\index{entropy} in the droplets\\index{droplet!picture} picture scales with the size of the droplet\\index{droplet} as $S = \\sigma(T) L^{d_s\/2}$, being $\\sigma(T)$ the \\textit{entropy\\index{entropy!stiffness} stiffness}. \n\nThe key to understanding the \\gls{TC}\\index{temperature chaos} in the droplets\\index{droplet!picture} picture is to focus on the scaling behavior of the free energy\\index{free energy}. One would \\textit{naively} expect, for domains\\index{magnetic domain!compact} of spin of volume $\\sim L^{d_s}$, that the free-energy\\index{free energy} would also scales as $\\sim L^{d_s}$. However, as we have mentioned before, the free-energy\\index{free energy} scales as $F_L(T) \\sim L^{\\theta}$ with $\\theta < d_s$. This happens due to large cancellations of the contribution to the free\\index{free energy} energy from different parts of the boundary of the droplet\\index{droplet}, and this delicate equilibrium is the key to understand the \\gls{TC}\\index{temperature chaos} phenomenon. In this picture, \\gls{TC}\\index{temperature chaos} appears if the free energy\\index{free energy} of a droplet\\index{droplet} changes its sign upon a small change in the temperature. The length scale in which this happens is the so-called \\textit{chaotic length} $\\ell_c$.\n\nThe computation of $\\ell_c$ is performed through thermodynamic arguments. The free\\index{free energy} energy is $F(T) = U(T) - TS(T)$ being $U(T)$ the internal energy\\index{energy} of the droplet\\index{droplet} and $S(T)$ the entropy\\index{entropy} of the same droplet\\index{droplet}, both at temperature $T$. However, when the changes of temperature are small enough, the internal energy\\index{energy} can be considered as an independent quantity with respect to the temperature, and therefore\n\\begin{equation}\nF(T_2) = U(T_2) - T_2S(T_2) \\approx U(T_1) - T_2S(T_2) = F(T_1) + T_1S(T_1) - T_2S(T_2) \\, .\n\\end{equation}\nTaking into account the scaling behavior of the free\\index{free energy} energy and the entropy\\index{entropy} in a droplet\\index{droplet}, we can compute the length scale $\\ell_c$ at which $F(T_2)$ inverts its sign\n\\begin{equation}\n\\ell_c = \\left(\\dfrac{\\Upsilon(T_1)}{T_2 \\sigma(T_2) - T_1 \\sigma(T_1)}\\right)^{1\/\\zeta} \\quad \\mathrm{being} \\quad \\zeta = d_s\/2 - \\theta \\, . \\labeq{def_chaotic_length}\n\\end{equation}\nThe usual approach\\footnote{In~\\cite{katzgraber:07} the authors avoid this simplification with small changes in the final result.} is to take $\\sigma(T_2) \\approx \\sigma(T_1)$ when $\\lvert T_2 - T_1 \\rvert \\ll 1$ and, therefore\n\\begin{equation}\n\\ell_c \\sim \\lvert T_2 - T_1 \\rvert^{-1\/\\zeta} \\, . \\labeq{scaling_chaotic_length}\n\\end{equation}\nThe meaning of $\\ell_c$ is clear: small changes in the temperature make domains\\index{magnetic domain} of spins of length scales greater than $\\ell_c$ to flip, leading to two separate regimes. On the one side, in the short-length regime, the chaos is absent or it is rather weak. On the other side, in the large-length regime, two equilibrium configurations\\index{configuration} at temperatures $T_1$ and $T_2$ are completely uncorrelated, leading to a strong chaos phenomenon.\n\nIn this framework, a multitude of numerical work~\\cite{ney-nifle:97,ney-nifle:98,aspelmeier:02,krzakala:04,sasaki:05,katzgraber:07,monthus:14} has been performed. Indeed, the scaling of the chaotic length showed in~\\refeq{scaling_chaotic_length} was numerically found and the exponent $\\zeta$ was computed. However, still this approach presents major problems. The equilibrium simulations performed at that time were limited to $L \\sim 10$ which made the system to be in the $L \\ll \\ell_c$ regime, where the chaos is almost absent. Moreover, the scaling of~\\refeq{scaling_chaotic_length} extends beyond the critical\\index{critical temperature} temperature $\\ensuremath{T_\\mathrm{c}}\\xspace$ where \\gls{TC}\\index{temperature chaos} should not occur. Thus, the numerical evidence supporting this picture is quite weak.\n\n\\section{Temperature Chaos in Mean Field}\nIn Mean-Field\\index{Mean-Field!model} models\\footnote{Those models that can be exactly solved through Mean-Field approximations.}, the \\gls{TC}\\index{temperature chaos} has proved to be particularly elusive. Specifically, the \\gls{SK}\\index{Sherrington-Kirkpatrick} model stoically resisted numerical attempts to characterize the \\gls{TC}\\index{temperature chaos} phenomenon~\\cite{billoire:00,billoire:02}, later solved by~\\cite{billoire:14} as we discuss below. The lack of numerical evidence of \\gls{TC}\\index{temperature chaos} and the publication of studies which, indeed, presented evidence against it~\\cite{mulet:01,rizzo:01} led to the conclusion that \\gls{TC}\\index{temperature chaos} did not take place in the \\gls{SK}\\index{Sherrington-Kirkpatrick} model.\n\nHowever, in 2003, a \\textit{tour de force}~\\cite{rizzo:03} showed that the \\gls{SK}\\index{Sherrington-Kirkpatrick} model presented an exceedingly small \\gls{TC}\\index{temperature chaos}, and it was necessary to compute up to the ninth order in a perturbative expansion in the replica\\index{replica!symmetry breaking (RSB)} framework\\footnote{Actually,~\\cite{rizzo:01} found no \\gls{TC}\\index{temperature chaos} because the computations in this paper were performed ``only'' until the fifth order in the perturbation expansion.} to find it.\n\nThis study is based on the use of a large-deviation functional. The idea is that, under the \\gls{TC}\\index{temperature chaos} hypothesis, the overlap\\index{overlap} between any pair of equilibrium states at temperatures $T_1$ and $T_2$ ($T_1 \\neq T_2$) should be zero. Therefore, the shape of the probability distribution of overlaps\\index{overlap!distribution} $q$ between equilibrium configurations\\index{configuration} at different temperatures should tend to a Dirac's delta function peaked on $q=0$ as the size of the system grows. Moreover, the scaling of this probability distribution is given by the large-deviations formula\n\\begin{equation}\nP(q) \\sim \\exp \\left[-N\\Delta F(q)\\right] \\, ,\n\\end{equation}\nwhere $N$ is the size of the system and $\\Delta F(q)$ takes account of the free-energy\\index{free energy} cost of constraining two replicas\\index{replica} to have a given mutual overlap\\index{overlap} at equilibrium. This functional $\\Delta F(q)$ is computed in~\\cite{rizzo:03} through a perturbative approach. It is necessary to reach the ninth order to find a non-vanished term, hence, it was demonstrated that \\gls{SK}\\index{Sherrington-Kirkpatrick} model presents, though pathologically, the \\gls{TC}\\index{temperature chaos} phenomenon.\n\nNonetheless, the \\gls{SK}\\index{Sherrington-Kirkpatrick} model has not been the only Mean-Field\\index{Mean-Field!model} model in which \\gls{TC}\\index{temperature chaos} has been studied. For example, it has been found that diluted Mean-Field\\index{Mean-Field!model} \\gls{SG}s present much stronger \\gls{TC}\\index{temperature chaos}~\\cite{parisi:10}. Besides, in $p$-spin models, which are just a generalization of the \\gls{SK}\\index{Sherrington-Kirkpatrick} model in which interactions occur between $p \\geq 3$ spins, different behaviors have been found. On the one hand, we have a recent mathematical proof of the absence of \\gls{TC}\\index{temperature chaos} in the homogeneous spherical $p$-spin model~\\cite{subag:17}, in agreement with a previous claim based on physical arguments~\\cite{kurchan:93}. On the other hand, \\gls{TC}\\index{temperature chaos} should be expected when one mixes several values of $p$~\\cite{barrat:97}, as confirmed by a quite recent mathematical analysis~\\cite{chen:14,panchenko:16,chen:17,arous:20}.\n\n\\section{Memory and rejuvenation}\\labsec{memory_rejuvenation_introduction_chaos}\nThe reader may note that all the previous discussion about \\gls{TC}\\index{temperature chaos} assumes that it is an equilibrium phenomenon (since its very definition), however, most of the experimental work in spin-glasses is carried out under non-equilibrium conditions as we have already discussed in~\\refsubsec{aging_memory_rejuvenation}\\footnote{With the notable exception of\nexperiments in a thin-film geometry, see~\\cite{guchhait:14}. In fact, the experimental study of \\gls{TC}\\index{temperature chaos} in thin films has been initiated~\\cite{guchhait:15b}.}.\n\nThe spectacular rejuvenation\\index{rejuvenation} and memory\\index{memory effects} effects~\\cite{jonason:98,lundgren:83,jonsson:00,hammann:00} (described in~\\refsubsec{aging_memory_rejuvenation}) have been commonly related to the phenomenon of \\gls{TC}\\index{temperature chaos}~\\cite{komori:00,berthier:02,picco:01,takayama:02,maiorano:05,jimenez:05}. Yet, the situation is far from clear.\n\nThe idea is that, due to the \\gls{TC}\\index{temperature chaos} phenomenon, the equilibrium configurations\\index{configuration} at two slightly different temperatures $T_1$ and $T_2$ would be completely different, thus, the aging\\index{aging} performed at temperature $T_1$ would not be useful at the temperature $T_2$, and the aging\\index{aging} process restarts, leading to the so-called \\textit{rejuvenation}\\index{rejuvenation} phenomenon. The opposite behavior, the \\textit{cumulative aging\\index{aging!cumulative}}, means that the relaxation\\index{relaxation} work carried out at temperature $T_1$ is still useful (partly useful at least) when the temperature is varied to $T_2$.\n\nIndeed, some experiments~\\cite{jonsson:02,bert:04} and most of the numerical work~\\cite{komori:00,berthier:02,picco:01,takayama:02,maiorano:05,jimenez:05} trying to simulate temperature-varying\\index{temperature-varying protocol} experimental protocols can be interpreted as cumulative aging\\index{aging!cumulative}. At this point, opinions are split. Some authors find only cumulative aging\\index{aging!cumulative} in their simulations~\\cite{picco:01,takayama:02,maiorano:05}, while others find some traces of aging\\index{aging} restart~\\cite{komori:00,berthier:02}. However, this restarting of aging\\index{aging} occurs on exceedingly short times~\\cite{jimenez:05}. Perhaps more worryingly, it has been found numerically that a site-diluted ferromagnet (where no \\gls{TC}\\index{temperature chaos} is expected) behaves analogously to the spin glass~\\cite{jimenez:05}.\n\nIf a strong connection between \\gls{TC}\\index{temperature chaos} and the experiments of memory\\index{memory effects} and rejuvenation\\index{rejuvenation} exists, some work is needed in order to establish it.\n\n\\section{Last steps}\nThis historical tour about \\gls{TC}\\index{temperature chaos} allows us to understand the general feeling in the field at the beginning of the 2010s. The \\gls{TC}\\index{temperature chaos} phenomenon seemed to be extremely weak, with gradual increasing effects that we were not able to perceive due to the difficulty of equilibrating large systems in the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass}. However, an alternative weak-\\gls{TC}\\index{temperature chaos} scenario~\\cite{sales:02} could be compatible with the results. In this scenario, almost all the samples\\index{sample} exhibit no \\gls{TC}\\index{temperature chaos} at all but a few of them suffer dramatic effects upon temperature changes. Actually, this scenario was not completely unknown and some numerical studies mentioned that situation~\\cite{katzgraber:07}, but a quantitative study was lacking.\n\nThe main idea is that very few samples\\index{sample} undergo \\textit{chaotic events} i.e. at well-defined temperatures $T^*$, the samples\\index{sample} suffer first-order\\index{phase transition!first order} like transitions (rounded in finite-systems) such that the typical spin configurations\\index{configuration} below and above $T^*$ differ. While the majority of samples\\index{sample} do not have any chaotic event, some of them display one (or more) with a $T^*$ which seems to be located randomly within the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass}. Yet, the fraction of samples\\index{sample} lacking chaotic events decreases upon increasing the system size. Indeed, one expects~\\cite{rizzo:03,parisi:10} that the fraction\nof samples\\index{sample} lacking TC will decrease exponentially (in the system size).\n\nIt was in 2013 when the \\gls{TC}\\index{temperature chaos} was quantitatively studied as a rare-event-driven phenomenon~\\cite{fernandez:13}. This numerical study was based on the study of large-deviations functional which was fundamental in order to deal with the wild sample-to-sample fluctuations\\index{sample-to-sample fluctuations}. The mean problem in the numerical study of \\gls{TC}\\index{temperature chaos} in short-ranged \\gls{SG} was, indeed, the statistical methods used to deal with the chaotic observables: the majority of non-chaotic samples\\index{sample} killed any chaos signal when taking the disorder\\index{disorder!average} average\\footnote{As quoted by ``The Buggles'': \\textit{Average Killed The Chaos Signal}.}.\n\nLater, further works~\\cite{billoire:14,martin-mayor:15,fernandez:16,billoire:18,janus:21} showed that the rare-event analysis was the appropriate protocol in order to study the \\gls{TC}\\index{temperature chaos} phenomenon. In the subsequent chapters (\\refch{equilibrium_chaos} and \\refch{out-eq_chaos}), we will develop two different rare-event analysis in 3D Ising\\index{Ising} \\gls{SG}s in order to study the \\gls{TC}\\index{temperature chaos} phenomenon.\n \n\n\n\n\n\\chapter{Metastate} \\labch{metastate}\n\\setlength\\epigraphwidth{.5\\textwidth}\n\n\\epigraph{\\textit{Si paso por Florida te recuerdo\\\\\nSi paso por la Valle me es igual,\\\\\nQue si estoy en Corrientes, que si estoy en Palermo\\\\\nPor todo Buenos Aires conmigo siempre est\u00e1s.}}{-- Julio Jaramillo, \\textit{No me toquen ese vals} }\n\n\nThis chapter is dedicated to discussing the metastate\\index{metastate}. We start by introducing the concepts of mixed and pure states in lattice systems, see \\refsec{mixed_pure_states_metastate}, which would be needed in order to understand further discussions. Then, we describe the problem of taking the thermodynamic limit\\index{thermodynamic limit} in disordered\\index{disorder!systems} systems in \\refsec{the_problem_metastate} and we introduce the proposed solution in \\refsec{the_solution_metastate}. The different theoretical pictures described in \\refsubsec{theoretical_pictures} provide different predictions for some observables in the metastate\\index{metastate} formalism, we introduce those observables and discuss the different scenarios in \\refsec{observables_predictions_metastate}. \n\nAt this point, we present an original contribution to the metastate\\index{metastate} problem developed during this thesis~\\cite{billoire:17} by explaining the numerical setup \\refsec{simulation_parameters_metastate} and by exploring the metastate\\index{metastate} from the numerical point of view in \\refsec{results_metastate}. Finally, we relate our numerical results to theory in~\\refsec{relating_numerical_theory_metastate}.\n\n\\section{Mixed and pure states} \\labsec{mixed_pure_states_metastate}\n\nConsider a spin system in a lattice $\\Lambda \\subset \\mathbb{Z}^d$ where the vertices correspond to the spins $s_i$ and the edges correspond to the couplings\\index{couplings} between spins $J_{ij}$, leading to nearest-neighbor interactions defined through a Hamiltonian\\index{Hamiltonian} $\\mathcal{H}_{\\mathcal{J}} = \\sum_{\\braket{i,j}} J_{ij} s_i s_j$. As far as we are dealing with general definitions, we are neither restricting the values of the spins, nor the values of the couplings\\index{couplings}. \n\nOne \\textit{configuration}\\index{configuration} of the system is determined by the value of the set of all the spins $\\mathcal{S} = \\{s_i\\}$ as $i$ runs over all the lattice sites. In the same way, a concrete \\textit{sample}\\index{sample} is determined by the value of the set of all the couplings\\index{couplings} $\\mathcal{J} = \\{J_{ij}\\}$ as the pair $(i,j)$ runs over all the pairs of spins. \n\nThe restriction of the lattice to finite size $L$ is simply done by considering a cubic lattice $\\Lambda_L$ composed of $L^d$ spins. However, in the case of finite systems, an additional problem arises with the choice of the boundary conditions\\index{boundary conditions}. It is possible to define several boundary conditions\\index{boundary conditions} but we will consider a very common choice, the \\gls{PBC}\\index{boundary conditions!periodic}.\n\nAt a given temperature $T=1\/\\beta$, a Gibbs state $\\Gamma_{L,\\mathcal{J}}$ for a finite system $\\Lambda_L$ is a probability distribution over the configurations\\index{configuration} $\\mathcal{S}$ where each configuration\\index{configuration} has a probability to appear equal to \n\\begin{equation}\n\\Gamma_{L,\\mathcal{J}}(\\mathcal{S}_{\\Lambda_L}) = \\dfrac{\\exp\\left(-\\beta \\mathcal{H}_{L,\\mathcal{J}}(\\mathcal{S})\\right)}{Z_L} \\, , \\labeq{Gibbs_probability_state}\n\\end{equation}\nbeing $\\mathcal{H}_{L,\\mathcal{J}}$ the Hamiltonian\\index{Hamiltonian} of the system restricted to the lattice $\\Lambda_L \\subset \\mathbb{Z}^d$ and $Z_L$ the partition function\\index{partition function}\\footnote{For the sake of simplicity, we tacitly assume that each spin can only take a finite set of possible values and, therefore, the set of configurations\\index{configuration} $\\mathcal{S}$ is countable and we can perform the sum. In the case of infinite-uncountable possible values for each spin, the trade between the sum and an integral is needed.} $Z_L=\\sum_{\\mathcal{S}} \\exp \\left( -\\beta \\mathcal{H}_{L,\\mathcal{J}}\\right)$.\n\nWhen considering an infinite lattice, this definition of state is not useful anymore because the Hamiltonian\\index{Hamiltonian} $\\mathcal{H}_{\\mathcal{J}}$ involves sums of infinite terms that do not converge. Nonetheless, there exist a well-established definition for state in infinite lattice: the \\gls{DLR} states \\cite{ruelle:04,sinai:14,friedli:17}. A probability distribution of states in an infinite-size lattice is a Gibbs state $\\Gamma_{\\mathcal{J}}$ if, for any finite subset $\\Lambda_W \\subset \\mathbb{Z}^d$, two conditions are fulfilled:\n\\begin{enumerate}\n\\item The Hamiltonian\\index{Hamiltonian} for a spin configuration\\index{configuration} inside the subset $\\Lambda_W$ conditioned to the values of the rest of spins in the infinite lattice $\\mathbb{Z}^d \\setminus \\Lambda_W$, namely $\\mathcal{H}_{W,\\mathcal{J}} (\\mathcal{S}_{\\Lambda_W} \\lvert \\mathcal{S}_{\\mathbb{Z}^d \\setminus \\Lambda_W})$, and the partition function\\index{partition function} restricted to that subset $Z_{W}$ are finite for almost every $\\mathcal{S}$.\\\\\n\nIn the case of the \\gls{EA}\\index{Edwards-Anderson!model} model the expression for $\\mathcal{H}_{W,\\mathcal{J}} (\\mathcal{S}_{\\Lambda_W} \\lvert \\mathcal{S}_{\\mathbb{Z}^d \\setminus \\Lambda_W})$ is very easy due to the short-ranged nature of the Hamiltonian\\index{Hamiltonian}. We have to take care only with the frontier $\\partial \\Lambda_W$ where the Hamiltonian\\index{Hamiltonian} includes terms with spins out of $\\Lambda_W$ and spins inside. Therefore, we have\n\\begin{equation}\n\\mathcal{H}^{\\mathrm{EA}}_{W,\\mathcal{J}} (\\mathcal{S}_{\\Lambda_W} \\lvert \\mathcal{S}_{\\mathbb{Z}^d \\setminus \\Lambda_W}) = \\sum_{J_{ij} \\in \\Lambda_W \\setminus \\partial \\Lambda_W} J_{ij} s^{\\mathrm{int}}_i s^{\\mathrm{int}}_j + \\sum_{J_{ij} \\in \\partial \\Lambda_W} J_{ij} s^{\\mathrm{int}}_i s^{\\mathrm{out}}_j \\, , \\labeq{hamiltonian_conditioned}\n\\end{equation}\nwhere the superindex \\textit{int} stands for spins belonging to $\\Lambda_W$ and the superindex \\textit{out} stands for spins outside of $\\Lambda_W$. The reader should notice that, if $\\Lambda_W$ contains a finite number of spins, the Hamiltonian\\index{Hamiltonian} of~\\refeq{hamiltonian_conditioned} only involves finite sums.\n\n\\item The conditional probability $\\Gamma_{W,\\mathcal{J}}$ of a configuration\\index{configuration} $\\mathcal{S}$ in $\\Lambda_W$ given the rest of the spins $\\mathbb{Z}^d \\setminus \\Lambda_W$ is absolutely continuous\\footnote{The reader may be confuse about the term ``absolutely continuous'' in this context. This continuity is defined over a measure of the random variables, the spins in our particular case. The reader should consult~\\cite{sinai:14} for a much more detailed discussion with rigorous proofs.} and it is defined by the expression\n\\begin{equation}\n\\Gamma_{W,\\mathcal{J}}(\\mathcal{S}_{\\Lambda_W} \\lvert \\mathcal{S}_{\\mathbb{Z}^d \\setminus \\Lambda_W})  = \\dfrac{\\exp \\left[-\\beta \\left( \\mathcal{H}_{W,\\mathcal{J}}(S_{\\Lambda_W}) + \\mathcal{H}_{W,\\mathcal{J}}(S_{\\Lambda_W} \\lvert \\mathcal{S}_{\\mathbb{Z}^d \\setminus \\Lambda_W}) \\right)  \\right]}{Z_{W}} \\, . \\labeq{def_DLR}\n\\end{equation}\n\\end{enumerate}\n\nFrom \\refeq{def_DLR} one can conclude that, for any two spin configurations\\index{configuration} $\\mathcal{S}_{1,\\Lambda_W}$ and $\\mathcal{S}_{2,\\Lambda_W}$ with the rest of the spins fixed $\\mathcal{S}_{\\mathbb{Z}^d \\setminus \\Lambda_W}$, the ratio of their conditional probabilities would be, simply\n\\begin{equation}\n\\dfrac{\\Gamma_{W,\\mathcal{J}}(\\mathcal{S}_{1,\\Lambda_W} \\lvert \\mathcal{S}_{\\mathbb{Z}^d \\setminus \\Lambda_W})}{\\Gamma_{W,\\mathcal{J}}(\\mathcal{S}_{2,\\Lambda_W} \\lvert \\mathcal{S}_{\\mathbb{Z}^d \\setminus \\Lambda_W})} = \\exp \\left[ -\\beta \\left( \\mathcal{H}_{W,\\mathcal{J}}(\\mathcal{S}_{1,\\Lambda_W})-\\mathcal{H}_{W,\\mathcal{J}}(\\mathcal{S}_{2,\\Lambda_W}) \\right) \\right] \\, . \\labeq{ratio_DLR}\n\\end{equation}\n\nDoes this definition hold for our particular case? It is easy to prove that the \\gls{EA}\\index{Edwards-Anderson!Hamiltonian} Hamiltonian\\index{Hamiltonian} defined in \\refeq{EA_Hamiltonian} is finite for any finite lattice $\\Lambda_W$ and, therefore, the partition function\\index{partition function}, which involves a finite number of summands since we are considering Ising\\index{Ising} spins, is simply a finite sum of finite terms. The second condition is also fulfilled because $\\mathcal{H}_{W,\\mathcal{J}}(\\mathcal{S}_{1,\\Lambda_W})-\\mathcal{H}_{W,\\mathcal{J}}(\\mathcal{S}_{2,\\Lambda_W})$ is finite, since the lattice $\\Lambda_W$ is finite by definition. The reader may find rigorous proof of the existence of the Gibbs states in the \\gls{EA}\\index{Edwards-Anderson!model} model in~\\cite{ruelle:04}.\n\nActually, given a Hamiltonian\\index{Hamiltonian} $\\mathcal{H}_{\\mathcal{J}}$ and given a temperature $T=1\/\\beta$, the previous definition of infinite state allows the existence of many different Gibbs states $\\Gamma_\\mathcal{J}$. The set of all of the possible Gibbs states is convex and compact, so there exist \\textit{extremal} Gibbs states that are not convex combinations of any other Gibbs states. The extremal Gibbs states are called \\textit{pure} states while the non-extremal Gibbs states are called \\textit{mixed} states and can be decomposed uniquely into a convex combination of pure states in the following way\\footnote{We assume, for the sake of simplicity, that the decomposition is discrete.}\n\\begin{equation}\n\\braket{\\cdots}_{\\Gamma_{\\mathcal{J}}} = \\sum_{\\alpha} \\omega_{\\alpha,\\Gamma_{\\mathcal{J}}} \\braket{\\cdots}_{\\alpha} \\, , \\labeq{decomposition_pure_states}\n\\end{equation}\nbeing $\\alpha$ a pure state and $\\omega_{\\alpha,\\Gamma_{\\mathcal{J}}}$ an appropriate weight for that pure state that fulfills the condition $\\sum_\\alpha \\omega_{\\alpha,\\Gamma_{\\mathcal{J}}} = 1$, with $\\omega_{\\alpha,\\Gamma_{\\mathcal{J}}} \\geq 0$ $\\forall \\alpha$.\n\nMoreover, the overlap\\index{overlap} between two pure states labeled as $\\alpha$ and $\\beta$ can be defined in the lattice $\\Lambda_W$ as\n\\begin{equation}\nq_{\\alpha \\beta} = \\dfrac{1}{W^d} \\sum_{x \\in \\Lambda_W} \\braket{s_x}_{\\alpha}\\braket{s_x}_{\\beta} \\, , \\labeq{overlap_pure_states}\n\\end{equation}\nbeing $s_x$ the spin at the position $x$. The \\gls{pdf} of the overlap\\index{overlap!distribution} can, therefore, be defined as\n\\begin{equation}\nP_{\\Gamma_{\\mathcal{J}}}(q) = \\sum_{\\alpha,\\beta}\\omega_{\\alpha,\\Gamma_{\\mathcal{J}}} \\omega_{\\beta,\\Gamma_{\\mathcal{J}}} \\delta(q-q_{\\alpha \\beta}) \\, . \\labeq{pq_pure_states}\n\\end{equation}\n\n\n\n\\section{The problem: Chaotic Size Dependence} \\labsec{the_problem_metastate}\n\nIn the previous section, we have properly defined the infinite-size states through the \\gls{DLR} states, however, that definition is not physically relevant because experiments are always conducted in large but finite systems. It would be desirable from the physical point of view, to connect the \\gls{DLR} states with a sequence of growing systems of linear size $L$ as $L \\to \\infty$.\n\nThis connection can be made for transitional-invariant Hamiltonians\\index{Hamiltonian} like the ferromagnet one. However, in systems with quenched disorder\\index{disorder!quenched}, such as the \\gls{EA}\\index{Edwards-Anderson!order parameter} model, that connection is still much of a mystery \\cite{aizenman:90}. The problem of taking the $L \\to \\infty$ limit have been already introduced in \\refsubsec{theoretical_pictures}: the \\gls{CSD}. \n\nLet us consider a system of linear size $L$ in a lattice $\\Lambda_L$ and fix an internal region of linear size $W$, $\\Lambda_W \\subset \\Lambda_L$. If we take the limit $L \\to \\infty$, remaining constant the couplings\\index{couplings} of the lattice $\\Lambda_W$, the state $\\Gamma_{W,\\mathcal{J}}$ changes chaotically as $L$ grows and also the observables measured in $\\Lambda_W$. \n\nThis extreme sensitivity of the system to the addition of couplings\\index{couplings} at the boundaries as long as $L$ tends to infinity is called \\gls{CSD}. This problem remained apparently oblivious to the \\gls{SG} literature and was pointed out for the first time by \\cite{newman:92}.\n\n\\section{The solution: the Metastate} \\labsec{the_solution_metastate}\nThe solution to that problem is the concept of Metastate\\index{metastate!Aizenman-Wehr} which was introduced by Aizenman and Wehr in~\\cite{aizenman:90} when studying first-order transitions\\index{phase transition!first order} in general disordered\\index{disorder!systems} systems. Two years later, Newman and Stein introduced this concept of metastate\\index{metastate!Newman-Stein} to solve the \\gls{CSD} problem in the particular case of \\gls{SG}s~\\cite{newman:92}. The concept of the metastate\\index{metastate} \\cite{aizenman:90,newman:92,newman:96,newman:98,newman:03} is just a generalization of the concept of Gibbs state that we have exposed in \\refsec{mixed_pure_states_metastate}. A Gibbs state in a finite system can be regarded as a probability distribution of configurations\\index{configuration} $\\mathcal{S}$, each one with an associated probability given by \\refeq{Gibbs_probability_state}. In the same way, the metastate\\index{metastate}, which we denote as $\\kappa_{\\mathcal{J}}(\\Gamma_\\mathcal{J})$, is a probability distribution over the states $\\Gamma_{\\mathcal{J}}$. \n\nThe reader may notice that the description of the metastate\\index{metastate} concept considers infinite-size states $\\Gamma_{\\mathcal{J}}$, this poses the problem of the construction of that metastate\\index{metastate} from finite-size systems. There exist two definition of metastates\\index{metastate} in the literature that propose the solution to that problem: the \\gls{AW} metastate\\index{metastate!Aizenman-Wehr} \\cite{aizenman:90} and the \\gls{NS} metastate\\index{metastate!Newman-Stein} \\cite{newman:92}. Although, it has been argued that both proposals should be equivalent (see \\cite{read:14} for a detailed discussion).\n\n\\subsection{Newman-Stein Metastate}\nWe first introduce the \\gls{NS} metastate\\index{metastate!Newman-Stein}, so-named in \\cite{newman:96}. Consider a growing sequence of $n$ systems of size $L_0<L_1<L_2< \\dots L_n$. For the smallest system of size $L_0$, we fix the couplings\\index{couplings} $\\mathcal{J}$ randomly in the lattice $\\Lambda_{L_0}$ and we define a hypercube of linear size $W < L_0$, usually called \\textit{window}, in the lattice $\\Lambda_W \\subset \\Lambda_L$. \n\nIn this system, we are able to define the Gibbs state $\\Gamma_{L_0,\\mathcal{J}}$ and to restrict this state to the window $W$, obtaining a probability distribution on the spins in $\\Lambda_W$ consisting in $2^{W^d}$ real numbers. This probability distribution is a Gibbs state $\\Gamma_{W,\\mathcal{J}}$. If we consider now the space of finite states $\\Gamma_{W,\\mathcal{J}}$, each time that a specific state $\\Gamma_{W,\\mathcal{J}}$ appears, we can ``record'' it as a Dirac delta $\\delta$ of integral $1\/n$. Repeating this procedure for the $n$ systems would lead to an empirical probability distribution over the space of finite states $\\Gamma_{W,\\mathcal{J}}$.\n\nThen, the limit $n \\to \\infty$ is taken and that empirical distribution may tend to a limit. If this limit exists for every window $W$ as $W \\to \\infty$, then, the resulting distribution of states $\\Gamma_\\mathcal{J}$ is the so-called \\gls{NS} metastate\\index{metastate!Newman-Stein}.\n\n\\subsection{Aizenman-Wehr Metastate} \\labsubsec{aw_metastate}\nWe introduce now the \\gls{AW} metastate\\index{metastate!Aizenman-Wehr} that was indeed introduced earlier \\cite{aizenman:90} in the context of first-order transitions\\index{phase transition!first order} but was later related with this concept of metastate\\index{metastate} as a limiting object for quenched-disordered\\index{disorder!quenched} systems \\cite{newman:92}.\n\nThe definition of the \\gls{AW} metastate\\index{metastate!Aizenman-Wehr} requires the introduction of three length scales $W \\ll R \\ll L$ which will be the linear size of three lattices $\\Lambda_W \\subset \\Lambda_R \\subset \\Lambda_L$ as sketched in \\reffig{sketch_AW_metastate}.\n\n\\captionsetup{justification=centering}\n\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{metastate\/sketch_AW_metastate.png}\n\\caption[\\textbf{Sketch of the Aizenman-Wehr metastate construction.}]{\\textbf{Sketch of the Aizenman-Wehr metastate\\index{metastate!Aizenman-Wehr} construction.}}\n\\labfig{sketch_AW_metastate}\n\\end{figure}\n\\captionsetup{justification=raggedright}\n\nConsider the lattice $\\Lambda_L$ (with a given boundary conditions\\index{boundary conditions}, usually \\gls{PBC}\\index{boundary conditions!periodic}). This lattice $\\Lambda_L$ is divided into an inner region $\\Lambda_R$ and an outer region $\\Lambda_L \\setminus \\Lambda_R$. Now, consider the state $\\Gamma_{L,\\mathcal{J}}$ for a set of couplings\\index{couplings} $\\mathcal{J}$ in the lattice $\\Lambda_L$. \n\nThe first step in the construction of the \\gls{AW} metastate\\index{metastate!Aizenman-Wehr} is to take an average over the states $\\Gamma_{L,\\mathcal{J}}$. We keep constant all the couplings\\index{couplings} that have both extremes inside $\\Lambda_R$ while changing the rest of them, obtaining a set of states $\\{\\Gamma_{L,\\mathcal{J}_1},\\Gamma_{L,\\mathcal{J}_2},\\dots\\}$. Then, we take the average of this set of states.\n\nFinally, we take the limit of the three-length scales to infinity while respecting the relation $W \\ll R \\ll L$. The obtained distribution of states should not depend on the arbitrary choice of the fixed internal couplings\\index{couplings} and will, hopefully, have a smooth limit: the metastate\\index{metastate}.\n\nThe lattice $\\Lambda_W$ plays a fundamental role in the \\gls{AW} metastate\\index{metastate!Aizenman-Wehr} as a \\textit{measuring window}, we are restricting our attention to that hypercube when measuring the relevant quantities in order to avoid boundary effects that may appear as long as $R$ is finite.\n\n\\section{Theoretical pictures in the metastate formalism} \\labsec{pictures_metastate}\nAs we have introduced in~\\refsubsec{theoretical_pictures}, the mathematical concept of metastate\\index{metastate} irrupted in the physics debate of the nature of a \\gls{SG} at finite dimensions. The metastate\\index{metastate} came to the \\gls{SG} field to solve some mathematical ambiguities in the early formalism of \\gls{RSB}\\index{replica!symmetry breaking (RSB)}\\index{replica!symmetry breaking (RSB)}.\n\nThe introduction of the concept of metastate\\index{metastate}, although, brought also a deep debate about the validity of the \\gls{RSB}\\index{replica!symmetry breaking (RSB)}\\index{replica!symmetry breaking (RSB)} metastate\\index{metastate!RSB} picture. According to \\cite{newman:03}, the existence of states composed of a mixture of infinitely many pure states would not be allowed. The Newman-Stein statement was not fully accepted in mathematical physics and it was debated in subsequent works. For example, see the work of Talagrand~\\cite{talagrand:06}, who demonstrated that Parisi's formula reproduced exactly the free-energy\\index{free energy} for the \\gls{SK}\\index{Sherrington-Kirkpatrick} model, and the work of Read, who questioned the arguments of Newman and Stein to rule out the \\gls{RSB}\\index{replica!symmetry breaking (RSB)}\\index{replica!symmetry breaking (RSB)} picture (part IV of \\cite{read:14}).\n\nThis debate leads to the proposal of an alternative picture: the \\textit{chaotic pairs}\\footnote{The term ``chaotic pair'' is intimately related with the droplet\\index{droplet!picture} picture, but, as quoted by Newman and Stein \\cite{newman:03} \\textit{The term 'chaotic pairs' was chosen in reference to spin-symmetric boundary conditions\\index{boundary conditions}, such as periodic. If one considers a fixed boundary condition metastate\\index{metastate}, then it would be more appropriate to refer to this picture as 'chaotic pure states' because the Gibbs state in a typical large volume $\\Lambda_L$ with fixed boundary conditions\\index{boundary conditions} will be (approximately) a single pure state that varies chaotically with L.}}.\n\nHere, we (very) briefly introduce both pictures that together with the classical Droplet\\index{droplet!picture} picture stands as the mathematical consistent formalism to describe the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass}.\n\n\\subsection{The RSB metastate}\n \nThe adequacy of the \\gls{RSB}\\index{replica!symmetry breaking (RSB)}\\index{replica!symmetry breaking (RSB)} formalism to the metastate\\index{metastate!RSB} concept lead to the so-baptized as \\textit{nonstandard \\gls{RSB}\\index{replica!symmetry breaking (RSB)}\\index{replica!symmetry breaking (RSB)} picture} \\cite{newman:03}. However, as quoted by \\cite{read:14} we find this name unfortunate because this picture should be the standard in the \\gls{RSB}\\index{replica!symmetry breaking (RSB)}\\index{replica!symmetry breaking (RSB)}. Thus, we adopt the terminology proposed by Read and we call this picture the \\textit{\\gls{RSB}\\index{replica!symmetry breaking (RSB)}\\index{replica!symmetry breaking (RSB)} metastate}\\index{metastate!RSB}. \n\nIn this picture, each of the states that form the metastate\\index{metastate!RSB} is \\gls{RSB}\\index{replica!symmetry breaking (RSB)}\\index{replica!symmetry breaking (RSB)}-like, that is, each state is composed of a hierarchical structure of pure states and presents all the properties exposed in~\\refsubsec{theoretical_pictures}. Moreover, the metastate\\index{metastate!dispersed} is \\textit{disperse} i.e. the metastate\\index{metastate!dispersed} is composed of infinitely many states. This dispersion has, indeed, deep consequences. For instance, the disperse metastate\\index{metastate!dispersed} is responsible for the lack of the self-averaging\\index{self-averaging!non-} property in certain quantities for the \\gls{SG}s in the low-temperature phase\\index{phase!low-temperature\/spin-glass} (see, for example \\cite{binder:86}, for the concept of non-self-average\\index{self-averaging}). The reader may notice that, despite the deep meaning of these results, the study of a finite fixed-size system would not change the predictions of \\gls{RSB}\\index{replica!symmetry breaking (RSB)}\\index{replica!symmetry breaking (RSB)} because, for that system, a single state is present and we have already stated that, in the \\gls{RSB}\\index{replica!symmetry breaking (RSB)}\\index{replica!symmetry breaking (RSB)} metastate\\index{metastate!RSB} picture, each state is \\gls{RSB}\\index{replica!symmetry breaking (RSB)}\\index{replica!symmetry breaking (RSB)}-like.\n\n\\subsection{Chaotic pairs picture}\nChaotic pairs can be regarded as a generalization of the Droplet\\index{droplet!picture} picture in the metastate\\index{metastate!chaotic pairs} formalism, therefore, each of the states composing the metastate\\index{metastate!chaotic pairs} would be trivial i.e. each state will be composed of a single pair of spin-flip related pure states. However, in this picture, the pair of pure states will change chaotically with $L$ due to the chaotic size dependence.\n\n\\section{Observables and predictions} \\labsec{observables_predictions_metastate}\nThe formalism of the metastate\\index{metastate} allows the computation of several quantities that would behave differently in the different theoretical pictures described above. Should we obtain these quantities with enough accuracy, it would be possible to discriminate (at least partially) between those competing pictures. This section is devoted to introducing the observables that we have measured in our numerical simulations and to recalling the predictions made for those observables.\n\n\\subsection{The observables} \\labsubsec{observables_metastate}\nThe overlap\\index{overlap} in the measure window $\\Lambda_W$ is defined taking advantage of the metastate\\index{metastate} setup. Consider one realization of the internal couplings\\index{couplings} $i$ and a pair of external realizations of the outer couplings\\index{couplings} $o$ and $o'$. One possible definition of the overlap\\index{overlap} (that will be useful later) is\n\\begin{equation}\nq^{i;o,o'} = \\dfrac{1}{W^d} \\sum_{x \\in \\Lambda_W} s_x^{\\sigma;i,o}s_x^{\\tau;i,o'}  = \\dfrac{1}{W^d} \\sum_{x \\in \\Lambda_W} q^{i,o,o'}_x \\, , \\labeq{metastate overlap}\n\\end{equation}\nwhere $\\sigma$ and $\\tau$ denotes replicas\\index{replica} of the system as in \\refeq{def_overlap}.\n\nAt this point, we should introduce a new type of average that will be useful in the study of the metastate\\index{metastate}. We have introduced the notion of metastate\\index{metastate} $\\kappa_{\\mathcal{J}}(\\Gamma_\\mathcal{J})$, which is a distribution over states, and therefore, we have a new kind of average. In addition to the usual thermal average\\footnote{In this chapter, we add the subscript $\\Gamma_{\\mathcal{J}}$ in order to avoid confusion with other averages.} $\\braket{\\cdots}_{\\Gamma_{\\mathcal{J}}}$ we can now average over the metastate\\index{metastate!averaged state} $\\left[ \\cdots \\right]_{\\kappa}$. \n\nThose averages can be combined in several ways but one particular combination is of great interest here. The \\gls{MAS} $\\rho$ is defined via the average $\\braket{\\cdots}_{\\rho} = \\left[ \\braket{\\cdots}_{\\Gamma_{\\mathcal{J}}} \\right]_{\\kappa}$. \n\nThe \\gls{MAS} is a mixed Gibbs state that can be decomposed in terms of pure states\\footnote{As stated in \\cite{read:14}, the organization of the pure states in the \\gls{MAS} is not ultrametric, on the contrary to each one of the states that compounds the metastate\\index{metastate} according to the \\gls{RSB}\\index{replica!symmetry breaking (RSB)}\\index{replica!symmetry breaking (RSB)} picture.}, as explained in \\refsec {mixed_pure_states_metastate}.\n\nThe main objects in our numerical study will be the \\gls{pdf} of the overlaps\\index{overlap!distribution}\n\\begin{equation}\nP(q) = \\overline{\\left[\\braket{\\delta(q-q^{i;o,o})}_{\\Gamma_\\mathcal{J}}\\right]_{\\kappa}}\\, , \\labeq{usual_pq}\n\\end{equation}\nwhere we remark that the overlap\\index{overlap} is defined with the same outer part $o$ for both replicas\\index{replica} and $\\overline{\\left(\\cdots\\right)}$ denotes the disorder average\\index{disorder!average}, as usual. This probability density function corresponds to the usual $P(q)$ in \\gls{SG}s because we are forcing the outer couplings\\index{couplings} to be the same, but we can also define a \\index{metastate!averaged state}metastate-averaged $P(q)$ as\n\\begin{equation}\nP_{\\rho}(q) = \\overline{\\left[ \\braket{\\delta(q-q^{i;o,o'})}_{\\Gamma_\\mathcal{J}} \\right]_{\\kappa}} \\, . \\labeq{metastate_pq}\n\\end{equation}\n\nThe \\gls{MAS} spin correlation\\index{correlation function!four point} function is\n\\begin{equation}\nC_{\\rho}(x) = \\overline{\\left[ \\braket{s^{\\sigma;i,o}_0 s^{\\sigma;i,o}_x}_{\\Gamma_\\mathcal{J}}\\right]^2_{\\kappa}} = \\overline{\\left[ \\braket{q^{i,o,o'}_0 q^{i,o,o'}_x}_{\\Gamma_\\mathcal{J}}\\right]_{\\kappa}} \\, .\n\\end{equation}\nThis correlation\\index{correlation function!four point} function has been studied in other contexts \\cite{dedominicis:98} (see also \\cite{dedominicis:06,marinari:00}) and it has been found that $C_{\\rho}(x) \\sim \\lvert x \\rvert^{-(d-\\zeta)}$ for large $\\lvert x \\rvert$. The exponent $\\zeta$ has a central importance in the Read's discussion of \\cite{read:14} as we discuss below.\n\nFinally, from this correlation\\index{correlation function!four point} function, we can define\n\\begin{equation}\n\\chi_{\\rho} = \\sum_{x \\in \\Lambda_W} C_{\\rho} = \\sum_{x \\in \\Lambda_W} \\overline{\\left[ \\braket{s^{\\sigma;i,o}_0 s^{\\sigma;i,o}_x}_{\\Gamma_\\mathcal{J}}\\right]^2_{\\kappa}} =  \\sum_{x \\in \\Lambda_W} \\overline{\\left[ \\braket{q^{i,o,o'}_0 q^{i,o,o'}_x}_{\\Gamma_\\mathcal{J}}\\right]_{\\kappa}}\\, .\n\\end{equation}\nWe can make some computations with this quantity, by decomposing it into pure states by using \\refeq{decomposition_pure_states}.\n\\begin{equation}\n\\chi_{\\rho} = \\sum_{x \\in \\Lambda_W} \\overline{\\left[ \\sum_{\\alpha} \\omega_{\\alpha,\\Gamma_{\\mathcal{J}}} \\braket{s^{\\sigma;i,o}_0 s^{\\sigma;i,o}_x}_{\\alpha}\\right]^2_{\\kappa}}  \\, . \\labeq{susc_intermediate}\n\\end{equation}\nIt is easy to prove that, because the pure states exhibit the clustering property $\\braket{s_0 s_x}_\\alpha \\sim \\braket{s_0}_{\\alpha}\\braket{s_x}_{\\alpha}$ as $\\lvert x \\rvert \\to \\infty$ and because as long as $W \\to \\infty$ the dominant terms in the sum are those with large $x$, the expression \\refeq{susc_intermediate} can be written as:\n\\begin{equation}\n\\chi_{\\rho} \\approx \\sum_{x \\in \\Lambda_W} \\overline{\\left[ \\sum_{\\alpha,\\beta} \\omega_{\\alpha,\\Gamma_{\\mathcal{J}}} \\omega_{\\beta,\\Gamma_{\\mathcal{J}}} \\braket{s^{\\sigma;i,o}_0}_{\\alpha} \\braket{s^{\\tau;i,o'}_0}_{\\beta}  \\braket{s^{\\sigma;i,o}_x}_{\\alpha} \\braket{s^{\\tau;i,o'}_x}_{\\beta}\\right]_{\\kappa}} \\, .\n\\end{equation}\nNow, using the relation $\\int_{-\\infty}^{\\infty} \\delta(x-a) dx = 1$ $\\forall a \\in \\mathbb{R}$ and the definitions of \\refeq{overlap_pure_states} and \\refeq{pq_pure_states} we can rewrite the previous expression \n\\begin{equation}\n\\chi_{\\rho} \\approx W \\int_{-\\infty}^\\infty q^2 P_{\\rho}(q) {\\mathrm{d}} q \\, .\n\\end{equation}\n\nThe quantity $\\chi_{\\rho}$ turns out to be the \\gls{MAS} susceptibility\\index{susceptibility}, and hence, it is related with the variance of the distribution function of the metastate\\index{metastate}. Recalling the long term behavior of the correlation\\index{correlation function!four point} function, the exponent $\\zeta$ emerges again\n\\begin{equation}\n\\chi_{\\rho} \\sim W^\\zeta \\, . \\labeq{susceptibility_scaling}\n\\end{equation}\n\n\\subsection{Theoretical predictions of the metastate} \\labsubsec{theoretical_predictions_metastate}\nAs introduced in \\refsubsec{theoretical_pictures} and in \\refsec{pictures_metastate} we have three mathematical consistent pictures for the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass}. First, the droplet\\index{droplet!picture} model whose metastate\\index{metastate} is concentrated on a single trivial state\\footnote{Recall that we name \\textit{trivial state} to a mixture of two pure states related by global spin-flip symmetry.}. Second, the chaotic pictures \\cite{newman:92,newman:96,newman:98,newman:03}, predicting a metastate\\index{metastate!chaotic pairs} composed of infinitely many states i.e. disperse metastate\\index{metastate!dispersed}, yet each state is trivial. Finally, the \\gls{RSB}\\index{replica!symmetry breaking (RSB)}\\index{replica!symmetry breaking (RSB)} metastate\\index{metastate!RSB} \\cite{read:14}, which is disperse and every state drawn from it contains the Parisi hierarchical tree of pure states. There might exist alternatives to these three pictures but are much limited by recent rigorous results \\cite{arguin:15}.\n\nThe Read's proposal \\cite{read:14} to partially discriminate between them is to compute the exponent $\\zeta$ because the logarithm of the number of pure states that one can discriminate in a window $\\Lambda_W$ of linear size $W$ scales as $\\sim W^{d-\\zeta}$. Therefore, $\\zeta<d$ implies an infinitely large number of states as $W \\to \\infty$, which in turn implies a disperse metastate\\index{metastate!dispersed}.\n\n\n\\section{Numerical construction of the metastate} \\labsec{simulation_parameters_metastate}\nWe simulate the \\gls{EA}\\index{Edwards-Anderson!order parameter} model introduced in \\refsubsec{3D_EA_model} for the sizes $L=\\{8,12,16,24\\}$ using Monte\\index{Monte Carlo} Carlo simulations consisting in Metropolis single spin-flip updates and \\gls{PT} exchanges. Due to the \\gls{PT} method, we have a temperature mesh of $n$ different values, however, in this work, we use only the lowest one $T=0.698 \\approx 0.64\\ensuremath{T_\\mathrm{c}}\\xspace$, well below the critical\\index{critical temperature} temperature $\\ensuremath{T_\\mathrm{c}}\\xspace = 1.102(3)$~\\cite{janus:13}.\n\nThe thermalization\\index{thermalization} protocol is explained with great detail in \\refsec{time_scales_eq_chaos} and \\refsec{variational_method_eq_chaos}, and for the largest systems, it required the use of multispin coding (see, \\refch{AP_multispin_coding}). Moreover, deeper explanation of the simulation can be found in~\\refsec{numerical_simulations_eq_chaos}.\n\nOur computation is performed for $N_i = 10$ different realizations of the internal couplings\\index{couplings} in the lattice $\\Lambda_R$ (see \\reffig{sketch_AW_metastate}) and, for each internal realization, we simulate a total of $N_o=128$ different outer-coupling\\index{couplings} realizations. Therefore, we have a total of $N_\\mathcal{J}=1280$ different samples\\index{sample} and for each one, we simulate $\\ensuremath{N_{\\text{Rep}}}\\xspace=4$ different replicas\\index{replica}.\n\nWe take $N_i \\ll N_o$ because we expect all the internal disorder\\index{disorder} to be ``typical''~\\cite{read:14} when computing metastate\\index{metastate!averaged state} averages at $R\\gg 1$. However, we find sizable sample-to-sample fluctuations\\index{sample-to-sample fluctuations} for the system sizes we consider. \n\nNumerically, the averages used in the computation of observables (see \\refsubsec{observables_metastate}) are approached in the following way:\n\\begin{itemize}\n\\item Thermal averages over the Gibbs state, that we denote as $\\braket{\\cdots}_{\\Gamma_{\\mathcal{J}}}$ are estimated via Monte\\index{Monte Carlo} Carlo thermal averages $\\braket{\\cdots}_{\\text{MC}}$ in a fixed sample\\index{sample}.\n\\item The metastate\\index{metastate!averaged state} average $\\left[ \\cdots \\right]_{\\kappa}$ is estimated by averaging over the outer disorder\\index{disorder!average} $\\sum_o \\sum_{o'} (\\cdots)\/N^2_o$.\n\\item The disorder\\index{disorder!average} average $\\overline{\\left(\\cdots\\right)}$ is estimated by averaging over the internal disorder\\index{disorder!average} $\\sum_i (\\cdots)\/N_i$.\n\\end{itemize}\nWith these estimations, the observables would be\n\\begin{equation}\nP(q) = \\dfrac{\\sum_i \\sum_o \\braket{\\delta(q-q_{i;o,o})}_{\\text{MC}}}{N_i N_o} \\, ,\n\\end{equation}\n\\begin{equation}\nP_{i,\\rho}(q) = \\dfrac{\\sum_{o,o'} \\braket{\\delta(q-q_{i;o,o'})}_{\\text{MC}}}{N^2_o} \\quad , \\quad P_{\\rho}(q) = \\dfrac{\\sum_i P_{i,\\rho}(q)}{N_i} \\, ,\n\\end{equation}\n\\begin{equation}\nC_{\\rho}(x) = \\dfrac{1}{N_i}\\sum_i \\dfrac{1}{N_o^2}\\sum_{o,o'} \\braket{s_0^{\\sigma;i,o}s_x^{\\sigma;i,o}s_0^{\\tau;i,o'}s_x^{\\tau;i,o'}}_{\\text{MC}} \\, .\n\\end{equation}\nFinally, due to the number of replicas\\index{replica} $\\ensuremath{N_{\\text{Rep}}}\\xspace=4$, it is worthy to note that, for each set of $\\{i,o,o'\\}$ we have $\\ensuremath{N_{\\text{Rep}}}\\xspace(\\ensuremath{N_{\\text{Rep}}}\\xspace-1)\/2=6$ different estimations of the observables if $o=o'$ and $\\ensuremath{N_{\\text{Rep}}}\\xspace^2=16$ estimations otherwise.\n\n\n\\section{Exploring the numerical metastate} \\labsec{results_metastate}\nIn this section, we will focus on the main results obtained from our work in the construction of the numerical metastate\\index{metastate}. First of all, the exponent $\\zeta$ is computed numerically in the metastate\\index{metastate} setup. Moreover, one of the requisites for the \\gls{AW} metastate\\index{metastate!Aizenman-Wehr} is the relation of the three-length scales $W \\ll R \\ll L$. We investigate which are the ratios $W\/R$ and $R\/L$ that allow safe computations (given an accuracy level). \n\nFinally, we explore the scaling of the functions $P_{\\rho}(q)$ and $P(q)$ as $W$ grows.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.85\\textwidth]{metastate\/RL_ratio.pdf}\n\\caption[\\textbf{\\boldmath $\\mathbf{R\/L}$ ratio in the metastate.}]{\\textbf{\\boldmath $\\mathbf{R\/L}$ ratio in the metastate\\index{metastate}.} Main plot: The \\gls{MAS} overlap\\index{overlap!distribution} distribution $P_{\\rho}(q)$ with $R=12$ at $T=0.698 \\approx 0.64\\ensuremath{T_\\mathrm{c}}\\xspace$ shows no statistically significant dependence on the lattice size $L$ (the error bars\\index{error bars}, computed from fluctuations on the inner disorder\\index{disorder}, are shown for only one value of $q$ for the sake of clarity). \\textbf{Insets:} $P_{i,\\rho}(q)$ for two specific configurations\\index{configuration} of the inner disorder\\index{disorder} (the error bars\\index{error bars}, computed from fluctuations on the outer disorder\\index{disorder}, are smaller than the data points).}\n\\labfig{RL_ratio}\n\\end{figure}\n\n\\subsection[The $R\/L$ ratio]{The \\boldmath $R\/L$ ratio}\nIn the main plot of~\\reffig{RL_ratio} we see that the \\gls{MAS} $P_{\\rho}(q)$ measured with $R=12$ and both $L=24$ and $L=16$ are statistically compatible, suggesting that the $R\/L = 3\/4$ is already a safe choice. As we have mentioned before, the error bars\\index{error bars} are quite large due to an unexpected high dependence of $P_{i,\\rho}(q)$ on the internal disorder\\index{disorder} for the values of $W$ and $R$ that we simulate. The reader can check in the inset of~\\reffig{RL_ratio} two examples of $P_{i,\\rho}(q)$, where the error bars\\index{error bars} are smaller than the data point and can appreciate the sample-to-sample\\index{sample-to-sample fluctuations} fluctuation. \n\nAn additional check can be performed by representing the susceptibility\\index{susceptibility} $\\chi_{\\rho}$ against the size of the measuring window $W$ for different values of $L$ by keeping constant $R$, see~\\reffig{RL_WR}. We see that, for $R\/L \\leq 0.75$ the data are statistically compatible, while data with $R\/L = 6\/7$ show significant deviations.\n\nIn view of the previous results, we establish the ratio $R\/L=0.5$ as a safe choice for the following analysis.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.85\\textwidth]{metastate\/RL_WR.pdf}\n\\caption[\\textbf{Susceptibility $\\mathbf{\\chi_{\\rho}}$ for different length scales $\\mathbf{\\{W,R,L\\}}$.}]{\\textbf{Susceptibility\\index{susceptibility} $\\mathbf{\\chi_{\\rho}}$ for different length scales $\\mathbf{\\{W,R,L\\}}$.} The susceptibility\\index{susceptibility} $\\chi_{\\rho}$ is plotted against $W$ for a fixed $R=12$ in three different curves $L=14,16,24$. Deviation from the asymptotic behavior $R\/L \\ll 1$ are evident only for $R\/L>0.75$.} \n\\labfig{RL_WR}\n\\end{figure}\n\n\\subsection[The $W\/R$ ratio]{The \\boldmath $W\/R$ ratio}\nThe susceptibility\\index{susceptibility} scaling with $W$ in the $W\/R \\ll 1$ limit has been already expressed in~\\refeq{susceptibility_scaling}. Therefore, the~\\reffig{susceptibility_scaling} gives us relevant information about the validity range for $W\/R$. We note that the expected power-law behavior in the $W\\ll R$ limit actually extends up to $W\/R \\approx 0.75$, where corrections to the asymptotic power-law appear. Therefore, similarly to the $R\/L$ case, the $W\/R \\approx 0.75$ stands as a safe choice.\n\n\\subsection{The exponent \\boldmath $\\zeta$}\nIn this section, we fix $R=L\/2$ (which is in the safe side, given our bound $R< 3L\/4$) and $W\/R \\approx 0.75$. We can now compute the exponent $\\zeta$ by taking advantage of the relation~\\refeq{susceptibility_scaling}. Fitting the data with $W\/R \\leq 0.75$ we found $\\zeta = 2.3 \\pm 0.3$. Moreover, we are concerned about the finite-size effects\\index{finite-size effects} that our small-lattices simulations could suffer.\n\nA finite-size scaling~\\cite{cardy:12}\\index{finite size scaling} is needed to study the size effects\\index{finite-size effects}. Indeed, we expect for finite $R$ and $W$ a scaling behavior\n\\begin{equation}\n\\chi_\\rho(W,R) = R^\\zeta f(W\/R)\\, , \\labeq{R-finite}\n\\end{equation}\nwhich is compatible to~\\refeq{susceptibility_scaling} for $f(x) \\propto x^\\zeta$ in the $x\\to 0$ limit. \\refeq{R-finite} is expected to be exact only in the limit of large $W$ and $R$~\\cite{cardy:12}, hence one needs to check for size corrections. We do so with the quotients method\\index{quotients method}~\\cite{nightingale:76,ballesteros:96,amit:05} which produces effective $\\zeta$ estimates at a well defined length scale. The size dependence can be assessed later on. Specifically, we take two sizes-pairs $(W_1,R_1)$, $(W_2,R_2)$ with the same value of $W\/R$, which ensures the cancellation of scaling functions in the quotient\n\\begin{equation}\n\\frac{\\chi_\\rho(W_2=x R_2,R_2)}{\\chi_\\rho(W_1=x R_1,R_1)} = \\frac{\\left(W_2\/x\\right)^\\zeta f(x) }{\\left(W_1\/x\\right)^\\zeta f(x)}=\\left(\\frac{W_2}{W_1}\\right)^\\zeta\\, . \\labeq{quotients}\n\\end{equation}\n\nThe resulting determination of $\\zeta$, see \\reftab{zeta_exponent_metastate}, is fully compatible with the computed result $\\zeta=2.3 \\pm 0.3$. Furthermore, no significant size-dependence emerges\nfrom \\reftab{zeta_exponent_metastate}. \n\n\\begin{table}\n\\centering\n\\begin{tabular}{cclc}\n \\toprule \n \\toprule\n$W\/R$ & $L\/R$ & $(W_1,W_2)$ & $\\zeta^\\mathrm{eff}$\\\\\\hline\\hline\n1\/2 & 2 & (4,6) & 2.18(40)\\\\\\hline\n2\/3 & 2 & (4,8) & 2.59(22)\\\\\\hline\n1   & 2& (8,12) & 2.37(26)\\\\\n    &  &(6,8)  & 2.14(37)\\\\\n    &  &(6,12) & 2.28(18)\\\\\n\\bottomrule\n\\end{tabular}\n\\caption[\\textbf{Effective $\\mathbf{\\zeta}$ exponent.}]{\\textbf{Effective $\\mathbf{\\zeta}$ exponent.} The effective $\\zeta$ exponent depends on the two lengths $W_1$ and $W_2$ and\non the ratio $W_1\/R_1=W_2\/R_2$. }\n\\labtab{zeta_exponent_metastate}\n\\end{table}\n\nBesides, in~\\reffig{susceptibility_scaling}, the \\gls{MAS} susceptibility\\index{susceptibility} has been rescaled by using the previously defined scaling relations. A power-law behavior is exhibited for $W\/R <0.75$ as expected and our $\\zeta$ estimation interpolates the data nicely in that region.\n\n\\begin{figure}[h!]\n\\includegraphics[width=1.0\\columnwidth]{metastate\/susceptibility_scaling}\n\\caption[\\textbf{Scaling behavior of the \\gls{MAS} susceptibility.}]{\\textbf{Scaling behavior of the \\gls{MAS} susceptibility\\index{susceptibility}.} \\gls{MAS} susceptibility\\index{susceptibility} data measured with fixed $R\/L=1\/2$ at $T=0.698 \\approx 0.64 \\ensuremath{T_\\mathrm{c}}\\xspace$ as a function of the $W\/R$ ratio.}\n\\labfig{susceptibility_scaling}\n\\end{figure}\n\n\\subsection{Size dependence of \\boldmath $P(q)$ and \\boldmath $P_{\\rho}(q)$}\nWe show in~\\reffig{pq_metastate}, for $L=24$ and $R\/L=1\/2$ the dependence of the functions $P_{\\rho}(q)$ and $P(q)$ on $W$. The expectation for a dispersed metastate\\index{metastate!dispersed}~\\cite{read:14} is that both distributions are different in the thermodynamic limit\\index{thermodynamic limit}. We found here that they are distinct objects even for moderate sizes of $W$.\n\n\\begin{figure}[h!]\n\\includegraphics[width=1.0\\columnwidth]{metastate\/pq.pdf}\n\\caption[\\textbf{Size dependence of $\\mathbf{P(q)}$ and $\\mathbf{P_{\\rho}(q)}.$}]{\\textbf{Size dependence of $\\mathbf{P(q)}$ and $\\mathbf{P_{\\rho}(q)}.$} Functions $P_{\\rho}(q)$ and $P(q)$ for $L=24$, $R=L\/2$ and $T=0.698 \\approx 0.64 \\ensuremath{T_\\mathrm{c}}\\xspace$. Different panels corresponds to different measuring window size $W=4,8,12$.}\n\\labfig{pq_metastate}\n\\end{figure}\n\n\\section{Relating numerical results and theory} \\labsec{relating_numerical_theory_metastate}\nTo the best of our knowledge, this is the first numerical construction of the metastate\\index{metastate} carried out in equilibrium simulations. Nonetheless, there exists one previous study in the non-equilibrium regime~\\cite{manssen:15}, however, a deep relation between this \\textit{dynamic metastate}\\index{metastate!dynamic} and the \\gls{AW} metastate\\index{metastate!Aizenman-Wehr} is still to be explored. \n\nWe have shown that the actual state of the art in the numerical \\gls{SG}s allows the simulation of the \\gls{AW} metastate\\index{metastate!Aizenman-Wehr} in the \\gls{EA}\\index{Edwards-Anderson!order parameter} for $d=3$. We cannot extrapolate safely to the thermodynamic limit\\index{thermodynamic limit} and the unexpected dependence of the \\gls{MAS} on the inner disorder\\index{disorder} is still important at the accessible system sizes. Nevertheless, we have found a convincing scaling law for the \\gls{MAS} susceptibility\\index{susceptibility} and we have estimated the exponent $\\zeta(d=3)=2.3 \\pm 0.3$, which strongly suggest $\\zeta < d$, in addition, we have found no substantial size-dependence for this exponent.\n\n\\begin{figure}\n\\includegraphics[width=0.9\\columnwidth]{metastate\/zeta_exponent.pdf}\n\\caption[\\textbf{The exponent \\boldmath $\\zeta$ as a function of  $d$.}]{\\textbf{The exponent \\boldmath$\\zeta$ as a function of $d$.} Different predictions of the exponent $\\zeta$ for $d=3$ and $d=4$ are plotted. Above $d=6$, the mean-field\\index{Mean-Field!solution} solution $\\zeta=4$ is correct. The line $\\zeta=d$ separates the disperse metastate\\index{metastate!dispersed} for the trivial one.}\n\\labfig{zeta_exponent_metastate}\n\\end{figure}\n\nIn~\\reffig{zeta_exponent_metastate} we have summarized our knowledge about the $\\zeta$ exponent. Below the lower critical dimension\\index{critical dimension!lower}\\footnote{The lower critical dimension\\index{critical dimension!lower} is the dimension below which there is no phase transition\\index{phase transition} at finite temperature $T$.} $d_\\mathrm{L}$ (at zero magnetic-field), the droplets\\index{droplet!picture} picture is expected to be valid, and, therefore, the exponent $\\zeta$ should be $\\zeta \\geq d$. In this work we have found $\\zeta = 2.3 \\pm 0.3$ (green-squared point in~\\reffig{zeta_exponent_metastate}). Moreover, alternative estimations of the $\\zeta$ exponent come from the decay of the four-spins spatial correlation\\index{correlation function!four point} function at equilibrium.\n\nIn the equilibrium systems, the spins configurations\\index{configuration} of different replicas\\index{replica} follow a distribution function $P(q)$ (see~\\refsubsec{theoretical_models}) and, therefore, the $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$ has contributions of pair of replicas\\index{replica} with all the possible values of $q$, weighted with the $P(q)$. However, the metastate\\index{metastate} is in the $q=0$ by definition, and therefore, in order to compare both determinations, we have to restrict the computation of the correlation\\index{correlation function!four point} functions to the zero overlap\\index{overlap!zero sector} sector in the equilibrium results.\n\nThe four-point correlation\\index{correlation function!four point} function conditioned to the $q=0$ sector is\n\\begin{equation}\nC_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace | q=0) \\sim \\lvert x \\rvert^{-\\vartheta} \\, ,\n\\end{equation}\nwith $\\vartheta=d-\\zeta_{q=0}$, see~\\refeq{long_distance_C4}. Previous studies found $\\zeta_{q=0}(d=3) = 2.62 \\pm 0.02$~\\cite{janus:09b,janus:10} and $\\zeta_{q=0}(d=4) = 2.62 \\pm 0.02$~\\cite{nicolao:14} (blue circles in~\\reffig{zeta_exponent_metastate}). \n\nFinally, in $d\\geq6$, where the mean-field\\index{Mean-Field} computations are correct, we found $\\zeta=4$~\\cite{dedominicis:98,dedominicis:99}. A gentle extrapolation with the values of $\\zeta_{q=0}(d=3,4)$ and the value of $z(d=6)=4$ (dashed line in~\\reffig{zeta_exponent_metastate}) seems to meet, as expected, the yellow line corresponding to $d=\\zeta$ around $d=2.5$, which is a general accepted estimation for the lower critical dimension\\index{critical dimension!lower} from numerical and experimental results (see e.g.~\\cite{franz:94,boettcher:04,boettcher:04b,boettcher:05,guchhait:14,maiorano:18}).\n\nAs we have previously discussed in~\\refsubsec{theoretical_predictions_metastate}, the exponent $\\zeta$ is related with the number of states that can be discriminated in a measuring window of size $W$, scaling that number with $\\sim W^{d-\\zeta}$~\\cite{read:14}. This numerical estimation of $\\zeta$ for $d=3$ supports the pictures of the metastate\\index{metastate} with infinitely many states, namely \\gls{RSB}\\index{replica!symmetry breaking (RSB)} metastate\\index{metastate!RSB} and chaotic pairs\\index{metastate!chaotic pairs}.\n\n\n\n\n\n\n\n\n\\chapter[The Mpemba effect]{The Mpemba effect} \\labch{mpemba}\n\n\\epigraph{\\textit{Aqu\u00ed el que no carneguea, borreguea.}}{-- Acervo popular}\n\nConsider two beakers of water that are identical to each other except for the fact that one is hotter than the other. If we put both of them in contact with a thermal reservoir (for example, a freezer) at some temperature lower than the freezing point of the water, under some circumstances, it can be observed that the, initially, hotter water freezes faster than the colder one. This phenomenon is known as the Mpemba\\index{Mpemba effect} effect~\\cite{mpemba:69}.\n\nThe history of this phenomenon is, indeed, very curious and constitutes one paradigmatic example of the importance of the scientific method in the development of science. Although the first written record comes from Aristotle, it would be probably a well-known fact for most of the people~\\cite{aristotle-lee:89}. This effect was sporadically mentioned through the ages~\\cite{bacon-burke:62,bacon:11,descartes:65} but it received little attention from the scientific community until the second half of the XX century.\n\nIn 1969 this phenomenon was brought back to the scientific debate by Erasto Mpemba, a young student in Tanzania, and Denis Osborne, a teacher of the University College Dar es Salaam, Tanzania~\\cite{mpemba:69}. The same year, Dr. Kell reported the same phenomenon in an independent publication~\\cite{kell:69}. \n\nDifferent arguments were given to explain this phenomenon~\\cite{kell:69,deeson:71,firth:71,walker:77}, but there is no consensus neither in the explanations~\\cite{osborne:79,freeman:79,wojciechowski:88} nor in the very existence of the effect.~\\cite{burridge:16}. We will briefly discuss the situation later in \\refsec{water_complicated_mpemba}.\n\nThis phenomenon is not specific to water and has been reported in other systems like nanotube resonators~\\cite{greaney:11}, clathrate hydrates~\\cite{ahn:16}, granular fluids~\\cite{lasanta:17} and colloidal systems~\\cite{kumar:20}. This chapter is devoted to discussing the Mpemba\\index{Mpemba effect} effect in \\gls{SG}s. Besides, the Mpemba\\index{Mpemba effect} effect constitutes a great example to stress the importance of the coherence length\\index{coherence length} as a fundamental quantity to describe the off-equilibrium phenomena in \\gls{SG}s.\n\nWe begin with a brief historical introduction\\footnote{The reader may consult also a fantastic historical review in~\\cite{jeng:06}.} to the phenomenon in \\refsec{historical_introduction_mpemba} and with some of the proposed explanations in~\\refsec{water_complicated_mpemba}. Then, we explain the numerical simulation, performed in the Janus\\index{Janus} II custom-built computer, that has allowed the study of the Mpemba\\index{Mpemba effect} effect in \\gls{SG} (see~\\refsec{numerical_simulation_mpemba}). At this point, we are ready to discuss the results. We first identify in \\refsec{identifying_mpemba} the Mpemba\\index{Mpemba effect} effect in \\gls{SG} by choosing the adequate quantity to represent the \\textit{temperature} of our system. We found in \\refsec{coherence_length_mpemba} that the quantity controlling the phenomenon is, indeed, the coherence length\\index{coherence length} $\\xi$. Finally, we study the Inverse Mpemba\\index{Mpemba effect} effect in \\refsec{inverse_mpemba}.\n\nThe results described in this chapter were published in~\\cite{janus:19}.\n\n\\section{A historical introduction}\\labsec{historical_introduction_mpemba}\n\nThe first record of the Mpemba\\index{Mpemba effect} effect is attributed to Aristotle in his \\textit{Metereologica} around 350 B.C.~\\cite{aristotle-lee:89}. His discussion there, suggests that the phenomenon was a well-known fact and he used it as an example to illustrate his theory of \\textit{antiperistasis}\\footnote{The concept of antiperistasis refers to the reaction between two opposite \\textit{forces}, when one increases, the other have to do it.}:\n\n\\textit{``If the water has been previously heated, this contributes to the rapidity with which it freezes [sic] for it cools more quickly (Thus so many people when they want to cool water quickly first stand it in the sun and the inhabitants of Pontus when they encamp on the ice to fish --they catch fish through a hole which they make in the ice-- pour hot water on their rods because it freezes quicker, using the ice like solder to fix their rods.) \\dots ''}\n\\\\[5pt]\n\\rightline{{ --- Aristotle, Metereologica Book I, Chapter XII}}\n\nThe Mpemba\\index{Mpemba effect} effect probably remained in the popular heritage through the centuries, but actually, did not receive too much attention from academics. Yet, it was mentioned in some important texts, for example in the \\textit{Opus Majus} of Roger Bacon~\\cite{bacon-burke:62} or in the \\textit{Novum Organum} of Francis Bacon~\\cite{bacon:11}. \n\nSome years after Francis Bacon mentioned the phenomenon, Descartes wrote about it in \\textit{Les Meteores}~\\cite{descartes:65}. Indeed, he stressed the importance of the experiments, and actually, he proposed a specific experiment that is not exactly the standard Mpemba effect (which is the most commonly studied). \n\nDescartes proposed to fill a beaker (and he also specified that should have a long straight neck) with hot water that has been kept over the fire for a long time, then the water should be let to reach room temperature. He proposed to do the same with another beaker of water but now, without boiling it. Then, both beakers should be put in contact with the [sic] ``freezing cold air'' and one observes that the beaker which has held for a long time over the fire, freezes first.\n\nHe stated in a letter to Mersenne (1638) that he performed that experiment and he defended that there was nothing incorrect in his methods. \n\nHowever, the Mpemba\\index{Mpemba effect} effect was relegated to oblivion in mainstream physics by the emergence of thermodynamics, which was supported by an unprecedented success, both in describing reality and in the creation of modern machines. Apparently, the Mpemba\\index{Mpemba effect} effect contradicts the knowledge provided by thermodynamics. It has been suggested~\\cite{kuhn:12} that the theoretical views of scientists may condition the experiments that they decide to perform, and this could be the answer to the small numbers of experiments researching this interesting effect before the XX century. Actually, this points also to one of the weaknesses of the human being, that affects the application of the scientific method: the confirmation bias.\n\nIn 1963, one young student of a secondary school in Tanzania, Erasto Mpemba, put this phenomenon under the scrutiny of the scientific community~\\cite{mpemba:69}. In his school, he used to make ice-cream by boiling milk, mixing it with sugar, and by putting the mix into the freezer. One day, some students were doing ice-cream but the space in the freezer was scarce. Despite the warnings for not introducing the hot mix directly in the freezer because that could damage it, Mpemba decided to do it anyway in order to do not lose his space in it.\n\nHe observed that his mix had frozen before that of other boys who had followed the \\textit{standard protocol} and had let the mix cool at the room's temperature before introducing it in the freezer. Persistent questions of Mpemba to different physics teachers about this fact led to the same answer ``That is impossible'', one of the teachers said that ``That is Mpemba's physics, not universal physics''.\n\nNonetheless, Mpemba did not surrender and he took the opportunity to ask this question to a university professor, Denis Osborne, that went to Mpemba's High School to give a talk. From the first time that Mpemba observed the phenomenon, he did great advances on building up a specific protocol to develop the experiment in a reproducible way and he asked a very concrete question:\n\n\\textit{``If you take two beakers with equal volumes of water, one at 35\u00baC and the other at 100\u00baC, and put them into a freezer, the one that started at 100\u00baC freezes first. Why?''}~\\cite{mpemba:69}.\n\nFortunately, Osborne did not dismiss the Mpemba's claim although he confessed that he thought at the first moment that Mpemba was wrong. Indeed, he asks a technician to make the experiment. The result was the following~\\cite{mpemba:69}:\n\n\\textit{``The technician reported that the water that started hot did indeed freeze first and added in a moment of unscientific enthusiasm: But we'll keep on repeating the experiment until we get the right result.''}\n\nOf course, further tests led to the same results and they started to think about an explanation that we will briefly sketch in the next section. The same year of the publication of the article of Mpemba and Osborne, Dr. Kell in Canada reported the same experiment~\\cite{kell:69}.\n\nA curious fact is that this phenomenon, though very far from academic physics, was conserved in the popular heritage through the years. Indeed, Mpemba remembers that the ice-cream makers in his town were aware of this effect and they used it commonly to obtain their ice-cream faster~\\cite{mpemba:69}.\n\nSeveral explanations came up in the following years (see~\\refsec{water_complicated_mpemba}) and the phenomenon was found to be not exclusive from water. Indeed, it was found in nanotube resonators~\\cite{greaney:11}, clathrate hydrates~\\cite{ahn:16}, granular fluids~\\cite{lasanta:17}, colloidal systems~\\cite{kumar:20} and, as we will expose in this chapter, in spin glasses~\\cite{janus:19}.\n\n\\section{Water is too complex} \\labsec{water_complicated_mpemba}\nAfter the work of Mpemba and Osborne~\\cite{mpemba:69}, several explanations were proposed, however, one of the main difficulties in the study of the Mpemba\\index{Mpemba effect} effect is the high number of parameters that may affect the phenomenon. The shape of the beaker, the composition of the water, its temperature distribution \\dots All of these parameters might play an important role in the explanation of the effect and should be taken into account.\n\nThe mass of the water may be one of these parameters that could (at least partially) explain the effect. The hotter system shall lose more mass due to evaporation processes than the colder one and it has been suggested~\\cite{kell:69} that this would be the reason for the Mpemba\\index{Mpemba effect} effect to occur. However, other experiments claimed that this mass loss would be insufficient to explain Mpemba\\index{Mpemba effect} effect~\\cite{osborne:79,freeman:79,wojciechowski:88}.\n\nAnother parameter that may play a role in the phenomenon is the temperature distribution of the water. Cold water is denser than hot water\\footnote{Above 4\u00baC.} and, when preparing the hot beaker, if the heating process is not uniform that may induce convection currents in it. Due to those convection currents and the different densities of the water as a function of the temperature, the top part of the hot beaker would be at a lower temperature than the bottom part. This could favor the creation of a layer of ice on the top of the hot beaker before than in the colder one. Besides, the convection currents may work together with other factors, like the above-mentioned evaporation, to provoke the Mpemba\\index{Mpemba effect} effect in the water. These convection currents would in turn be affected by other parameters like the shape of the beaker. Actually, experiments in which the hotter beaker was stirred in order to make the temperature gradient disappear, showed a sizable raise of the time of freezing~\\cite{deeson:71}.\n\nThe above examples are only two of the variety of explanations proposed for the Mpemba\\index{Mpemba effect} effect. However, the situation is far from clear and there exist experiments claiming the nonexistence of the phenomenon, see for example~\\cite{burridge:16}.\n\nIt is clear that the complexity of water makes it too difficult to study the Mpemba\\index{Mpemba effect} effect and it would be desirable to have a much better-controlled system to study the phenomenon. Here, we take advantage of the numerical simulations in \\gls{SG}s to address the Mpemba\\index{Mpemba effect} effect and studying it with total control of the system.\n\n\\section{Numerical simulation}\\labsec{numerical_simulation_mpemba}\nIn this work, we use the same simulations performed for the study of the aging\\index{aging!rate} rate (\\refch{aging_rate}) and we add some simulations with temperature-varying\\index{temperature-varying protocol} protocols. We briefly remind here the parameters of the simulation for the reader's convenience.\n\nWe simulate in the \\gls{FPGA}-based\\index{FPGA} computer Janus\\index{Janus} II an \\gls{EA}\\index{Edwards-Anderson!model} model in three-dimensional spin glasses (\\refsubsec{3D_EA_model}) for several temperatures $T$ in a lattice of linear size $L=160$, which is aimed to represent a system of infinite size. \n\nWe shall perform two different protocols. The isothermal\\index{isothermal} protocol consists of a direct quench from configurations\\index{configuration} of spins randomly initialized (which corresponds to infinite temperature) to the working temperature $T$, where the system is left to relax for a time $\\ensuremath{t_\\mathrm{w}}\\xspace$. This relaxation\\index{relaxation} corresponds to the (very slow) growth of glassy magnetic domains\\index{magnetic domain} of size $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$. To uniquely identify this protocol, it is enough to label it with its temperature $T$.\n\nThe temperature-varying\\index{temperature-varying protocol} protocol begins in the same way that the isothermal\\index{isothermal} one, by quenching the system from configurations\\index{configuration} of randomly initialized spins to the working temperature $T_1$. When the system reaches a certain coherence length\\index{coherence length} $\\xi'(\\ensuremath{t_\\mathrm{w}}\\xspace)$ we change the temperature of the thermal reservoir to a temperature $T_2$. Hence, this protocol should be labeled with a pair of temperatures (the initial one $T_1$ and the final one $T_2$), and with the coherence length\\index{coherence length} at which the temperature-change was produced $\\xi'(\\ensuremath{t_\\mathrm{w}}\\xspace)$. We use the following notation $T_1,\\xi' \\to T_2$.\n\nWe compute a total of $\\ensuremath{N_{\\text{S}}}\\xspace = 16$ different samples\\index{sample}. For each sample\\index{sample}, we shall consider $\\ensuremath{N_{\\text{Rep}}}\\xspace=256$ replicas\\index{replica}. As said in~\\refch{aging_rate}, this simulation had the original aim to study the temperature chaos\\index{temperature chaos} phenomenon under non-equilibrium conditions (see~\\refch{out-eq_chaos}), however, the reader may notice that in that study we use a total number of replicas\\index{replica} $\\ensuremath{N_{\\text{Rep}}}\\xspace=512$. Indeed, this study about the Mpemba\\index{Mpemba effect} effect was performed much earlier and we had at our disposal ``only'' $\\ensuremath{N_{\\text{Rep}}}\\xspace=256$.\n\nThe main observables of this study are the coherence length\\index{coherence length} $\\xi(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$ and the energy\\index{energy!density} density $e(t)$. The coherence length\\index{coherence length} is estimated by $\\xi_{12}(T,\\ensuremath{t_\\mathrm{w}}\\xspace)$, computed from integral estimators of the correlation function $C_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace)$. These two observables have been described with great detail in \\refsubsec{observables_introduction}. The energy\\index{energy!density} density is defined as\n\\begin{equation}\ne(t,{\\mathcal{J}}) = \\dfrac{1}{L^3} \\braket{\\mathcal{H}_{\\mathcal{J}}(t)} \\quad  \\, \\quad e(t) = \\overline{e(t,\\mathcal{J})} \\, , \\labeq{energy-density_definition}\n\\end{equation}\nwhere $\\overline{(\\cdots)}$ is the exact mean over the disorder\\index{disorder!average}. Although this estimation is perfectly correct, we decided to increase the accuracy of our estimations by using a control variate\\index{control variate}~\\cite{fernandez:09,ross:14} ${\\varepsilon_{\\mathrm{cv}}(\\mathcal{J})}$ depending on the sample\\index{sample} $\\mathcal{J}$ and with an exact disorder\\index{disorder!average} average $\\mu_{{\\varepsilon_{\\mathrm{cv}}(\\mathcal{J})}}$. \n\nThe studied quantity would be now\n\\begin{equation}\n\\tilde{e}(t,\\mathcal{J}) = e(t,{\\mathcal{J}}) - \\left[ {\\varepsilon_{\\mathrm{cv}}(\\mathcal{J})} - \\mu_{{\\varepsilon_{\\mathrm{cv}}(\\mathcal{J})}} \\right] \\quad \\, \\quad \\tilde{e}(t) = \\overline{\\tilde{e}(t,\\mathcal{J})} \\, . \\labeq{energy-density_control_variate}\n\\end{equation}\n\nThis new quantity $\\tilde{e}(t,\\mathcal{J})$ has the same disorder\\index{disorder!average} mean that the usual energy-density\\index{energy!density} $e(t,\\mathcal{J})$ but has a significantly lower variance. The reader may find further details of the implementation of the control variate\\index{control variate} in \\refsec{improving_statistics}.\n\n\\section{Identifying the Mpemba Effect}\\labsec{identifying_mpemba}\nThe aim of identifying the Mpemba\\index{Mpemba effect} effect in \\gls{SG} is composed of two main tasks. Firstly, we have to identify what ``temperature'' means in an out-of-equilibrium \\gls{SG}. Secondly, we have to establish a protocol to mimic the traditional protocol of the classic Mpemba\\index{Mpemba effect} effect. \n\nThe natural candidate to take the place of the temperature, which is telling us if a system is hotter than another, is the energy-density\\index{energy!density} $e(t)$ [or equivalently, $\\tilde{e}(t)$] because it is the observable conjugated with (the inverse of the) temperature. Furthermore, at equilibrium, where the temperature $T$ of the thermal reservoir corresponds to the temperature of the system by definition, there exists a monotonically increasing correspondence between the energy-density\\index{energy!density} and the temperature.\n\nThe protocol followed in our numerical experiment strongly resembles the original Mpemba's protocol~\\cite{mpemba:69}. We study the evolution of three different off-equilibrium systems. The first one is quenched from infinite temperature (random configuration\\index{configuration}) to a temperature (of the thermal reservoir) $T_1=1.3$, which is above the critical\\index{critical temperature} temperature $\\ensuremath{T_\\mathrm{c}}\\xspace = 1.102(3)$~\\cite{janus:13}. This system is labeled with the number $1$. We let it evolve until it reaches an energy\\index{energy!density} $\\tilde{e}_1(t=0)\\approx -1.6428$ and we set this time as our starting point of the (numerical) experiment. \n\nThe second system (labeled with the number $2$) is prepared in a similar way but the temperature of the reservoir is now $T_2=1.2$ and we let the system reach a much lower energy\\index{energy!density} $\\tilde{e}_2(t=0)\\approx -1.6714$. \n\nIn the last one, labeled with the number $3$, the temperature of the reservoir is again $T_3=1.2$ but the starting point is at even lower energy\\index{energy!density} $\\tilde{e}_3(t=0) \\approx -1.6738$. \n\nAt that point, we quenched the three systems to a temperature $T_{\\mathrm{f}} =0.7 \\approx 0.64\\ensuremath{T_\\mathrm{c}}\\xspace$, we let them evolve and we record their energies. The results are shown in \\reffig{first_mpemba} where we can appreciate the classical Mpemba\\index{Mpemba effect} effect. The \\textit{hot start} system (system 1) crosses the other two curves in a way indicating a faster cooling process.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{mpemba\/first_mpemba.pdf}\n\\caption[\\textbf{Classical Mpemba protocol}]{\\textbf{Classical Mpemba protocol}. We show the time evolution of the energy\\index{energy!density} of spin-glass systems initially prepared at a higher temperature ($T=1.3$, yellow line) or a lower temperature ($T=1.2$, blue and green lines), but always in the paramagnetic (high-temperature) phase\\index{phase!high-temperature\/paramagnetic} ($T_\\text{c}\\approx1.102$). In all three cases, the systems are initially left to evolve out of equilibrium until they reach the internal energies shown in the figure key. At $t=0$ all preparations are quenched, that is, put in contact with a thermal reservoir at temperature $T=0.7\\approx0.64 T_\\text{c}$.  As discussed in the text, the instantaneous energy-density\\index{energy!density} is a measure of the (off-equilibrium) sample\\index{sample} temperature. In agreement with the original Mpemba experiment~\\cite{mpemba:69}, the system originally at the higher energy\\index{energy!density} cools faster. \\textbf{Bottom left inset:} Closeup of the first crossing between energy\\index{energy!density} curves, showing the very small error bars\\index{error bars}, equal to the thickness of the lines. \\textbf{Top right inset:} Closeup of the second crossing between energy\\index{energy!density} curves.}\n\\labfig{first_mpemba}\n\\end{figure}\n\nThe reader should notice that the crossing of $\\tilde{e}_1(t)$ and $\\tilde{e}_2(t)$ takes place at much longer times that the crossing of $\\tilde{e}_1(t)$ and $\\tilde{e}_3(t)$, even if the initial energies of $\\tilde{e}_2(t)$ and $\\tilde{e}_3(t)$ differ only by a $0.15\\%$. We need a control parameter that helps us to quantitatively characterize the Mpemba\\index{Mpemba effect} effect.\n\n\\section{Coherence length controls the Mpemba Effect in spin glasses}\\labsec{coherence_length_mpemba}\nThe natural candidate, that characterizes the dynamical state of an off-equilibrium \\gls{SG}, is the coherence length\\index{coherence length} $\\xi(t)$. Indeed, in terms of the coherence length\\index{coherence length} $\\xi(t)$ our three systems are very different. The \\textit{hot start} system [$T_1=1.3$ and $\\tilde{e}_1(t=0)\\approx -1.6428$] has $\\xi_1(t=0)=12$, the \\textit{cold start} system [$T_2=1.2$ and $\\tilde{e}_2(t=0)\\approx -1.6714$] has $\\xi_2(t=0)=5$ and the \\textit{colder start} system [$T_3=1.2$ and $\\tilde{e}_3(t)\\approx -1.6738$] has $\\xi_3(t=0)=8$.\n\nThis perspective is pointing us that the out-equilibrium \\gls{SG}s for the study of the Mpemba\\index{Mpemba effect} effect should not be labeled only with the temperature of the thermal reservoir, not even with the temperature of the thermal reservoir plus the energy density\\index{energy!density} at some time $t$. We need the coherence length\\index{coherence length} to fully characterize the state of the system and understand the Mpemba\\index{Mpemba effect} effect in \\gls{SG}s. Next, we test this hypothesis.\n\n\\subsection{A first test} \\labsubsec{first_test_mpemba}\nIf our hypothesis is correct, crossing the critical\\index{critical temperature} temperature should not matter in order to observe the Mpemba\\index{Mpemba effect} effect. We only require that both starting points fulfill the next conditions: $T_{\\mathrm{A}} > T_{\\mathrm{B}}$ and $\\xi_{\\mathrm{A}} > \\xi_{\\mathrm{B}}$.\n\nTo test this hypothesis we focus on the low-temperature phase\\index{phase!low-temperature\/spin-glass}. We set the final temperature $T_\\mathrm{f}=0.7$ and we simulate $4$ different systems, $3$ of them with the temperature-varying\\index{temperature-varying protocol} protocol described in \\refsec{numerical_simulation_mpemba} and the other with the isothermal\\index{isothermal} protocol. We use the temperature-varying\\index{temperature-varying protocol} notation also for the isothermal\\index{isothermal} protocol in this case to stress that we set the time $t=0$ at a given coherence length\\index{coherence length} $\\xi$. The protocols are\n\\begin{itemize}\n\\item \\textbf{Preparation A:} $T=0.7, \\xi=6 \\to T=0.7$ (this is the isothermal\\index{isothermal} protocol).\n\\item \\textbf{Preparation B:} $T=0.9, \\xi=5 \\to T=0.7$.\n\\item \\textbf{Preparation C:} $T=0.9, \\xi=8 \\to T=0.7$.\n\\item \\textbf{Preparation D:} $T=0.9, \\xi=15 \\to T=0.7$.\n\\end{itemize}\n\nThe time at which we quench the four systems to $T=0.7$ (in the case of the isothermal\\index{isothermal} protocol simply corresponds to the time at which the system reaches $\\xi=6$) will be $t=0$. Then, we plot $\\tilde{e}(t)$ against time for the four preparations and we show the results in \\reffig{test_mpemba}.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{mpemba\/test_mpemba.pdf}\n\\caption[\\textbf{Mpemba effect in the spin-glass phase.}]{\\textbf{Mpemba\\index{Mpemba effect} effect in the spin-glass phase\\index{phase!low-temperature\/spin-glass}.} As in \\reffig{first_mpemba}, but all four initial preparations are now carried out in the spin-glass phase\\index{phase!low-temperature\/spin-glass} ($T<T_\\text{c}$). The preparation that cools faster is not the initially coldest one, but the one with the largest initial coherence length\\index{coherence length}. \\textbf{Inset:} A zoom of the \\emph{second} crossing point between the curves for preparations $(T=0.9,\\xi=8$; green curve) and $(T=0.7,\\xi=6$; yellow curve). This second crossing is not the Mpemba\\index{Mpemba effect} effect. Rather, the Mpemba\\index{Mpemba effect} effect arises at the \\emph{first} crossing at $t\\approx 5\\times 10^4$. The second crossing disappears if one plots parametrically $\\tilde{e}(t)$ as a function of $\\xi(t)$.}\n\\labfig{test_mpemba}\n\\end{figure}\n\n\nBefore initiating our discussion, we want to clarify the figure. The reader may wonder about the peculiar behavior of the curve belonging to the isothermal\\index{isothermal} protocol. This peculiar behavior is just an optical artifact. Indeed, the times at which we record the configuration\\index{configuration} of our system and measure the relevant quantities are exponential ($t=\\Floor{2^{n\/8}}$, with $n$ integer). Thus, for $T=0.7,\\xi=6$ the measuring times are quite far from each other and the first point roughly corresponds to $t\\approx 7 \\cdot 10^5$. However, between two consecutive measurements, the change of the energy-density\\index{energy!density} is very small and that explains the almost horizontal line.\n\nWe observe that preparations at initial temperature $T=0.9$ with coherence length\\index{coherence length} $\\xi>6$, namely preparation C and preparation D, cool faster than preparation A. Indeed, we observe the Mpemba\\index{Mpemba effect} effect for preparation D at time $t\\approx 10^3$, when it crosses the isothermal\\index{isothermal} preparation. For preparation C we observe the same effect at time $t \\approx 5 \\cdot 10^4$. Nevertheless, for preparation B we observe no Mpemba\\index{Mpemba effect} effect.\n\nIt is worthy to mention that a second crossing can be observed for higher times $t\\approx 5\\cdot 10^8$. This crossing does not correspond to the Mpemba\\index{Mpemba effect} effect. Actually, we will observe that this crossing disappears under an appropriate representation.\n\n\n\\subsection[The $\\tilde{e}-\\xi$ phase-diagram]{The \\boldmath $\\tilde{e}-\\xi$ phase-diagram}\\labsubsec{e_xi_phase_diagram_mpemba}\n\nAlthough we have identify the coherence length\\index{coherence length} as the hidden parameter controlling the Mpemba\\index{Mpemba effect} effect, we need to explore the relation between them to make our interpretation quantitative. Numerical and heuristic arguments \\cite{marinari:96,parisi:97,janus:09b} suggest\n\\begin{equation}\n\\tilde{e}(t) = \\tilde{e}_{\\infty}(T) + \\dfrac{e_1}{\\xi^{d_\\mathrm{L}}(t)} + \\cdots \\, , \\labeq{energy_coherence_length_relation}\n\\end{equation}\nwhere $d_{\\mathrm{L}}\\approx 2.5$~\\cite{boettcher:05,maiorano:18} is the lower critical dimension\\index{critical dimension!lower} at zero magnetic field, and the dots stand for scaling corrections, subdominant for large $\\xi$. This relation makes sense only for the \\gls{SG} phase\\index{phase!low-temperature\/spin-glass}~\\cite{parisi:88}.\n\n\n\n\\subsubsection{The isothermal protocols}\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{mpemba\/phase_diagram_isothermal.pdf}\n\\caption[\\textbf{\\boldmath Relationship between the energy density\\index{energy!density} $\\tilde{e}$ and the  $\\xi$ for isothermal\\index{isothermal} protocols.}]{\\textbf{\\boldmath Relationship between the energy density\\index{energy!density} $\\tilde{e}$ and the coherence length\\index{coherence length} $\\xi$ for isothermal\\index{isothermal} protocols.} As suggested by~\\refeq{energy_coherence_length_relation} for isothermal\\index{isothermal} relaxations\\index{relaxation} $\\tilde{e}$ is an essentially linear function of $1\/\\xi^{2.5}$, (at least for the plotted range of  $\\xi>4.8$). Furthermore, the dependence of the slope on temperature is marginal.}\n\\labfig{phase_diagram_isothermal}\n\\end{figure}\n\nWe test the relation defined by \\refeq{energy_coherence_length_relation} in \\reffig{phase_diagram_isothermal} by plotting the energy-density\\index{energy!density} $\\tilde{e}$ against $1\/\\xi^{d_{\\mathrm{L}}}$, arrows indicate the direction of the points for increasing $t$. First, we observe that the isothermal\\index{isothermal} protocols ($T=0.7$ and $T=0.9$) are (almost) straight lines in our representation. In addition, both isothermal\\index{isothermal} protocols are (almost) parallel to each other. Of course, we need to make these observations quantitative. To that purpose we fit our data to\n\\begin{equation}\n\\tilde{e}(t) = \\tilde{e}_{\\infty}(T) + \\dfrac{e_1}{\\xi^{d_\\mathrm{L}}(t)} + \\dfrac{e_2}{\\xi^{2d_\\mathrm{L}}(t)} \\, , \\labeq{quadratic_fit_mpemba}\n\\end{equation}\nwhich is just~\\refeq{energy_coherence_length_relation} with a simple quadratic correction in $1\/\\xi^{d_\\mathrm{L}}(t)$ that would be negligible for large $\\xi$. \n\nBecause this subdominant term (the quadratic one) becomes less important for the interesting limit (the large-$\\xi$ limit), we decide to establish an objective criterion to select the fitting range. We perform the fit for the range $[\\xi_{\\min},\\xi_{\\max}]$ by setting $\\xi_{\\max}$ to the maximum $\\xi$ simulated and by varying $\\xi_{\\min}$. We set $\\xi_{\\min}$ to be the lowest value of $\\xi$ that stabilizes the values of $\\tilde{e}_{\\infty}$, $e_1$ and $e_2$ (within the error bars\\index{error bars}), and, for the desired temperatures $T=0.7$ and $T=0.9$ we found $\\xi_{\\min}=6$. In addition, to describe the quality of the fit, we report the figure of merit $\\chi^2$\/d.o.f.\\index{degree of freedom} The results can be consulted in \\reftab{fit_results_mpemba}\n\n\\begin{table}[b!]\n\\begin{tabular}{ccccc}\n\\toprule\n\\toprule\n$T$ & $\\tilde{e}_{\\infty}$ & $e_1$ & $e_2$ & $\\chi^2\/$d.o.f.\\index{degree of freedom} \\\\\n\\hline\n0.7 & $-1.7708070(7)$ & $0.2217(3)$ & $1.17(2)$ & $20.9(1)\/119$ \\\\\n0.9 & $-1.7443347(6)$ & $0.2251(3)$ & $1.08(2)$ & $11.0(4)\/118$ \\\\\n\\bottomrule\n\\end{tabular}\n\\caption[\\textbf{Mpemba parameters of the quadratic fit.}]{\\textbf{Mpemba parameters of the quadratic fit.} We report the results of the fits to \\refeq{quadratic_fit_mpemba}. For each fit, the figure of merit $\\chi^2$\/d.o.f.\\index{degree of freedom} is also reported. Errors are computed by using the Jackknife\\index{Jackknife} method.}\n\\labtab{fit_results_mpemba}\n\\end{table}\n\nAs we said, both curves are almost parallel ($e^{T=0.7}_1\/e^{T=0.9}_1 \\approx 0.9849$) and are also straight lines, because the effect of the curvature is around 1\\% of the effect of the linear term $e_1\/\\xi^{d_{\\mathrm{L}}}$ for $\\xi \\approx8$ that is a typical coherence length\\index{coherence length} in our data.\n\n\\subsubsection{The temperature-varying protocols}\n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{mpemba\/phase_diagram.pdf}\n\\caption[\\textbf{\\boldmath Relationship between the energy density\\index{energy!density} $\\tilde{e}$ and the coherence length\\index{coherence length} $\\xi$ for temperature-varying\\index{temperature-varying protocol} protocols.}]{\\textbf{\\boldmath Relationship between the energy\\index{energy!density} density $\\tilde{e}$ and the coherence length\\index{coherence length} $\\xi$ for temperature-varying\\index{temperature-varying protocol} protocols.} Temperature-varying\\index{temperature-varying protocol} protocols are seen to be essentially vertical moves between the straight lines corresponding to isothermal\\index{isothermal} relaxations\\index{relaxation} at the initial and final temperatures. These vertical moves are very quick initial transients, in which (in moves to higher temperatures only), $\\xi$ slightly decreases and then increases again.}\n\\labfig{phase_diagram}\n\\end{figure}\n\nWe add the temperature-varying\\index{temperature-varying protocol} protocols to the analysis, see \\reffig{phase_diagram}. Those protocols where the temperature of the thermal reservoir decreases correspond to preparations B, C, and D in \\reffig{test_mpemba}. We can see that the time-scales of the energy-density\\index{energy!density} and the coherence length\\index{coherence length} are totally decoupled. The energy-density\\index{energy!density} $\\tilde{e}$ is a \\textit{fast variable} and, as a first approximation, when a quick temperature change takes place, $\\tilde{e}$ instantaneously takes the value of the energy-density\\index{energy!density} corresponding to its new thermal reservoir. However, the coherence length\\index{coherence length} $\\xi$ is a \\textit{slow variable} that basically remains unchanged when a temperature change takes place. The combination of both effects is translated into almost vertical movements between isothermal\\index{isothermal} protocols in \\reffig{phase_diagram}.\n\nIn this representation, the crossing points in \\reffig{test_mpemba} are not so evident. Now, the temperature-varying\\index{temperature-varying protocol} protocols experiment a very fast decrease of the energy-density\\index{energy!density}, while the isothermal\\index{isothermal} protocols need longer times (equivalently, longer coherence length\\index{coherence length}) to reach those values of the energy-density\\index{energy!density} and, therefore, the temperature-varying\\index{temperature-varying protocol} protocol ``cools'' faster. Of course, the previous analysis is a simplification, and measurable (still small) transient effects can be seen in \\reffig{phase_diagram}, however, it provides a very simple explanation of the Mpemba\\index{Mpemba effect} effect.\n\nThe curves corresponding to an increase in the temperature of the thermal reservoir are analyzed next.\n\n\\section{The Inverse Mpemba Effect} \\labsec{inverse_mpemba}\n\\begin{figure}[b!]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{mpemba\/inverse_me_ener.pdf}\n\\caption[\\textbf{A tiny inverse Mpemba effect.}]{\\textbf{A tiny inverse Mpemba\\index{Mpemba effect} effect.} Time evolution of the energy\\index{energy!density}, for the three different preparations (namely 1,2 and 3), compared with an isothermal\\index{isothermal} protocol with $T=1.4$ (top curve). In the three preparations, the initial temperature is in the spin-glass phase\\index{phase!low-temperature\/spin-glass}, and the final temperature is $T=1.4>T_\\mathrm{c}$. A very small Mpemba\\index{Mpemba effect} effect is found at the time pointed by the arrow, only when warming up samples\\index{sample} with similar starting energy\\index{energy!density}.}\n\\labfig{inverse_me_ener}\n\\end{figure}\n\nWe focus now on the inverse Mpemba\\index{Mpemba effect} effect protocol that was first suggested in~\\cite{lu:17,lasanta:17}. Now, the final temperature of the thermal reservoir is chosen to be higher than the starting one. We see in \\reffig{phase_diagram} that both curves corresponding to that protocol behave in a symmetrical way concerning to the classical protocol. In \\reffig{phase_diagram} all the temperatures are below the critical one, and the natural question is, does the inverse Mpemba\\index{Mpemba effect} effect survive for $T>\\ensuremath{T_\\mathrm{c}}\\xspace$? The question is not trivial because \\refeq{energy_coherence_length_relation} is not expected to hold for $T>\\ensuremath{T_\\mathrm{c}}\\xspace$.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{mpemba\/inverse_me_xi.pdf}\n\\caption[\\textbf{Coherence length: Undershooting and convergence to a master curve.}]{\\textbf{coherence length\\index{coherence length}: Undershooting and convergence to a master curve.} coherence length\\index{coherence length}s $\\xi$ of the experiments described in \\reffig{inverse_me_ener}. The time evolution of $\\xi$ tends to converge towards the curve corresponding to isothermal\\index{isothermal} protocol with $T=1.4$ (bottom curve), giving rise to an undershoot of $\\xi$ when its initial value is higher than the equilibrium $\\xi$ at $T=1.4$.}\n\\labfig{inverse_me_xi}\n\\end{figure}\n\nTo answer this question, we use our temperature-varying\\index{temperature-varying protocol} protocol, but this time the final temperature will be at $T>\\ensuremath{T_\\mathrm{c}}\\xspace$. We propose three starting conditions:\n\\begin{itemize}\n\\item \\textbf{Preparation 1:} $T=0.7, \\xi=2.5 \\to T=1.4$.\n\\item \\textbf{Preparation 2:} $T=0.7, \\xi=11.7 \\to T=1.4$.\n\\item \\textbf{Preparation 3:} $T=0.8, \\xi=15.8 \\to T=1.4$.\n\\end{itemize}\nThe reader should be aware that, although for $T<\\ensuremath{T_\\mathrm{c}}\\xspace$ the coherence length\\index{coherence length} grows without bonds\\footnote{In an infinite system.}, this is not the case for $T>\\ensuremath{T_\\mathrm{c}}\\xspace$. Specifically, for $T=1.4$ the asymptotic equilibrium value for the coherence length\\index{coherence length} is $\\xi=8.95(5)$. We also compare these temperature-varying\\index{temperature-varying protocol} protocols with the isothermal\\index{isothermal} protocol at $T=1.4$.\n\nIf we study the relaxation\\index{relaxation} of the energy-density\\index{energy!density} we can observe a small Mpemba\\index{Mpemba effect} effect between protocols 2 and 3 for $t\\approx 20$ (see \\reffig{inverse_me_ener}). However, between protocols 1 and 3 or 1 and 2, the Mpemba\\index{Mpemba effect} effect is clearly absent.\n\nWe can study also the relaxation\\index{relaxation} of the coherence length\\index{coherence length} $\\xi$, see \\reffig{inverse_me_xi}. In the paramagnetic phase\\index{phase!high-temperature\/paramagnetic}, the growth of the magnetic domains\\index{magnetic domain} does not follow~\\refeq{xi_powerlaw} as it becomes evident in the figure. In addition, we observe that all the protocols tend to the isothermal\\index{isothermal} one very fast (for $t \\leq 10^5$).\n\nWe can see here again that both time scales, the $\\xi$ one and the $\\tilde{e}$ one, are clearly decoupled. In \\reffig{inverse_me_ener} and \\reffig{inverse_me_xi} we can see that both quantities tend to their equilibrium values at very different time scales. Furthermore, if we focus on protocol 2 we can see that the undershoot present for the coherence length\\index{coherence length} does not correspond to a similar behavior for the energy-density\\index{energy!density}. Although for $T>\\ensuremath{T_\\mathrm{c}}\\xspace$ the Mpemba\\index{Mpemba effect} effect is strongly suppressed, this decoupling between both time scales seems to be necessary for the Mpemba\\index{Mpemba effect} effect to take place.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\chapter[Temperature Chaos phenomenon in off-equilibrium Spin Glasses]{Temperature Chaos phenomenon in \\\\ off-equilibrium spin glasses} \\labch{out-eq_chaos}\n\n\\gls{TC} phenomenon is described in terms of equilibrium configurations\\index{configuration}, and therefore, all the previous work studying it is focused on equilibrated spin glasses. However, experiments are almost always carried out in non-equilibrium conditions. Moreover, the concept of \\gls{TC}\\index{temperature chaos} is not totally alien to the non-equilibrium regime in \\gls{SG}. Actually, it has been related to the memory\\index{memory effects} and rejuvenation\\index{rejuvenation} effects~\\cite{komori:00,berthier:02,picco:01,takayama:02,maiorano:05,jimenez:05}.\n\nHowever, at this point, the relation of \\gls{TC}\\index{temperature chaos} with memory\\index{memory effects} and rejuvenation\\index{rejuvenation} effects is far from clear and no quantitative description of \\gls{TC}\\index{temperature chaos} under off-equilibrium conditions has been provided. In this chapter, we present a numerical work in off-equilibrium conditions that try to be the first step to fill that gap. All the simulations have been performed in the dedicated computer Janus\\index{Janus} II~\\cite{janus:14} and the high-accuracy achieved could not be possible without its computational power.\n\nIn the course of the chapter, we will explain why traditional approaches to study \\gls{TC}\\index{temperature chaos} do not work and we need, again, a rare-event analysis in order to fully understand the phenomenon. Moreover, the statics-dynamics equivalence\\index{statics-dynamics equivalence}~\\cite{barrat:01,janus:08b,janus:10b,janus:17} shows us the path to quantitatively study an equilibrium phenomenon in a non-equilibrium system.\n\nThe results of this work show us how, again, the coherence length\\index{coherence length} $\\xi$ rules the off-equilibrium phenomena in \\gls{SG}s. In particular, a crossover behavior between a weak chaos regime and a strong chaos regime is found when $\\xi$ grows. The characteristic length scale where this occurs, $\\xi^*$, is related to its equilibrium counterpart, the chaotic length $\\ell_c$ defined in~\\refeq{def_chaotic_length}.\n\nIn \\refsec{numerical_simulations_tc}, we give all the information about the performed numerical simulations. We explore the first \\textit{naive} attempt to characterize \\gls{TC}\\index{temperature chaos} in off-equilibrium dynamics in~\\refsec{average_killed_chaos_signal}. The computed observables to perform our rare-event analysis are introduced in~\\refsec{observables_tc}. The rare-event analysis can be found in~\\refsec{char} and its results in~\\refsec{results}. In~\\refsec{scaling_fixed_r} we explore the scaling behavior of our chaos-related quantities and, in \\refsec{T-changes}, we focus on temperature changing protocols in order to make contact with the cumulative-aging\\index{aging!cumulative} controversy (described in \\refsec{memory_rejuvenation_introduction_chaos}) and lay the groundwork to future numerical works trying to relate simulations and experiments.\n\nAll the results provided in this chapter were originally published (in a reduced form) in~\\cite{janus:21}.\n\n\\section{Numerical parameters of the simulation} \\labsec{numerical_simulations_tc}\nIn this work, we simulate the \\gls{EA}\\index{Edwards-Anderson!model} model in three-dimensional spin glasses (\\refsubsec{3D_EA_model}) for several temperatures $T$ in a lattice of linear size $L=160$, which is aimed to represent a system of infinite size. This assumption is sound, provided that $L\\gg\\xi$ (see~\\reftab{xi_max}). Note that this condition limits the maximum time at which we can safely ignore finite-size effects\\index{finite-size effects}. The temperature remains constant through the whole simulation, with the only exception of the runs reported and discussed in~\\refsec{T-changes}.\n\nWe shall perform direct quenches from configurations\\index{configuration} of spins randomly initialized (which corresponds to infinite temperature) to the working temperature $T<\\ensuremath{T_\\mathrm{c}}\\xspace$, where the system is left to relax for a time $\\ensuremath{t_\\mathrm{w}}\\xspace$. This relaxation\\index{relaxation} corresponds with the (very slow) growth of glassy magnetic domains\\index{magnetic domain} of size $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$.\n\nWe compute a total of $\\ensuremath{N_{\\text{S}}}\\xspace = 16$ different samples\\index{sample}. For each sample\\index{sample} we shall consider $\\ensuremath{N_{\\text{Rep}}}\\xspace=512$ \\emph{replicas}\\index{replica} \\footnote{It has been noted in~\\cite{janus:18} (see also \\refsec{Nr_aging}) that, for global observables (see~\\refsubsec{observables-globales}), it is advantageous to have $\\ensuremath{N_{\\text{Rep}}}\\xspace\\gg \\ensuremath{N_{\\text{S}}}\\xspace$. However, working with $\\ensuremath{N_{\\text{Rep}}}\\xspace\\gg \\ensuremath{N_{\\text{S}}}\\xspace$ is not only a matter of numerical convenience for us. In fact, the local observables in~\\refsubsec{observables-locales} are well defined only in the limit of $\\ensuremath{N_{\\text{Rep}}}\\xspace\\to\\infty$.}.\n\nThe simulation has been performed in the dedicated \\gls{FPGA}-based\\index{FPGA} computer Janus\\index{Janus} II~\\cite{janus:14} by using Metropolis dynamics. \n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{l c r c c c r}\n\\toprule\n\\toprule\n$T$ &  \\hspace{1cm} & $\\ensuremath{t_\\mathrm{w}}\\xspace$ (MCs) & \\hspace{1cm} & $\\log_2(\\ensuremath{t_\\mathrm{w}}\\xspace)$ & \\hspace{1cm} & $\\xi_{\\max}(\\ensuremath{t_\\mathrm{w}}\\xspace)$ \\\\\n\\toprule\n$0.625$ & \\hspace{1cm} & 42669909513 & \\hspace{1cm} & 35.3 & \\hspace{1cm} & 9.52(1) \\\\\n\\hline\n$0.7$ & \\hspace{1cm} & 48592007999 & \\hspace{1cm} & 35.5 & \\hspace{1cm} & 12.02(2) \\\\\n\\hline\n$0.8$ & \\hspace{1cm} & 34359738368 & \\hspace{1cm} & 35\\phantom{.5} & \\hspace{1cm} & 15.84(5) \\\\\n\\hline \n$0.9$ & \\hspace{1cm} & 17179869184 & \\hspace{1cm} & 34\\phantom{.5} & \\hspace{1cm} & 20.34(6)  \\\\\n\\hline\n$1.0$ & \\hspace{1cm} & 4294967296 & \\hspace{1cm} & 32\\phantom{.5} & \\hspace{1cm} & 24.4(1)\\phantom{0} \\\\\n\\bottomrule\n\\end{tabular}\n\\caption[\\textbf{Parameters of the simulations.}]{\\textbf{Parameters of the simulations.} Maximum value of $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ simulated for each temperature. The central columns show the $\\ensuremath{t_\\mathrm{w}}\\xspace$ corresponding value for the computed $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$.}\n\\labtab{xi_max}\n\\end{center}\n\n\\end{table}\n\n\n\\section{Taking spatial averages kills the chaotic signal} \\labsec{average_killed_chaos_signal}\n\nThe first naive attempt to study \\gls{TC}\\index{temperature chaos} phenomenon in out-equilibrium \\gls{SG}s consisted of studying global quantities affecting the whole system. However, in close analogy with equilibrium studies~\\cite{fernandez:13}, we find that \\gls{TC}\\index{temperature chaos} is extremely weak when the full system is considered on average, see \\reffig{xi_T1T2}, although the effect increases when the coherence length\\index{coherence length} $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ grows (just as \\gls{TC}\\index{temperature chaos} becomes more visible in equilibrium when the system size increases).\n\nIn view of the above negative result, we have followed Ref.~\\cite{fernandez:13} and performed a rare-event analysis that provides a satisfactory quantification of the \\gls{TC}\\index{temperature chaos} phenomenon. The\nrationale for this approach is the statics-dynamics equivalence\\index{statics-dynamics equivalence}~\\cite{barrat:01,janus:08b,janus:10b,janus:17}: we expect to learn about the non-equilibrium dynamics of a spin glass (of infinite size), with a finite coherence length\\index{coherence length} $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$, by studying small samples\\index{sample} of size $L\\sim\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ which can be equilibrated. \n\n\\begin{figure}[t!] \n\\centering \n\\includegraphics[width=0.8\\textwidth]{off-eq_chaos\/xi_T1T2.pdf}\n\\caption[\\textbf{Non-equilibrium \\gls{TC}\\index{temperature chaos} is weak when averaging over the whole system.}]{\\textbf{Non-equilibrium \\gls{TC}\\index{temperature chaos} is weak when averaging over the whole system.} We compare typical spin configurations\\index{configuration} at temperature $T_1$ and time $t_{\\mathrm{w},1}$ with configurations\\index{configuration} at $T_2$ and time $t_{\\mathrm{w},2}$. The comparison is carried through a global estimator of the coherence length\\index{coherence length} of their overlap\\index{overlap} $\\xi^{T_1T_2}_{1,2}$, see \\refeq{def_correlation_length_2T} (physically, $\\xi^{T_1T_2}_{1,2}$ is the maximum length scale at which configurations\\index{configuration} at temperatures $T_1$ and $T_2$ still look similar). The two times $t_{\\mathrm{w},1}$ and $t_{\\mathrm{w},2}$ are chosen in such a way that the configurations\\index{configuration} at both temperatures have glassy-domains\\index{magnetic domain} of the same size, namely $\\xi_{1,2}(t_{\\mathrm{w},1},T_1)=\\xi_{1,2}(t_{\\mathrm{w},2},T_2)=\\xi$. The figure shows the ratio $\\xi^{T_1T_2}_{1,2}\/\\xi$ as a function of $\\xi$ for two pairs of temperatures $(T_1,T_2)$, recall that $\\ensuremath{T_\\mathrm{c}}\\xspace\\approx 1.1$, see~\\refsubsec{3D_EA_model}. Under the hypothesis of fully developed \\gls{TC}\\index{temperature chaos}, we would expect $\\xi^{T_1T_2}_{1,2}$ to be negligible as compared to $\\xi$. Instead, our data show only a small decrease of $\\xi^{T_1T_2}_{1,2}\/\\xi$ upon growing $\\xi$ (the larger the difference $T_2-T_1$ the more pronounced the decrease).}\n \\labfig{xi_T1T2}\n\\end{figure}\n\nIn our case, we shall be considering spatial regions (spheres) of linear size $\\sim \\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$, chosen randomly within a very large spin glass. Just as found with the small samples\\index{sample} in equilibrium~\\cite{fernandez:13}, we expect that a small fraction of our spheres will display strong \\gls{TC}\\index{temperature chaos}. The important question will be how this rare-event phenomenon evolves as $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ grows. In fact, we expect that our description of non-equilibrium \\gls{TC}\\index{temperature chaos} will allow us to perform sensible extrapolations to values of $\\xi$ of experimental interest (for comparison, a typical experimental value is $\\xi\\sim 100$ lattice spacings, while in our simulations $\\xi\\sim 10$ lattice spacings).\n\n\n\\section{Observables} \\labsec{observables_tc}\nIn this section, we briefly introduce the quantities that will help us to characterize \\gls{TC}\\index{temperature chaos} in off-equilibrium systems. The global observables (\\refsubsec{observables-globales}) will be fundamental in order to characterize the relevant length (equivalently time) scales of the system. On the contrary, local observables (\\refsubsec{observables-locales}) will be necessary in order to perform a rare-event analysis of the \\gls{TC}\\index{temperature chaos}.\n\n\\subsection{Global observables}\\labsubsec{observables-globales}\nThe out-equilibrium time evolution is usually characterized by the growth of the coherence length\\index{coherence length} $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ at temperature $T$, see~\\refsubsec{observables_introduction}. In order to compute it, two basic observables are needed: the overlap\\index{overlap!field} field and the four-point\\index{correlation function!four point} spatial correlation function [see~\\refeq{def_overlap} and~\\refeq{def_C4}]. We repeat here the definitions for the reader's convenience\n\\begin{equation}\nq^{\\sigma,\\tau}(x,\\ensuremath{t_\\mathrm{w}}\\xspace) = s_{x}^\\sigma(\\ensuremath{t_\\mathrm{w}}\\xspace) s^\\tau_{x}(\\ensuremath{t_\\mathrm{w}}\\xspace) \\, ,\n\\end{equation}\n\\begin{equation} \nC_4(T,r,\\ensuremath{t_\\mathrm{w}}\\xspace) = \\overline{\\langle q^{\\sigma,\\tau}(x,\\ensuremath{t_\\mathrm{w}}\\xspace) q^{\\sigma,\\tau}(x+r,\\ensuremath{t_\\mathrm{w}}\\xspace)\\rangle_T} \\, . \\labeq{def_corr_func}\n\\end{equation} \nIn the previous definitions $ s_{x}^\\sigma(\\ensuremath{t_\\mathrm{w}}\\xspace)$ is the spin of the replica\\index{replica} $\\sigma$ in the lattice position $x$ at time $\\ensuremath{t_\\mathrm{w}}\\xspace$, $\\langle\\dots\\rangle_T$ is the average over thermal noise at temperature $T$, and $\\overline{(\\cdots)}$ is the average over the disorder\\index{disorder!average}. Of course, the two replica\\index{replica} indices $\\sigma$ and $\\tau$ should be different.\n\nThe correlation function in~\\refeq{def_corr_func} can be extended for a pair of temperatures $T_1$ and $T_2$ in the following\nway\n\\begin{equation}\nC_4^{T_1T_2}(T_1,T_2,t_{\\mathrm{w}1},t_{\\mathrm{w}2},r) = \\overline{\\langle q^{\\sigma(T_1),\\tau(T_2)} (x,t_{\\mathrm{w}1},t_{\\mathrm{w}2}) q^{\\sigma(T_1),\\tau(T_2)}(x+r,t_{\\mathrm{w}1},t_{\\mathrm{w}2})\\rangle_{T}} \\, , \\labeq{def_corr_func_2T}\n\\end{equation}\nwhere now the thermal averages are taken at temperature $T_1$ for the replica\\index{replica} $\\sigma$, and at temperature $T_2$ for the replica\\index{replica} $\\tau$. From the four-point\\index{correlation function!four point} correlation function we can compute the coherence length\\index{coherence length} as have been explained in~\\refsubsec{observables_introduction} and \\refsec{finite_size_effects}.\n\nOf course, the coherence length\\index{coherence length} can be straightforwardly extended to a pair of temperatures $T_1$ and $T_2$ by using  $C_4^{T_1T_2}$ instead of $C_4$:\n\\begin{equation} \nI^{T_1T_2}_k( t_{\\mathrm{w}1},t_{\\mathrm{w}2}) = \\int_{0}^{\\infty}r^k\\,C^{T_1T_2}_4(r,t_{\\mathrm{w}1},t_{\\mathrm{w}2})\\,\\mathrm{d} r \\, , \\labeq{def_integral_2T}\n\\end{equation}\nand\n\\begin{equation}\n\\xi^{T_1T_2}_{k,k+1}(t_{\\mathrm{w}1},t_{\\mathrm{w}2}) = \\dfrac{I^{T_1T_2}_{k+1}(t_{\\mathrm{w}1},t_{\\mathrm{w}2})}{I^{T_1T_2}_k(t_{\\mathrm{w}1},t_{\\mathrm{w}2})} \\, . \\labeq{def_correlation_length_2T}\n\\end{equation}\nAs a rule, we shall fix the two times $t_{\\mathrm{w}1}$ and $t_{\\mathrm{w}2}$ through the condition\\footnote{Because our $\\ensuremath{t_\\mathrm{w}}\\xspace$ are in a discrete grid, we solve~\\refeq{el_reloj_doble} for the \\emph{global} overlaps\\index{overlap} defined in~\\refsubsec{observables-globales} through a (bi)linear interpolation.}:\n\\begin{equation}\n\\xi(t_{\\mathrm{w}1},T_1)=\\xi(t_{\\mathrm{w}2},T_2)=\\xi\\,, \\labeq{el_reloj_doble}\n\\end{equation}\nthat ensures that we are comparing spin-configurations\\index{configuration} which are ordered on the same length scale.\n\n\\subsection{Local observables}\\labsubsec{observables-locales}\n\nIn order to explore the heterogeneity of the system, we construct here local observables that will allow us to generalize the rare-event analysis in~\\cite{fernandez:13}. Specifically, we shall be studying the properties of spherical regions.\n\nWe start by choosing $N_{\\mathrm{sph}}=8000$ centers for the spheres, on each sample\\index{sample}. The sphere centers are chosen randomly, with uniform probability, on the dual lattice~\\footnote{The dual lattice of a cubic lattice with \\gls{PBC}\\index{boundary conditions!periodic} is another cubic lattice of the same size, and with \\gls{PBC}\\index{boundary conditions!periodic} as well. The nodes of the dual lattice are the centers of the elementary cells of the original lattice.}. The radius of the spheres is varied, but their centers are held fixed. Let $B_{s,r}$ be the $s$-th ball of radius $r$. \n\nSimilarly as in the previous chapter (\\refch{equilibrium_chaos}), our basic observable will be the overlap\\index{overlap} between temperatures $T_1$ and $T_2$\n\\begin{equation}\nq_{T_1,T_2}^{s,r}(\\xi) = \\dfrac{1}{N_r} \\sum_{x\\in B_{s,r}} s_{x}^{\\sigma,T_1}(t_{\\mathrm{w}1}) s_{x}^{\\tau,T_2}(t_{\\mathrm{w}2}) \\>\\> , \\labeq{sphere_overlap}\n\\end{equation}\nwhere $N_r$ is the number of spins within the ball of radius $r$, and the two times $t_{\\mathrm{w}1}$ and $t_{\\mathrm{w}2}$ are chosen according to~\\refeq{el_reloj_doble}\\footnote{Because the local overlaps\\index{overlap} in~\\refeq{sphere_overlap} have much larger fluctuations than the global overlaps\\index{overlap} in~\\refsubsec{observables-globales}, in this case we solve~\\refeq{el_reloj_doble} in a cruder way. We just select the value of $t_{\\mathrm{w}1}$ that yields the $\\xi(t_{\\mathrm{w}1},T_1)$ nearest to our target $\\xi$ value. The same procedure is followed with $t_{\\mathrm{w}2}$.}.\nNext, again as in the previous chapter, we introduce the so called \\textit{chaotic parameter}~\\cite{ritort:94,ney-nifle:97,fernandez:13,billoire:14} which now is restricted to the balls $B_{s,r}$\n\\begin{equation} \nX^{s,r}_{T_1,T_2}(\\xi) = \\dfrac{\\langle [q_{T_1,T_2}^{s,r}(\\xi)]^2\\rangle_T}{\\sqrt{\\langle[q_{T_1,T_1}^{s,r}(\\xi)]^2\\rangle_T \\,\\langle[q_{T_2,T_2}^{s,r}(\\xi)]^2\\rangle_T}} \\, . \\labeq{def_chaotic_parameter}\n\\end{equation} \nThe extreme values of the chaotic parameter, just in close analogy with the equilibrium case, have a very clear interpretation: $X^{s,r}_{T_1,T_2}=1$ corresponds to a situation in which spin configurations\\index{configuration} in the ball $B_{s,r}$, at temperatures $T_1$ and $T_2$, are completely indistinguishable (absence of chaos) while $X^{s,r}_{T_1,T_2}=0$ corresponds to completely different configurations\\index{configuration} (which means strong \\gls{TC}\\index{temperature chaos}). A representative example of our results is shown in \\reffig{hetereogeneity_chaos}.\n\n\\begin{figure}[h!] \n\\centering \n\\includegraphics[width=0.48\\textwidth]{off-eq_chaos\/paraview.png}\n\\caption[\\textbf{Dynamic temperature chaos is spatially heterogeneous.}]{\\textbf{Dynamic temperature chaos is spatially heterogeneous.} The 8000 randomly chosen spheres in a sample\\index{sample} of size $L=160$ are depicted with a color code depending on $1-X$ [$X$ is the chaotic parameter, \\refeq{def_chaotic_parameter}, as computed for spheres of radius $r=12$, $\\xi=12$ and temperatures $T_1=0.7$ and $T_2=1.0$]. For visualization purposes, spheres are represented with a radius $12(1-X)$, so that only fully chaotic spheres (i.e., $X=0$) have their real size.}\n\\labfig{hetereogeneity_chaos}\n\\end{figure}\n\nWe shall focus our attention on the distribution function\n\\begin{equation}\nF(X,T_1,T_2,\\xi,r)=\\text{Probability}[X^{s,r}_{T_1,T_2}(\\xi)<X] \\, ,\\labeq{F-def}\n\\end{equation}\nand its inverse $X(F,T_1,T_2,\\xi,r)$.\n\nThe alert reader will note that the computation of $X^{s,r}_{T_1,T_2}(\\xi)$ requires the \\emph{exact} computation of thermal expectation values, which would require an infinite number of replicas\\index{replica}.  Fortunately, we have been able of extrapolating $X(F,T_1,T_2,\\xi)$ to the limit $\\ensuremath{N_{\\text{Rep}}}\\xspace\\to\\infty$. Details about this extrapolation are provided in~\\refsubsec{extrapolation}.\n\nA final note: as explained in~\\refsec{cambio_de_r}, we have found it advantageous to trade the sphere radius $r$ by $N_r^{1\/3}$. Of course, $N_r$ equals $4\\pi r^3\/3$ for large $r$, but $N_r^{1\/3}$ provides smoother interpolations at small $r$.\n\n\\section{The temperature-chaos distribution functions} \\labsec{char}\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=\\textwidth]{off-eq_chaos\/distribution_function_xi.pdf}\n\\caption[\\textbf{Temperature chaos increases with \\boldmath $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$.}]{\\textbf{Temperature chaos increases with \\boldmath $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$.} The figure shows the distribution function $F(X,T_1,T_2,\\xi,r)$, see~\\refeq{F-def}, for $T_1=0.625$ and $T_2=0.9$, for spheres of radius $r=4$ (left) and $r=8$ (right), as computed for various values of $\\xi$. The distributions have been extrapolated to $\\ensuremath{N_{\\text{Rep}}}\\xspace=\\infty$. Error bars\\index{error bars} are horizontal, because we have actually extrapolated the inverse function $X(F,T_1,T_2,\\xi,r)$ (the extrapolation is not always safe for $F>0.3$, see~\\refsubsec{extrapolation}; the same caveat applies to all the distribution functions shown in~\\refsec{char} and~\\refsec{results}). Most of the spheres have a chaotic parameter very close to $X=1$.}\n\\labfig{distribution_function_xi}\n\\end{figure}\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=\\textwidth]{off-eq_chaos\/distribution_function_r.pdf}\n\\caption[\\textbf{Dependency of Temperature Chaos on the size of the observation region.}]{\\textbf{Dependency of Temperature Chaos on the size of the observation region.} The figure shows the distribution function $F(X,T_1,T_2,\\xi,r)$ for $T_1=0.625$ and $T_2=0.9$, for coherence length\\index{coherence length} $\\xi=5$ (left) and $\\xi=9$ (right), as computed for spheres of various radius $r$. If we focus on some low probability ($F=0.01$, for instance), we find that there is an optimal size for the observation of chaos (in the sense of a smallest chaotic parameter $X$).}\n \\labfig{distribution_function_r}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=\\textwidth]{off-eq_chaos\/plot_Xvsr_examples.pdf}\n\\caption[\\textbf{The complementary chaotic parameter $\\mathbf{1-X(F,T_1,T_2,\\xi,r)}$ for fixed $F$ as a function of $\\mathbf{r}$.}]{\\textbf{The complementary chaotic parameter $\\mathbf{1-X(F,T_1,T_2,\\xi,r)}$ for fixed $F$ as a function of $\\mathbf{r}$.} The difference $1-X(F,T_1,T_2,\\xi,r)$ [recall that $X(F,T_1,T_2,\\xi,r)$ is the inverse of the distribution function, see~\\refeq{F-def}] as a function of the cubic root of the number of points in the spheres $N_r^{1\/3}$, as computed for different values of $F$, $T_1$, $T_2$ and $\\xi$.  Our rationale for choosing $N_r^{1\/3}$ as independent variable, rather than the radius of the spheres $r$, is explained in~\\refsec{cambio_de_r}. In this representation, the size of the spheres which are optimal for the observation of chaos (for given parameters $F$, $T_1$, $T_2$ and $\\xi$) appears as the maximum of these curves. Continuous lines are fits to~\\refeq{functional_form}.}\n\\labfig{Xvsr_examples}\n\\end{figure}\n\nSome examples of the distribution functions $F(X,T_1,T_2,\\xi,r)$ can be found in~\\reffig{distribution_function_xi} and~\\reffig{distribution_function_r}, for typical (fixed) values of $T_1$ and $T_2$. Although most spheres are clearly non-chaotic ($X>0.9$), the situation is far more interesting for low probabilities (say $F=0.01$). For the sake of simplicity, consider first spheres of a fixed size (\\reffig{distribution_function_xi}). For small $F$, we find that $X$ decreases significantly (and monotonically) upon growing $\\xi$. The situation is more complex if we consider spheres of different sizes, for given $F$ and $\\xi$.  As \\reffig{distribution_function_r} shows, when the size of the spheres grows the chaotic parameter is non-monotonic.\n\nThe situation clarifies when we fix both the probability $F$ and the coherence length\\index{coherence length} $\\xi$, see~\\reffig{Xvsr_examples}. Rather than the chaotic parameter, let us consider the difference $1-X$ (which grows when \\gls{TC}\\index{temperature chaos} becomes stronger). We find that $1-X$ peaks for one size of the spheres which indicates the optimal length scale for the study of \\gls{TC}\\index{temperature chaos} (however, see~\\reffig{Xvsr_examples}, this peak is asymmetric and becomes broader when $\\xi$ increases). Our main analysis in~\\refsec{results} will correspond to the scaling with $\\xi$ of these peaks.\n\nLet us remark that, at least close to a maximum, any smooth curve is characterized by the position, height and width of the peak. In order to meaningfully compute these three parameters from our data (see e.g.~\\reffig{Xvsr_examples}), we fit $1-X$ to\n\\begin{equation}\nf(z) = \\dfrac{az^b}{1+cz^d} \\quad ,\\quad z=N_r^{1\/3}\\,, \\labeq{functional_form}\n\\end{equation}\n($a$, $b$, $c$, and $d$ are the parameters of the fit). We extract the position, width and height from the fitted function $f(z)$. In order to compute errors in (say) the peak position $N_{r,\\max}^{1\/3}$ we use a Jackknife\\index{Jackknife} method (see \\refsec{estimating_errorbars} for further details): we perform a separated fit for each Jackknife\\index{Jackknife} block, extract $N_{r,\\max}^{1\/3}$ from the fit for that block, and compute errors from the block fluctuations. Of course, Jackknife\\index{Jackknife} blocks are formed from our $\\ensuremath{N_{\\text{S}}}\\xspace=16$ samples\\index{sample}. Let us stress that~\\refeq{functional_form} is meant to be only a convenient way of characterizing the peak, without any deep meaning attached to it.\n\nHowever, the reader may question whether or not the local peak description (i.e. position, height, and width) is sensible for the full curve. We provide some positive evidence in this respect in~\\refsubsec{taylor}.\n\n\\subsection{Global versus local description of the peaks}\\labsubsec{taylor} \n\\begin{figure}[b!]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{off-eq_chaos\/taylor_0710_algunos.pdf}\\\\\n\n\\includegraphics[width=0.8\\textwidth]{off-eq_chaos\/taylor_examples.pdf}\n\\caption[\\textbf{Universality in $1-X$ extends beyond the trivial Taylor's Universality.}]{\\textbf{Universality in $1-X$ extends beyond the trivial Taylor's Universality.} The upper part shows $1-X$ in units of its peak value, for the temperatures $T_1=0.7$, $T_2=1.0$ and $F=0.01$. Taylor's theorem implies that, using the independent variable $y$ [see~\\refeq{Taylor-universality}], the different curves should coincide close to $y=0$. However, we see that the coincidence holds beyond the quadratic approximation (as evinced by the strong asymmetry of the master curve). The lower panel shows the same set of temperatures $T_1$ and $T_2$ and probabilities $F$ shown in~\\reffig{Xvsr_examples} (we have added data for several coherence length\\index{coherence length}). Mixing different values of $F$, $T_1$ and $T_2$ leads to significant discrepancies for large values of $|y|$. Nevertheless, the collapse of the curves is still present in the range $y \\in (-0.3,0.5)$ where the asymmetry is not negligible.}\n\\labfig{taylor}\n\\end{figure}\n\nConsider any smooth, positive function $H(z)$, with a local maximum at $z=z_{\\max}$. Close to this peak, Taylor's theorem implies some (trivial) Universality\n\\begin{equation}\n\\frac{H(z)}{H(z_{\\max})}=1-\\frac{1}{2} y^2+{\\cal O}(y^3)\\,,\\text{ where } \\,\ny=\\sqrt{\\frac{|H''(z_{\\max})|}{H(z_{\\max})}}(z-z_{\\max})\\,. \\labeq{Taylor-universality}\n\\end{equation}\nNote that, in the language of the previous paragraph, the peak position is $z_{\\text{max}}$, its heigth is $H(z_{\\max})$ and its (inverse) width is $\\sqrt{|H''(z_{\\max})|\/H(z_{\\max})}$. Of course, in principle, there is no reason for~\\refeq{Taylor-universality} to be accurate away from the peak. However, \\refeq{Taylor-universality} suggests yet another representation for our $1-X$ curves, see~\\reffig{taylor}. We note that, in this new representation, the $1-X$ curves are invariant under changes of coherence length\\index{coherence length} $\\xi$ (\\reffig{taylor} upper panel). However, when considering changes in the temperatures $T_1$ and $T_2$ and the probability $F$, the curves mildly differ away from the peak (see~\\reffig{taylor} lower panel). This (approximate) independence in $(T_1,T_2,F,\\xi)$ is a fortunate fact because the complexity of the problem gets reduced to the study of the scaling with $\\xi$ of the three peak parameters while keeping constant $(T_1,T_2,F)$.\n\n\n\n\n\\section{The off-equilibrium characterization of Temperature Chaos}\\labsec{results}\nIn this section we present the scaling of the peak position (\\refsubsec{peaks-position}), the peak height (\\refsubsec{peaks-height}) and the peak (inverse) width (\\refsubsec{peaks-width}) with the coherence length\\index{coherence length} $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$.\n\nDue to the difficulty of characterizing peaks which exhibit weak \\gls{TC}\\index{temperature chaos}, in the following analysis we exclude the data corresponding to the pair of temperatures ($T_1=0.625,T_2=0.7$) at the probability level $F=0.01$ (see~\\refsec{peak_characterization} for further details).\n\n\\subsection{The peak position}\\labsubsec{peaks-position}\n\\begin{figure}[b!]\n  \\centering\n  \\includegraphics[width=\\textwidth]{off-eq_chaos\/Nr_xi.pdf}\n  \\caption[\\textbf{Peak position increases with $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$.}]{\\textbf{Peak position increases with $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$.} Position of the peak against $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ is plotted for all the simulated pair of temperatures $T_1$ and $T_2$ at different probability levels, $F=0.001$ (left panel) and $F=0.01$ (right panel).}\n \\labfig{Nr_xi}\n\\end{figure}\n\nLet us recall that the peak position indicates the most convenient length-scale for studying \\gls{TC}\\index{temperature chaos} (for a given coherence length\\index{coherence length} $\\xi$, probability $F$ and temperatures $T_1$ and $T_2$). Dimensional analysis suggests the linear fit as the natural ansatz to study the scaling of the peak position $N_{r,\\max}^{1\/3}$ with the coherence length\\index{coherence length} $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ (indeed, both quantities are lengths):\n\\begin{equation}\nN_{r,\\max}^{1\/3} = a \\> \\xi(\\ensuremath{t_\\mathrm{w}}\\xspace) + b \\, \\, . \\labeq{Nr_xi}\n\\end{equation}\n\\reffig{Nr_xi} and~\\reftab{parametros_Nmax} show the fits to~\\refeq{Nr_xi}. In all cases, values of the parameter $b$ are compatible with $0$ (at the two-$\\sigma$ level). In addition, the proportional parameter $a$ shows a monotone increasing behavior with $T_2-T_1$ and with the probability $F$. Hence, our naive expectation $N_{r,\\max}^{1\/3} \\propto \\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ is confirmed.\n\n\\begin{table}[t!]\n\\begin{center}\n\\begin{tabular}{c c c c c c c c c c c}\n\\toprule\n\\toprule\n$F$ & \\hspace{1cm} & $T_1$ &  \\hspace{1cm} & $T_2$ & \\hspace{1cm} & $a$ & \\hspace{1cm} &$b$ & \\hspace{1cm} & $\\chi^2\/\\mathrm{d.o.f.}$\\index{degree of freedom} \\\\\n\\hline \\hline\n0.001 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.7 & \\hspace{1cm} & 0.60(12) & \\hspace{1cm} & 0.9(9) & \\hspace{1cm} & 22.12\/19 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.8 & \\hspace{1cm} & 0.81(7) & \\hspace{1cm} & 0.0(5) & \\hspace{1cm} & 11.52\/19 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.9 & \\hspace{1cm} & 0.93(10) & \\hspace{1cm} & 0.1(6) & \\hspace{1cm} & 5.35\/19 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 1.0 & \\hspace{1cm} & 1.13(13) & \\hspace{1cm} & -0.6(8) & \\hspace{1cm} & 3.99\/19 \\\\\n\\hline \n\\hline\n0.001 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 0.8 & \\hspace{1cm} & 0.86(8) & \\hspace{1cm} & -0.5(6) & \\hspace{1cm} & 43.40\/28 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 0.9 & \\hspace{1cm} & 0.98(8) & \\hspace{1cm} & -0.1(6) & \\hspace{1cm} & 14.90\/28 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 1.0 & \\hspace{1cm} & 1.08(7) & \\hspace{1cm} & -0.2(6) & \\hspace{1cm} & 22.32\/28 \\\\\n\\hline \n\\hline \\hline\n0.01 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.8 & \\hspace{1cm} & 1.29(5) & \\hspace{1cm} & -0.2(3) & \\hspace{1cm} & 22.30\/19 \\\\\n\\hline \n0.01 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.9 & \\hspace{1cm} & 1.47(6) & \\hspace{1cm} & -0.5(4) & \\hspace{1cm} & 7.32\/19 \\\\\n\\hline \n0.01 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 1.0 & \\hspace{1cm} & 1.65(6) & \\hspace{1cm} & -0.8(4) & \\hspace{1cm} & 4.83\/19 \\\\\n\\hline \n\\hline\n0.01 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 0.8 & \\hspace{1cm} & 1.19(6) & \\hspace{1cm} & 0.1(4) & \\hspace{1cm} & 53.23\/28 \\\\\n\\hline \n0.01 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 0.9 & \\hspace{1cm} & 1.48(9) & \\hspace{1cm} & -0.7(6) & \\hspace{1cm} & 17.19\/28 \\\\\n\\hline \n0.01 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 1.0 & \\hspace{1cm} & 1.63(9) & \\hspace{1cm} & -0.8(6) & \\hspace{1cm} & 10.81\/28 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption[\\textbf{Peak position characterization.}]{\\textbf{Peak position characterization.} Parameters obtained in the fits of our data for $N_{r,\\max}^{1\/3}$ to \\refeq{Nr_xi}. For each fit, we also report the figure of merit $\\chi^2\/\\mathrm{d.o.f.}$\\index{degree of freedom}}\n\\labtab{parametros_Nmax}\n\\end{center}\n\\end{table}\n\n\n\\subsection{The peak height}\\labsubsec{peaks-height}\n\n\\begin{figure}[!h]\n  \\centering\n  \\includegraphics[width=1\\textwidth]{off-eq_chaos\/fmax_xi.pdf}\n  \\caption[\\textbf{Peak height increases with $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$.}]{\\textbf{Peak height increases with $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$.} Height of the peak against $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ is plotted for all the simulated pair of temperatures $T_1$ and $T_2$ at different probability levels, $F=0.001$ (left panel) and $F=0.01$ (right panel). Curves display a monotone trend with the difference of temperatures $T_2-T_1$.}\n \\labfig{fmax_xi}\n\\end{figure}\n\n\\begin{figure}[!h]\n\t\\centering\n\t\\includegraphics[width=0.7\\textwidth]{off-eq_chaos\/zeta_exponent_fixed.pdf}\n\t\\caption[\\textbf{The exponent $\\mathbf{\\zeta_{\\text{NE}}}$ turns out to be independent of $\\mathbf{F}$ and $\\mathbf{T_1}$.}]{\\textbf{The exponent $\\mathbf{\\zeta_{\\text{NE}}}$ turns out to be independent of $\\mathbf{F}$ and $\\mathbf{T_1}$.} The characteristic length $\\xi^*$ is plotted against the temperature difference $T_2-T_1$ in a log-log scale. Each curve is uniquely identified by the probability level $F$ and the smallest temperature of each pair $T_1$. Fits to~\\eqref{eq:def_zeta}, enforcing a common exponent, are shown with continuous lines and result in a chaotic exponent $\\zeta_\\text{NE}=1.19(2)$.}\n \\labfig{zeta_exponent}\n\\end{figure}\n\nThe peak height $H_{\\max} \\equiv H(z_{\\max})$ is an indication of the strength of \\gls{TC}\\index{temperature chaos} (for a given coherence length\\index{coherence length} $\\xi$, probability $F$ and temperatures $T_1$ and $T_2$). In order to study the scaling with $\\xi$, we have considered the following ansatz:\n\\begin{equation}\n  H_{\\max}(\\xi) = \\dfrac{\\varepsilon(\\xi)}{1+ \\varepsilon(\\xi)} \\, , \\text{ with } \\varepsilon(\\xi)=(\\xi\/\\xi^*)^\\alpha\\,. \\labeq{fmax_xi}\n\\end{equation}\nThe fit parameters are the characteristic length scale $\\xi^*$ and the exponent $\\alpha$. The rationale behind~\\refeq{fmax_xi} is that, although in cases of extremely weak chaos $1-X$ may grow with $\\xi$ as a power law, $1-X$ should eventually approach its upper bound $1-X=1$ (when chaos becomes strong).\nNevertheless, a consistency check necessary to give some physical meaning to~\\refeq{fmax_xi}, is that exponent $\\alpha$ should not depend neither on temperatures $T_1$ and $T_2$ nor on the chosen probability $F$.\n\nWe find fair fits to~\\refeq{fmax_xi}, see~\\reffig{fmax_xi} and~\\reftab{parametros_fmax}. Fortunately, in all cases exponent $\\alpha$ turns out to be very close to $\\alpha \\approx 2.1$ (actually, all the $\\alpha$ obtained in the fits turn out to be compatible with $2.1$ at the two-$\\sigma$ level). Under these conditions, we can interpret $\\xi^*$ as a characteristic length indicating the crossover from weak to strong \\gls{TC}\\index{temperature chaos}, at the probability level indicated by $F$ (the relatively large value of exponent $\\alpha$ indicates that this crossover is sharp). The trends for the crossover-length $\\xi^*(F,T_1,T_2)$ are very clear: it grows upon increasing $F$ or upon decreasing $T_2-T_1$. At this point, we can try to be more quantitative. \n\nIndeed, because $\\xi^*$ indicates the crossover between weak and strong chaos, it must be the non-equilibrium analogue of the equilibrium chaotic length $\\ell_\\text{c}(T_1,T_2)$~\\cite{fisher:86,bray:87b} (see~\\refsec{origin_tc}). Now, the equilibrium $\\ell_\\text{c}(T_1,T_2)$ has been found to scale for the 3D Ising\\index{Ising} spin glass as \n\\begin{equation}\n\\ell_\\text{c}(T_1,T_2) \\propto (T_2-T_1)^{-1\/\\zeta} \\, \\, , \\labeq{def_zeta_equilibrium}\n\\end{equation}\nwith $\\zeta \\approx 1.07$~\\cite{katzgraber:07} and $\\zeta \\approx 1.07(5)$~\\cite{fernandez:13}. These considerations suggest the following ansatz for the non-equilibrium crossover length\n\\begin{equation}\n\\xi^*(T_1,T_2,F) = B(F) \\, (T_2-T_1)^{-1\/\\zeta_{\\text{NE}}} \\, \\, , \\labeq{def_zeta}\n\\end{equation}\nwhere $B(F)$ is an amplitude.\n\nWe have tested~\\refeq{def_zeta} by computing a joint fit for four $(T_1, F)$ pairs as functions of $T_2-T_1$, allowing each curve to have its own amplitude but enforcing a common $\\zeta_\\text{NE}$ (see~\\reffig{zeta_exponent}). The resulting $\\chi^2\/\\text{d.o.f.} = 7.55\/7$\\index{degree of freedom} validates our ansatz, with an exponent $\\zeta_\\text{NE}=1.19(2)$ fairly close  to the equilibrium result $\\zeta=1.07(5)$~\\cite{fernandez:13}. This agreement strongly supports our physical interpretation of the crossover length. We, furthermore, find that $B$ is only weakly dependent on $T_1$.\nNevertheless, the reader should be warned that it has been suggested~\\cite{fernandez:13} that the equilibrium exponent $\\zeta$ may be different in the weak- and strong-chaos regimes.\n\n\\begin{table}[t!]\n\\begin{center}\n\\begin{tabular}{c c c c c c c c c c c}\n\\toprule\n\\toprule\n$F$ & \\hspace{1cm} & $T_1$ &  \\hspace{1cm} & $T_2$ & \\hspace{1cm} & $\\xi^*$ & \\hspace{1cm} &$\\alpha$ & \\hspace{1cm} & $\\chi^2\/\\mathrm{d.o.f.}$\\index{degree of freedom} \\\\\n\\hline \\hline\n0.001 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.7 & \\hspace{1cm} & 55(4) & \\hspace{1cm} & 2.10(7) & \\hspace{1cm} & 14.10\/19 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.8 & \\hspace{1cm} & 23.5(7) & \\hspace{1cm} & 2.22(5) & \\hspace{1cm} & 38.00\/19\\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.9 & \\hspace{1cm} & 16.8(3) & \\hspace{1cm} & 2.09(4) & \\hspace{1cm} & 28.88\/19 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 1.0 & \\hspace{1cm} & 13.24(15) & \\hspace{1cm} & 2.04(3) & \\hspace{1cm} & 8.77\/19 \\\\\n\\hline \n\\hline\n0.001 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 0.8 & \\hspace{1cm} & 43.5(15) & \\hspace{1cm} & 2.12(5) & \\hspace{1cm} & 41.05\/28 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 0.9 & \\hspace{1cm} & 22.9(5) & \\hspace{1cm} & 2.09(4) & \\hspace{1cm} & 33.32\/28 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 1.0 & \\hspace{1cm} & 16.3(2) & \\hspace{1cm} & 2.04(4) & \\hspace{1cm} & 22.32\/28 \\\\\n\\hline \n\\hline \\hline\n0.01 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.8 & \\hspace{1cm} & 28.4(4) & \\hspace{1cm} & 2.26(2) & \\hspace{1cm} & 49.15\/19 \\\\\n\\hline \n0.01 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.9 & \\hspace{1cm} & 20.1(2) & \\hspace{1cm} & 2.16(2) & \\hspace{1cm} & 48.07\/19\\\\\n\\hline \n0.01 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 1.0 & \\hspace{1cm} & 15.87(16) & \\hspace{1cm} & 2.08(2) & \\hspace{1cm} & 23.93\/19 \\\\\n\\hline \n\\hline\n0.01 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 0.8 & \\hspace{1cm} & 51.4(12) & \\hspace{1cm} & 2.17(3) & \\hspace{1cm} & 8.06\/28 \\\\\n\\hline \n0.01 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 0.9 & \\hspace{1cm} & 27.7(4) & \\hspace{1cm} & 2.13(2) & \\hspace{1cm} & 65.66\/28 \\\\\n\\hline \n0.01 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 1.0 & \\hspace{1cm} & 19.9(2) & \\hspace{1cm} & 2.05(2) & \\hspace{1cm} & 31.78\/28 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption[\\textbf{Peak height characterization.}]{\\textbf{Peak height characterization.} Parameters obtained in the fits of our data for $H_{\\max}$ to~\\refeq{fmax_xi}. For each fit, we also report the figure of merit $\\chi^2\/\\mathrm{d.o.f.}$\\index{degree of freedom}}\n\\labtab{parametros_fmax}\n\\end{center}\n\\end{table}\n\n\\begin{table}[h!]\n\\begin{center}\n\\begin{tabular}{c c c c c c c c c}\n\\toprule\n\\toprule\n$F$ & \\hspace{0.25cm} & $T_1$ &  \\hspace{0.25cm} & $B(F)$ & \\hspace{0.25cm} & $\\zeta_{\\text{NE}}$ & \\hspace{0.25cm} & $\\chi^2\/\\mathrm{d.o.f.}$\\index{degree of freedom} \\\\\n\\hline \\hline\n0.001 & & 0.625 & & 5.77(11) & & 1.19(2) & & 2.14\/2\\\\\n\\hline\n0.01 & & 0.625 & & 6.94(14) & & 1.19(2) & & 1.57\/2\\\\\n\\hline\n0.001 & & 0.7 & & 5.99(13) & & 1.19(2) & & 2.46\/2\\\\\n\\hline\n0.01 & & 0.7 & & 7.28(16) & & 1.19(2) & & 1.38\/2\\\\\n\\hline\n\\hline\n0.001 & & 0.625, 0.7 & & 5.85(11) & & 1.19(2) & & 11.54\/5\\\\\\hline\n0.01 & & 0.625, 0.7 & & 7.08(14) & & 1.19(2) & & 21.19\/5\\\\\n\n\\bottomrule\n\\end{tabular}\n\\caption[\\textbf{The chaotic exponent $\\zeta$.}]{\\textbf{The chaotic exponent $\\zeta$.} The smallest temperature $T_1$ is fixed in each fit in the upper part of the table. The two last rows in the table correspond to the fit including all the available pairs of temperatures (i.e. in these fits we mix data with $T_1=0.625$ and $T_1=0.7$). Points with $(T_1=0.625,T_2=1.0)$ for both $F=0.001$ and $F=0.01$ are not considered in these fits.}\n\\labtab{zeta}\n\\end{center}\n\\end{table}\n\n\\subsection{The (inverse) peak width}\\labsubsec{peaks-width}\n\n\\begin{figure}[!h]\n  \\centering\n  \\includegraphics[width=1\\textwidth]{off-eq_chaos\/width_xi_power_law.pdf}\n  \\caption[\\textbf{Curvature $\\kappa$ decays as a power law when increasing $\\xi$.}]{\\textbf{Curvature $\\kappa$ decays as a power law when increasing $\\xi$.} The inverse peak width $\\kappa(\\xi)$ is plotted against the coherence length\\index{coherence length} $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ for all the simulated pairs of temperatures $T_1$ and $T_2$ at different probability levels, $F=0.001$ (left panel) and $F=0.01$ (right panel). Similar decaying exponent $\\beta$, actually compatible at the two-$\\sigma$ level (see~\\reftab{parametros_width}), is displayed for all the pairs of temperatures in both probability levels.}\n \\labfig{width_xi}\n\\end{figure}\n\nThe peak width provides the answer to the following question: how critical is it to select the right length-scale to study \\gls{TC}\\index{temperature chaos}? Obviously, if the peak width becomes larger than its position (see~\\refsubsec{peaks-position}), this choice is no longer critical.\n\nWe study the inverse peak width (i.e. the curvature)\n\\begin{equation}\n\\kappa(\\xi)=\\sqrt{\\dfrac{|H''(z_{\\max})|}{H(z_{\\max})}} \\, , \\labeq{def_curvature}\n\\end{equation}\nand propose a power law decaying with $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace)$ characterized by the ansatz\n\\begin{equation}\n\\kappa(\\xi) = A(F) \\, \\xi^{-\\beta} \\, , \\labeq{width_xi}\n\\end{equation}\nwhere $A(F)$ is an amplitude while $\\beta$ is the power law exponent. Results are shown in~\\reffig{width_xi} and~\\reftab{parametros_width}.\n\nThe value of $A(F)$ turns out to be compatible for all the pairs of temperatures $(T_1,T_2)$ at fixed probability $F$. Furthermore, at the current precision of the data, exponent $\\beta$ does not exhibit any significant dependency on the temperature pair ($T_1,T_2$) or the probability $F$.\n\nLet us now recall the linear relation between the peak position and the coherence length\\index{coherence length}, see~\\refeq{Nr_xi}. Consider the ratio between the position of the maximum and its width, $N_{r,\\max}\\kappa(\\xi) \\sim \\xi^{1-\\beta}$. The~\\reftab{parametros_width} mildly suggests that $\\beta$ is slightly greater than 1, which implies that the ratio goes to zero (very slowly) in the limit of large $\\xi$. The parameter $\\beta$ would have, indeed, a critical meaning for the large $\\xi$ limit. If greater than one, as mildly suggested by the results, the chaotic behavior would be present at any scale $\\xi_c$. On the contrary, if further works find $\\beta < 1$, in the $\\xi \\to \\infty$ limit, the chaos would only be visible at the coherence-length scale.\n\n\\begin{table}[h!]\n\\begin{center}\n\\begin{tabular}{c c c c c c c c c c c}\n\\toprule\n\\toprule\n$F$ & \\hspace{1cm} & $T_1$ &  \\hspace{1cm} & $T_2$ & \\hspace{1cm} & $A$ & \\hspace{1cm} &$\\beta$ & \\hspace{1cm} & $\\chi^2\/\\mathrm{d.o.f.}$\\index{degree of freedom} \\\\\n\\hline \\hline\n0.001 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.7 & \\hspace{1cm} & 0.8(3) & \\hspace{1cm} & 0.9(2) & \\hspace{1cm} & 18.72\/19 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.8 & \\hspace{1cm} & 1.6(4) & \\hspace{1cm} & 1.27(14) & \\hspace{1cm} & 8.07\/19 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.9 & \\hspace{1cm} & 1.4(3) & \\hspace{1cm} & 1.32(12) & \\hspace{1cm} & 10.05\/19 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 1.0 & \\hspace{1cm} & 1.3(2) & \\hspace{1cm} & 1.37(9) & \\hspace{1cm} & 5.60\/19 \\\\\n\\hline \n\\hline\n0.001 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 0.8 & \\hspace{1cm} & 1.1(3) & \\hspace{1cm} & 1.10(12) & \\hspace{1cm} & 35.26\/28 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 0.9 & \\hspace{1cm} & 1.26(16) & \\hspace{1cm} & 1.25(7) & \\hspace{1cm} & 25.90\/28 \\\\\n\\hline \n0.001 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 1.0 & \\hspace{1cm} & 1.19(17) & \\hspace{1cm} & 1.29(7) & \\hspace{1cm} & 23.01\/28 \\\\\n\\hline \n\\hline \\hline\n0.01 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.8 & \\hspace{1cm} & 0.63(9) & \\hspace{1cm} & 1.11(7) & \\hspace{1cm} & 20.44\/19 \\\\\n\\hline \n0.01 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 0.9 & \\hspace{1cm} & 0.59(10) & \\hspace{1cm} & 1.14(8) & \\hspace{1cm} & 6.08\/19 \\\\\n\\hline \n0.01 & \\hspace{1cm} & 0.625 &  \\hspace{1cm} & 1.0 & \\hspace{1cm} & 0.58(15) & \\hspace{1cm} & 1.21(12) & \\hspace{1cm} & 9.05\/19 \\\\\n\\hline \n\\hline\n0.01 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 0.8 & \\hspace{1cm} & 0.59(11) & \\hspace{1cm} & 1.05(11) & \\hspace{1cm} & 21.26\/28 \\\\\n\\hline \n0.01 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 0.9 & \\hspace{1cm} & 0.63(8) & \\hspace{1cm} & 1.15(7) & \\hspace{1cm} & 18.46\/28 \\\\\n\\hline \n0.01 & \\hspace{1cm} & 0.7 &  \\hspace{1cm} & 1.0 & \\hspace{1cm} & 0.59(12) & \\hspace{1cm} & 1.18(9) & \\hspace{1cm} & 17.93\/28 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption[\\textbf{Peak width characterization.}]{\\textbf{Peak width characterization.} Parameters obtained in the fits of our data for $\\kappa(\\xi)$ to~\\refeq{width_xi}. For each fit, we also report the figure of merit $\\chi^2\/\\mathrm{d.o.f.}$\\index{degree of freedom}}\n\\labtab{parametros_width}\n\\end{center}\n\\end{table}\n\n\\subsection{On the relation between experimental and numerical results} \\labsubsec{relation_experiment_numerical_simulations}\nThis characterization of \\gls{TC}\\index{temperature chaos} in the off-equilibrium dynamics of a large \\gls{SG} paves the way to a major interplay between numerical simulations and experiments.\n\nAlthough we have considered in this work fairly small values of the chaotic system fraction $F$, a simple extrapolation, linear in $\\log F$, predicts $\\xi^* \\approx 60$ for $F=0.1$ at $T_1=0.7$ and $T_2=0.8$ (our closest pair of temperatures in~\\reftab{parametros_fmax}). A spin-glass coherence length\\index{coherence length} well above $60 a_0$ is experimentally reachable\nnowadays~\\cite{zhai:19,zhai:20b,zhai-janus:20,zhai-janus:21} ($a_0$ is the typical spacing between spins), which makes our dynamic \\gls{TC}\\index{temperature chaos} significant.\n\nIndeed the \\gls{TC}\\index{temperature chaos}-closely related experimental study~\\cite{zhai:20b} reported a value for exponent\\footnote{The authors propose several schemes and different computations of the exponent are provided.} $\\zeta_{\\text{NE}}$ in fairly good agreement with our result of $\\zeta_\\text{NE}=1.19(2)$ in~\\reffig{zeta_exponent}.\n\nA deeper relation between the \\gls{TC}\\index{temperature chaos} phenomenon in experiments and in numerical simulations would be desirable. Actually, simple temperature-varying protocols\\index{temperature-varying protocol} (in which temperature sharply drops from $T_2$ to $T_1$, see, e.g.~\\cite{zhai:20b}) seems more accessible to a first analysis than memory\\index{memory effects} and rejuvenation\\index{rejuvenation} experiments~\\cite{jonason:98,lundgren:83,jonsson:00,hammann:00}.\n\nThe rupture point between both approaches is the difference in the measured observables. An important problem is that the correlation functions that are studied theoretically are not easily probed experimentally. Instead, experimentalists privilege the magnetization\\index{magnetization!density} density (which is a spatial average over the whole sample\\index{sample}). The study of the magnetization\\index{magnetization!density} density from a numerical point of view, on the other side, is clearly bounded by the computational power available nowadays since its global nature makes the chaos signal almost disappear for our achievable coherence lengths\\index{coherence length}. Therefore an important theoretical goal is to predict the behavior of the non-equilibrium time-dependent magnetization\\index{magnetization} upon a temperature drop.\n\n\\section{Scaling at fixed \\boldmath $r$}\\labsec{scaling_fixed_r}\n\\begin{figure}[!h]\n  \\centering\n  \\includegraphics[width=0.49\\textwidth]{off-eq_chaos\/extrapola_q2_T1T1_v2.pdf}\n  \\includegraphics[width=0.49\\textwidth]{off-eq_chaos\/log_extrapola_q2_T1T1_v2.pdf}\t\n  \\includegraphics[width=0.49\\textwidth]{off-eq_chaos\/extrapola_q2_T1T2_v2.pdf}\n  \\includegraphics[width=0.49\\textwidth]{off-eq_chaos\/log_extrapola_q2_T1T2_v2.pdf}\t\n  \\caption[\\textbf{Extrapolation of $\\mathbf{F(q^2)}$ to the $\\mathbf{\\xi \\to \\infty}$ limit.}]{\\textbf{Extrapolation of $F(q^2_{T_1T_1})$ (up) and  $F(q^2_{T_1T_2})$ (down) to the $\\xi \\to \\infty$ limit, both in linear scale (left) and logarithmic scale (right).} \\textbf{Main plots:} We extrapolate the distribution functions for spheres of radius $r=1$ to the $\\xi \\to \\infty$ limit. Both ans\\\"atze, the one in~\\refeq{extrapolacion_estatica_lineal}, golden curves, and the one in~\\refeq{extrapolacion_estatica_cuadratica}, blue curves, produce equivalent extrapolations. For the sake of clarity, we plot two curves for each extrapolation (with the same color) which corresponds to the computed central value plus (minus) the standard deviation. We compare the results with the equilibrium data from an \\gls{EA}\\index{Edwards-Anderson!model} model in a cubic lattice of linear size $L=32$ (red curve, see main text for further details). The equilibrium distribution and the extrapolated one are compatible for the $q^2_{T_1T_1}$ curves. Instead, $q^2_{T_1T_2}$ is compatible for percentiles of order one but not for probabilities smaller than $F=0.1$. \\textbf{Insets:} As in the main plots, for spheres of radius $r=2$.}\n\\labfig{extrapola_estatica_q2}\n\\end{figure}\n\nUntil this point, the analysis of the chaotic parameter (its characterization through the size of the sphere and the dependence of the peak with the coherence length\\index{coherence length}) have been inspired by the representation of the data given by the~\\reffig{distribution_function_r} and the idea of an optimal scale to observe \\gls{TC}\\index{temperature chaos}. However, a different approach can be performed by regarding the representation of the data in~\\reffig{distribution_function_xi}.\n\nFixing the two temperatures $T_1$ and $T_2$ and the size of the sphere, the distribution function $F(X,T_1,T_2,\\xi,r)$ exhibit a monotonic behavior with the coherence length\\index{coherence length} $\\xi$ (see~\\reffig{distribution_function_xi}). Specifically, regarding at the lowest possible size of the sphere $r=1$, we found there was no convergence (apparently) to a limit curve when increasing the coherence length\\index{coherence length} $\\xi$.\n\nThe absence of a limit curve led us to consider slow convergence as a hypothesis. In this section, we propose algebraic extrapolations to explain the convergence to the $\\xi \\to \\infty$ limit. Moreover, we compare the extrapolations with the equilibrium data obtained from an \\gls{EA}\\index{Edwards-Anderson!model} Ising\\index{Ising} model simulated with a \\gls{PT} algorithm in a cubic lattice of linear size $L=32$ (see~\\cite{janus:10,janus:10b}). We expect that, for the smallest sphere radius at least, $L=32$ will be representative of the thermodynamic limit\\index{thermodynamic limit}.\n\nLet us fix the probability level $F$ and the radius $r$ of the spheres, we propose two different extrapolations to the $\\xi(\\ensuremath{t_\\mathrm{w}}\\xspace) \\to \\infty$ limit\n\\begin{equation}\n\\Omega(T_1,T_2,\\xi,r,F) = \\Omega(T_1,T_2,\\xi= \\infty,r,F) +  \\Phi(F) \\left(\\dfrac{N_r^{1\/3}}{\\xi}\\right)^{\\delta(F)} \\, , \\labeq{extrapolacion_estatica_lineal}\n\\end{equation}\n\\begin{equation}\n\\Omega(T_1,T_2,\\xi,r,F) = \\Omega(T_1,T_2,\\xi= \\infty,r,F) + \\Psi_1 (F) \\left(\\dfrac{N_r^{1\/3}}{\\xi}\\right)^{\\epsilon(F)} + \\Psi_2(F) \\left(\\dfrac{N_r^{1\/3}}{\\xi}\\right)^{2\\epsilon(F)} \\, , \\labeq{extrapolacion_estatica_cuadratica}\n\\end{equation}\nwhere $\\Omega(T_1,T_2,\\xi,r)$ is the extrapolated quantity $\\Omega \\in \\lbrace q^2_{T_1T_1},q^2_{T_2T_2},q^2_{T_1T_2},X_{T_1T_2} \\rbrace$ at the fixed $F$ probability level, $\\Omega(T_1,T_2,\\xi= \\infty,r)$ is the value of that quantity in the $\\xi \\to \\infty$ limit, $\\Phi$, $\\Psi_1$ and $\\Psi_2$ are the coefficients of the fit, and $\\delta$ and $\\epsilon$ are the exponents of the fits.\n\nFrom now on, we focus on the $r=1$ and $r=2$ cases. The extrapolations for $q^2_{T_1T_1}$ and $q^2_{T_2T_2}$ show the agreement between the extrapolations to the $\\xi \\to \\infty$ limit and the equilibrium data, see~\\reffig{extrapola_estatica_q2} (up panels), for the whole range. Besides, the quadratic and the linear extrapolations are statistically equivalent.\n\nThe extrapolation for $q^2_{T_1T_2}$ keeps the agreement between the extrapolations to the $\\xi \\to \\infty$ limit and the equilibrium data for high values of the distribution function, but differs for short values of $F(q^2_{T_1T_2})$, see~\\reffig{extrapola_estatica_q2} (down panels). Again, linear and quadratic extrapolation show no difference within the statistical error.\n\nFor the chaotic parameter $X_{T_1T_2}$ we find no convergence at all at the current precision of the data and the simulated scales of the coherence length\\index{coherence length}.\n\nWe also explore the $r>2$ case, however, the greater the sphere size $r$, the bigger the simulated coherence length\\index{coherence length} $\\xi$ needed for a reliable fit to~\\refeq{extrapolacion_estatica_lineal} and~\\refeq{extrapolacion_estatica_cuadratica}. Extreme events stop providing reliable fits when increasing the sphere size $r$.\n\nIndeed, this approach provides reasonable results in the extrapolation to the $\\xi \\to \\infty$ limit for $q^2$ distribution functions when compared with the equilibrium data. However, the convergence is slow and the fits are difficult at our current precision.\n\nFinally, we examine our results in search of scale invariance. Due to the compatibility of the linear and quadratic extrapolations, we will focus on the former for the sake of simplicity, see~\\refeq{extrapolacion_estatica_lineal}. We study the $r=1$ and $r=2$ cases which more likely hold the limit $r\/\\xi \\ll 1$. \n\nIf the scale invariance is present in our results, we would expect the exponent $\\delta$ to be independent of the radius $r$. We find a similar behavior for the exponent in the extrapolations for spheres of radius $r=1$ and $r=2$ (\\reffig{factor_exponente} main plots), nevertheless the results are not compatible for all the percentiles $F(q^2_{T_1T_1})$ and $F(q^2_{T_1T_2})$.\n\nWe also expect the ratio of the coefficients $\\Phi(r=2)\/\\Phi(r=1)$ to be constant if the scale invariance holds. We find small changes in the ratio between the amplitudes $\\Phi(F,r)$, see~\\reffig{factor_exponente} insets. The ratio remains around $1$ for all the percentiles, however, the results are not compatible for the whole range with $1$.\n\nOur results mildly suggest the existence of scale invariance for the limit $r\/\\xi \\ll 1$, in our cases represented by the simulations with sphere radius $r=1$ and $r=2$. However, the difficulty of the extrapolations and the noise in the amplitude and the exponent, prevent us from making robust statements.\n\n\n\\begin{figure}[h!]\n  \\centering\t\n  \\includegraphics[width=1.0\\textwidth]{off-eq_chaos\/factor_exponente.pdf}\n  \\caption[\\textbf{Scale invariance test.}]{\\textbf{Scale invariance test for parameters in~\\refeq{extrapolacion_estatica_lineal}.} Indeed, scale invariance demands that both the exponent $\\delta(F)$ and the amplitude $\\Phi(F)$ should be independent of the sphere radius. The figure compares both quantities for the radius $r=1$ and $r=2$. \\textbf{Main panels:} exponent $\\delta(F)$ as computed for spheres of radius $r=1$ (golden curves) and $r=2$ (blue curves), as a function of $F(q^2_{T_1T_1})$ (left panel) or $F(q^2_{T_1T_2})$ (right panel). \\textbf{Insets: }Ratio of amplitudes $\\Phi(F,r=2)\/\\Phi(F,r=1)$ for the same fits to~\\refeq{extrapolacion_estatica_lineal} reported in the main panels. The horizontal black line is set at $1$ as a reference.}\n\\labfig{factor_exponente}\n\\end{figure}\n\n\n\\section{Temperature changing protocols}\\labsec{T-changes}\n\nIn this section, we explore the impact of temperature-varying\\index{temperature-varying protocol} simulations in the \\gls{TC}\\index{temperature chaos} results and we also address the cumulative aging\\index{aging!cumulative} controversy: Is the aging\\index{aging} performed at temperature $T_a$ useful at different temperature $T_b$?\n\nSpecifically, the cumulative aging\\index{aging!cumulative} hypothesis~\\cite{jonsson:02,bert:04} considers a protocol in which the system is first suddenly quenched from high temperature to temperature $T_a < \\ensuremath{T_\\mathrm{c}}\\xspace$, where it ages for time $t_a$. Then, the temperature is changed to $T_b < \\ensuremath{T_\\mathrm{c}}\\xspace$ and the system evolves for a time $t_b$. The hypothesis is made that this two-steps protocol is equivalent to isothermal\\index{isothermal} aging\\index{aging} at temperature $T_b$ for an effective time $t^{\\mathrm{eff}}_b$ defined through the equation \n\\begin{equation}\n\\xi (t_a ; t_b ; T_a \\rightarrow T_b) = \\xi(t^{\\mathrm{eff}}_b,T_b) \\> . \\labeq{cumulative_hypotesis}\n\\end{equation}\n\nFortunately, we now know how to compute the coherence length\\index{coherence length} $\\xi$~\\cite{janus:09b,fernandez:18b} accurately (see also \\refsec{finite_size_effects}). Therefore, we are able to compare the evolution of systems that have undergone either the two-steps protocol or the isothermal\\index{isothermal} one, choosing the time scales to precisely match~\\refeq{cumulative_hypotesis}.\n\nHere, we will consider two symmetric two-steps proposals. On the one hand, we let age the system at temperature $T_a=0.8$ (by \\textit{the system} we mean the $\\ensuremath{N_{\\text{S}}}\\xspace$ samples\\index{sample} and the $\\ensuremath{N_{\\text{Rep}}}\\xspace$ replicas\\index{replica}) until it reaches coherence length\\index{coherence length} $\\xi(t_a)=6$. At this point, we suddenly quench to the temperature $T_b=1.0$ and let the system evolve until $\\xi(t_a;t_b; T_a=0.8 \\rightarrow T_b=1.0)=9$. On the other hand we perform the symmetric two-steps protocol, starting at temperature $T_a=1.0$ and cooling the system to temperature $T_b=0.8$. On the technical side, let us mention that we compute the distribution function as explained in~\\refsec{procedure}.\n\nAs everywhere else in this chapter, we will compute the overlaps\\index{overlap} between configurations\\index{configuration} at two temperatures $T_1<T_2$. However, we shall need to introduce notation in order to specify the protocol. \n\nWe will put the superscript \\textit{iso} on a temperature to indicate that the system has always been at that temperature (\\textit{iso} is a short for isothermal\\index{isothermal}). For instance, the notation $\\ensuremath{T^{\\text{iso}}}\\xspace_1 = 0.8$ will mean that we are computing the overlap\\index{overlap} between a system that has always been at temperature $0.8$ and a system that currently is at some higher temperature. Instead, $\\ensuremath{T^{\\text{iso}}}\\xspace_2 = 0.8$ means that the overlap\\index{overlap} is computed between our isothermal\\index{isothermal} system and a system which currently is at some lower temperature.\n\nWe will put the superscript $T_a \\to T_b$ on a temperature to indicate that the system has followed the two-steps protocol from temperature $T_a$ to temperature $T_b$. Notice that the current temperature of the system is $T_b$. For instance, the notation $X(\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.8,T^{0.8 \\to 1.0}_2=1.0)$ indicates that we are computing overlaps\\index{overlap} between the configurations\\index{configuration} of a system that has always been at temperature $0.8$ and those of a system that has followed the two-steps protocol from temperature $T_a=0.8$ to $T_b=1.0$.\n\nWe organize the obtained results into two subsections. The first one (see~\\refsubsec{no-equivalence}) captures the asymmetric behavior when the system is cooled or heated. The second one (see~\\refsubsec{auxiliary}) takes advantage of an auxiliary temperature to study the effect of the two-steps protocol on the \\gls{TC}\\index{temperature chaos} phenomenon.\n\n\\subsection{On the asymmetry between cooling and heating}\\labsubsec{no-equivalence}\n\n\\begin{figure}[!h]\n  \\centering\n  \\includegraphics[width=1.0\\textwidth]{off-eq_chaos\/no_auxiliar.pdf}\n  \\caption[\\textbf{Comparison between isothermal and two-step protocol.}]{\\textbf{Comparison of the main parameters in the \\boldmath $1-X$ curves for the two-steps protocol and the isothermal\\index{isothermal} protocol. The two protocols approach to each other for large $\\xi$.} Position of the peak (up panels), height of the peak (middle panels) and curvature (down panels) against $\\xi(t_w)$ are plotted for the reference curve $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.8$, $\\ensuremath{T^{\\text{iso}}}\\xspace_2=1.0$ and for the temperature-varying\\index{temperature-varying protocol} simulations at two different probability levels $F=0.001$ (left) and $F=0.01$ (right). Heating and cooling two-steps protocols are not equivalent.}\n \\labfig{xi_noaux}\n\\end{figure}\n\nWe analyze the \\gls{TC}\\index{temperature chaos} phenomenon through the same three observables used in~\\refsec{results}: the peak position, the peak height, and the (inverse) peak width (i.e. the curvature). Each of those quantities are extracted from one of the following overlap\\index{overlap} combination. First we have the reference curve $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.8$ and $\\ensuremath{T^{\\text{iso}}}\\xspace_2=1.0$. Next we have the so called \\textit{cooling} curve $T_1^{1.0 \\to 0.8} = 0.8$ and $\\ensuremath{T^{\\text{iso}}}\\xspace_2=1.0$. Finally we have the so called \\textit{heating} curve $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.8$ and $T_2^{0.8 \\to 1.0} = 1.0$.\n\nThe prediction of the cumulative aging\\index{aging!cumulative}, see~\\refeq{cumulative_hypotesis}, is that the three curves, namely the reference, the \\textit{heating} and the \\textit{cooling} curves, should coincide. Indeed, we choose the times in such a way that the coherence length\\index{coherence length} is the same for all the four systems involved: $\\ensuremath{T^{\\text{iso}}}\\xspace=0.8$, $\\ensuremath{T^{\\text{iso}}}\\xspace=1.0$, $T^{0.8 \\to 1.0} = 1.0$ and $T^{1.0 \\to 0.8} = 0.8$. Instead, our data in~\\reffig{xi_noaux} show that cumulative aging\\index{aging!cumulative} holds only in an approximated way. In fact, both the \\textit{heating} and the \\textit{cooling} curves significantly differ from the reference curve at coherence length\\index{coherence length} $\\xi \\approx 6$, although the three curves tend to merge as $\\xi$ grows. Nevertheless, we notice that the \\textit{cooling} curve does not converge to the other two curves, at least in the simulated range of $\\xi$. In this sense, there exist an asymmetry between the \\textit{cooling} curve and the \\textit{heating} curve. Furthermore, by comparing the left and right panels in~\\reffig{xi_noaux}, we conclude that the larger the probability $F$, the stronger the asymmetry.\n\nFinally, the reader may wonder about the possibility of computing overlaps\\index{overlap} between configurations\\index{configuration} at temperature $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.8$ and $T_2^{1.0 \\to 0.8} = 0.8$ or configurations\\index{configuration} at temperature $\\ensuremath{T^{\\text{iso}}}\\xspace_1=1.0$ and $T_2^{0.8 \\to 1.0} = 1.0$. Indeed, we have attempted to compute those overlaps\\index{overlap} but we found very large values of the chaotic parameter that made impossible the characterization of the peak.\n\n\\subsection{Auxiliary-temperature analysis}\\labsubsec{auxiliary}\nIn this subsection, we study the behavior of the two-steps protocol simulations by comparing them with the most ordered configurations\\index{configuration} we have, namely, the simulations of temperature $\\ensuremath{T^{\\text{iso}}}\\xspace=0.625$. We establish two reference curves $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.625$, $\\ensuremath{T^{\\text{iso}}}\\xspace_2=0.8$ and $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.625$, $\\ensuremath{T^{\\text{iso}}}\\xspace_2=1.0$ and we compare them with the curves $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.625$, $T^{1.0 \\to 0.8}=0.8$, and $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.625$, $T^{0.8 \\to 1.0}=1.0$.\n\n\\begin{figure}[!h]\n  \\centering\n  \\includegraphics[width=1.0\\textwidth]{off-eq_chaos\/Nr_width_xi_changeT08.pdf}\t\n  \\caption[\\textbf{Temperature-varying\\index{temperature-varying protocol} simulations of the peak position and curvature exhibit non-symmetric behavior when symmetric cooling and heating protocols are applied.}]{\\textbf{Temperature-varying\\index{temperature-varying protocol} simulations of the peak position and curvature exhibit non-symmetric behavior when symmetric cooling and heating protocols are applied.} The peak position (upper panels) and the curvature (lower panels) are plotted against the coherence length\\index{coherence length} $\\xi(t_w)$. We use a constant auxiliary temperature $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.625$ combined with two fixed-temperature simulations as the reference curves ($\\ensuremath{T^{\\text{iso}}}\\xspace_2=0.8$ and $\\ensuremath{T^{\\text{iso}}}\\xspace_2=1.0$) and the constant auxiliary temperature $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.625$ combined with two temperature-varying\\index{temperature-varying protocol} simulations as the studied curves ($T^{0.8 \\to 1.0}_2=1.0$ and $T^{1.0 \\to 0.8}_2=0.8$). We compute all the curves at the probability levels $F=0.001$ (left panels) and $F=0.01$ (right panels).}\n \\labfig{Nr_width_xi_changeT}\n\\end{figure}\n\nWe can observe the migration, as the coherence length\\index{coherence length} grows, of the curve $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.625$, $T^{0.8 \\to 1.0}_2=1.0$ from the reference curve $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.625$ $\\ensuremath{T^{\\text{iso}}}\\xspace_2=0.8$ to the reference curve $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.625$ $\\ensuremath{T^{\\text{iso}}}\\xspace_2=1.0$ by increasing the position of the peak (\\reffig{Nr_width_xi_changeT} upper panels) and the curvature (\\reffig{Nr_width_xi_changeT} lower panels). However, the opposite temperature-change, namely the curve corresponding to $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.625$ $T^{1.0 \\to 0.8}_2=0.8$, does not lead to the symmetric behavior. The position of the peak (respectively, the curvature) does not decrease (respectively, increase) when cooling the temperature to merge with the reference curve $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.625$, $\\ensuremath{T^{\\text{iso}}}\\xspace_2=0.8$. Thus, the optimal scale to observe \\gls{TC}\\index{temperature chaos} is a monotonically increasing function of the coherence length\\index{coherence length} while the curvature seems a monotonically decreasing function of the coherence length\\index{coherence length}.\n\nAlthough we do not know the behavior of the peak position and the peak curvature for values of the coherence length\\index{coherence length} higher than those simulated, one could expect, as a naive ansatz, that the curve relative to the \\textit{cooling} protocol would merge with its corresponding reference curve ($\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.625$ $\\ensuremath{T^{\\text{iso}}}\\xspace_2=0.8$) when the coherence length\\index{coherence length} is increased and then it would behave as the isothermal\\index{isothermal} simulation.\n\n\\begin{figure}[!h]\n  \\centering\n  \\includegraphics[width=1.0\\textwidth]{off-eq_chaos\/fmax_xi_changeT08_twopanels.pdf}\t\n  \\caption[\\textbf{Peak height of cooled two-steps protocol simulations with an auxiliary temperature converges slower than the symmetric heating protocol.}]{\\textbf{Peak height of cooled two-steps protocol simulations with an auxiliary temperature converges slower than the symmetric heating protocol.} The height of the peak is plotted against the coherence length\\index{coherence length} $\\xi(t_w)$ for fixed-temperature simulations and for two-steps protocol simulations with an auxiliary temperature $\\ensuremath{T^{\\text{iso}}}\\xspace_1=0.625$ at the probability levels $F=0.001$ (left panel) and $F=0.01$ (right panel).}\n \\labfig{fmax_xi_changeT}\n\\end{figure}\n\nAs for the peak height (\\reffig{fmax_xi_changeT}), we may appreciate both studied curves approaching the reference curves. However, the same information provided in the previous subsection (\\refsubsec{no-equivalence}) emerges here: the curve with the \\textit{cooling} protocol needs a higher coherence length\\index{coherence length} to converge to the reference curve. \n\nRecalling the fact that the height of the peak is related to the strength of \\gls{TC}\\index{temperature chaos}, the system tends to erase the memory of its previous relaxation\\index{relaxation} state and exhibits the same amount of \\gls{TC}\\index{temperature chaos} as it could be expected from its current pair of temperatures. Nonetheless, the erasing process is not complete in the cooling protocol and a small amount of information of the previous state lasts in the system, at least for the simulated coherence lengths\\index{coherence length}.\n\nLet us conclude this section with a small summary of the results. The reader might imagine a \\gls{TC}\\index{temperature chaos} movie in which each frame corresponds to a function that represents $1-X(F,T_1,T_2,\\xi,r)$ against the size of the studied region $N_r^{1\/3}$, similar to those appearing in~\\reffig{Xvsr_examples}. When both temperatures $T_1$ and $T_2$ follow an isothermal\\index{isothermal} protocol, the function will, monotonically, increase the height of its peak, will move to the right (i.e. higher values of the sphere size), and will become \\textit{flatter} (i.e. the curvature decreases). \n\nThe same situation is expected to happen (qualitatively) when $T_2$ is suddenly heated following the two-steps protocol. However, in this situation, the increase of the peak position, the peak height, and the decrease of the curvature \\textit{accelerates} (i.e. requires shorter values of coherence length\\index{coherence length}) compared with the $T_2$-isothermal\\index{isothermal} movie.\n\nFinally, when $T_2$ is suddenly cooled following the two-steps protocol, the height of the peak will decrease, while the position of the peak and the curvature will \\textit{freeze}. The naive expectation is that the \\textit{freezing} of the position of the curve and the curvature will end at higher values of the coherence length\\index{coherence length} than the simulated ones and then, the curve will behave as the isothermal\\index{isothermal} one. \n\n\n\n\\section*{Preface}\n\n\\input{chapters\/aknowledgment.tex}\n\\checkoddpage\n\\ifoddpage{\n\t\\afterpage{\\blankpage}\n}\\fi\n\n\\oldmainmatter\n\n\\setlength{\\overflowingheadlen}{\\linewidth}\n\\addtolength{\\overflowingheadlen}{\\marginparsep}\n\\addtolength{\\overflowingheadlen}{\\marginparwidth}\n\n\\pagestyle{scrheadings}\n\\input{chapters\/general_introduction.tex}\n\\addpart{A brief introduction to Spin Glasses}\n\\input{chapters\/Introduction.tex}\n\\addpart{Metastate}\n\\input{chapters\/Metastate.tex}\n\\addpart{Off-equilibrium phenomena}\n\\input{chapters\/Aging}\n\\input{chapters\/Mpemba}\n\\addpart{Temperature Chaos}\n\\input{chapters\/Introduction_chaos}\n\\input{chapters\/Equilibrium_chaos}\n\\input{chapters\/Out_equilibrium_chaos}\n\\addpart{Conclusions}\n\\input{chapters\/Conclusions}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section*{Supplementary information}\n\n\\begin{table}[hbtp]\n\\begin{center}\n\\caption{The complete listing of the reflections used to index the Fast Fourier transforms of region (A) of the HRTEM image in Fig~\\ref{hrtem-23-a}. Reflections are assigned to  \\ce{Mg}, with beam direction, $\\boldsymbol{B}=[2\\overline{4}2\\overline{3}]$. The table lists the measured ($d$)  and calculated  ($d^\\prime$) interplanar spacings for a given reflection. The relative difference between these two values is shown by the value $|1-(d\/d^\\prime)|$. Interplanar angles are measured with respect to the $10\\overline{1}0$ reflection (indicated by an asterisk). The difference between the measured and expected angles ($\\Delta\\theta$) is also provided.\n\\label{tab-fft-hrtem-23a}}\n\\begin{tabular}{c@{ }c@{ }c@{ }ccccrrr}\n\\toprule\n\t&\t\t& \t&\t\t&\t\\multicolumn{3}{c}{$d_{hkl}$}\t\t\t\t\t&\t\\multicolumn{3}{c}{Angle}\t\t\t\t\t\\\\ \n\\cmidrule(r){5-7}\\cmidrule(l){8-10} \n\t&\t\t&&\t\t&\tMeas.($d$)\t&\tExpect.($d^\\prime$) &\t\t&\tMeas.\t&\tExpect.\t&\t$\\Delta\\theta$\t\\\\\nh\t&\tk\t&\t$\\cdot$\t&l\t&\t(\\AA)\t&\t(\\AA)\t\t&\t$|1-(d\/d^\\prime)|$\t&\t\t$(^\\circ)$\t\t\t&\t$(^\\circ)$\t\t\t\t&\t$(^\\circ)$\t\t\t\t\t\\\\ \\midrule\n1\t&\t0\t&\t$\\overline{1}$\t&\t0*\t&\t0.279\t&\t0.277\t&\t0.01\t&\t0.0\t&\t0.0\t&\t0.0\t\\\\\n$\\overline{1}$\t&\t0\t&\t1\t&\t0\t&\t0.271\t&\t0.277\t&\t0.02\t&\t179.7\t&\t180.0\t&\t0.3\t\\\\\n1\t&\t$\\overline{1}$\t&\t0\t&\t2\t&\t0.188\t&\t0.190\t&\t0.01\t&\t70.3\t&\t70.0\t&\t0.3\t\\\\\n0\t&\t$\\overline{1}$\t&\t1\t&\t2\t&\t0.188\t&\t0.190\t&\t0.01\t&\t110.9\t&\t110.0\t&\t0.9\t\\\\\n$\\overline{1}$\t&\t1\t&\t0\t&\t$\\overline{2}$\t&\t0.189\t&\t0.190\t&\t0.00\t&\t110.1\t&\t110.0\t&\t0.1\t\\\\\n0\t&\t1\t&\t$\\overline{1}$\t&\t$\\overline{2}$\t&\t0.189\t&\t0.190\t&\t0.00\t&\t70.4\t&\t70.0\t&\t0.4\t\\\\\n1\t&\t1\t&\t$\\overline{2}$\t&\t$\\overline{2}$\t&\t0.135\t&\t0.136\t&\t0.01\t&\t137.8\t&\t137.6\t&\t0.3\t\\\\\n$\\overline{1}$\t&\t$\\overline{1}$\t&\t2\t&\t2\t&\t0.136\t&\t0.136\t&\t0.00\t&\t42.8\t&\t42.4\t&\t0.4\t\\\\\n2\t&\t0\t&\t$\\overline{2}$\t&\t0\t&\t0.140\t&\t0.139\t&\t0.01\t&\t0.0\t&\t0.0\t&\t0.0\t\\\\\n$\\overline{2}$\t&\t0\t&\t2\t&\t0\t&\t0.138\t&\t0.139\t&\t0.01\t&\t180.0\t&\t180.0\t&\t0.0\t\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\\begin{table}\n\\begin{center}\n\\caption{A complete listing of the reflections used to index the Fast Fourier transforms of region (B) of the HRTEM image in Fig~\\ref{hrtem-23-a}. Reflections are assigned to the  \\ce{Mg4Zn7} phase, with beam direction, $\\boldsymbol{B}=[010]$. The table lists the measured ($d$)  and calculated  ($d^\\prime$) interplanar spacings for a given reflection. The relative difference between these two values is shown by the value $|1-(d\/d^\\prime)|$. Interplanar angles are measured with respect to the 206 reflection (indicated by an asterisk). The difference between the measured and expected angles ($\\Delta\\theta$) is also provided.\\label{tab-fft-hrtem-23b}}\n\\begin{tabular}{c@{ }c@{ }c@{ }ccccrrr}\n\\toprule\n\t&\t\t&\t\t&\t\\multicolumn{3}{c}{$d_{hkl}$}\t\t\t\t\t&\t\\multicolumn{3}{c}{Angle}\t\t\t\t\t\\\\ \n\\cmidrule(r){4-6}\\cmidrule(l){7-9} \n\t&\t\t&\t\t&\tMeas.($d$)\t&\tExpect.($d^\\prime$\t)&\t\t&\tMeas.\t&\tExpect.\t&\t$\\Delta\\theta$\t\\\\\nh\t&\tk\t&\tl\t&\t(\\AA)\t&\t(\\AA)\t\t&\t$|1-(d\/d^\\prime)|$\t&\t\t$(^\\circ)$\t\t\t&\t$(^\\circ)$\t\t \t\t&\t$(^\\circ)$\t\t\t\t\t\\\\ \\midrule\n\n2\t&\t0\t&\t6*\t&\t0.222\t&\t0.228\t&\t0.00\t&\t0\t&\t0.0\t&\t0\t\\\\\n$\\overline{4}$\t&\t0\t&\t6\t&\t0.222\t&\t0.244\t&\t0.06\t&\t33.2\t&\t32.2\t&\t1.0\t\\\\\n4\t&\t0\t&\t$\\overline{6}$\t&\t0.220\t&\t0.244\t&\t0.07\t&\t148.1\t&\t147.8\t&\t0.3\t\\\\\n$\\overline{2}$\t&\t0\t&\t$\\overline{6}$\t&\t0.218\t&\t0.228\t&\t0.02\t&\t179.5\t&\t180.0\t&\t0.5\t\\\\\n12\t&\t0\t&\t$\\overline{3}$\t&\t0.213\t&\t0.211\t&\t0.04\t&\t97.8\t&\t92.6\t&\t5.2\t\\\\\n$\\overline{12}$\t&\t0\t&\t3\t&\t0.210\t&\t0.211\t&\t0.02\t&\t83.0\t&\t87.4\t&\t4.4\t\\\\\n8\t&\t0\t&\t5\t&\t0.178\t&\t0.194\t&\t0.05\t&\t23.4\t&\t26.6\t&\t3.2\t\\\\\n$\\overline{8}$\t&\t0\t&\t$\\overline{5}$\t&\t0.176\t&\t0.194\t&\t0.06\t&\t156.1\t&\t153.4\t&\t2.7\t\\\\\n$\\overline{8}$\t&\t0\t&\t7\t&\t0.167\t&\t0.194\t&\t0.11\t&\t45.4\t&\t46.8\t&\t1.4\t\\\\\n8\t&\t0\t&\t$\\overline{7}$\t&\t0.164\t&\t0.194\t&\t0.12\t&\t135.5\t&\t133.2\t&\t2.2\t\\\\\n12\t&\t0\t&\t0\t&\t0.192\t&\t0.211\t&\t0.06\t&\t63.7\t&\t67.4\t&\t3.6\t\\\\\n$\\overline{12}$\t&\t0\t&\t0\t&\t0.191\t&\t0.211\t&\t0.07\t&\t116.6\t&\t112.6\t&\t3.9\t\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}\n\\begin{center}\n\\caption{A complete listing of the reflections used to index the Fast Fourier transforms of region (B) of the HRTEM image in Fig~\\ref{hrtem-n20}. The reflections are assigned to the \\ce{MgZn2} phase, with beam direction, $\\boldsymbol{B}=[0001]$. The table lists the measured ($d$)  and calculated  ($d^\\prime$) interplanar spacings for a given reflection. The relative difference between these two values is shown by the value $|1-(d\/d^\\prime)|$.     Interplanar angles are measured with respect to the $\\overline{1}2\\overline{1}0$ reflection  (indicated by an asterisk). The difference between the measured and expected angles ($\\Delta\\theta$) is also provided. \\label{tab-fft-_N20_6}}\n\\begin{tabular}{c@{ }c@{ }c@{ }ccccrrr}\n\\toprule\n\t&\t\t& \t&\t\t&\t\\multicolumn{3}{c}{$d_{hkl}$}\t\t\t\t\t&\t\\multicolumn{3}{c}{Angle}\t\t\t\t\t\\\\ \n\\cmidrule(r){5-7}\\cmidrule(l){8-10} \n\t&\t\t& &\t\t&\tMeas.($d$)\t&\tExpect.($d^\\prime$) &\t\t&\tMeas.\t&\tExpect.\t&\t$\\Delta\\theta$\t\\\\\nh\t&\tk\t&\t$\\cdot$\t&l\t&\t(\\AA)\t&\t(\\AA)\t\t&\t$|1-(d\/d^\\prime)|$\t&\t\t$(^\\circ)$\t\t\t&\t$(^\\circ)$\t\t\t\t&\t$(^\\circ)$\t\t\t\t\t\\\\ \\midrule\n$\\overline{1}$\t&\t2\t&\t$\\overline{1}$\t&\t0*\t&\t0.259\t&\t0.260\t&\t0.00\t&\t0.3\t&\t0.0\t&\t0.3\t\\\\\n1\t&\t1\t&\t$\\overline{2}$\t&\t0\t&\t0.226\t&\t0.260\t&\t0.13\t&\t63.7\t&\t60.0\t&\t3.7\t\\\\\n2\t&\t$\\overline{1}$\t&\t$\\overline{1}$\t&\t0\t&\t0.227\t&\t0.260\t&\t0.13\t&\t116.5\t&\t120.0\t&\t3.5\t\\\\\n1\t&\t$\\overline{2}$\t&\t1\t&\t0\t&\t0.252\t&\t0.260\t&\t0.03\t&\t177.1\t&\t180.0\t&\t2.9\t\\\\\n$\\overline{1}$\t&\t$\\overline{1}$\t&\t2\t&\t0\t&\t0.222\t&\t0.260\t&\t0.14\t&\t118.3\t&\t120.0\t&\t1.8\t\\\\\n$\\overline{2}$\t&\t1\t&\t1\t&\t0\t&\t0.237\t&\t0.260\t&\t0.09\t&\t64.5\t&\t60.0\t&\t4.5\t\\\\\n0\t&\t3\t&\t$\\overline{3}$\t&\t0\t&\t0.147\t&\t0.150\t&\t0.02\t&\t32.6\t&\t30.0\t&\t2.6\t\\\\\n3\t&\t0\t&\t$\\overline{3}$\t&\t0\t&\t0.133\t&\t0.150\t&\t0.11\t&\t89.7\t&\t90.0\t&\t0.3\t\\\\\n3\t&\t$\\overline{3}$\t&\t0\t&\t0\t&\t0.143\t&\t0.150\t&\t0.05\t&\t146.0\t&\t150.0\t&\t4.0\t\\\\\n0\t&\t$\\overline{3}$\t&\t3\t&\t0\t&\t0.144\t&\t0.150\t&\t0.04\t&\t147.9\t&\t150.0\t&\t2.1\t\\\\\n$\\overline{3}$\t&\t0\t&\t3\t&\t0\t&\t0.132\t&\t0.150\t&\t0.12\t&\t91.7\t&\t90.0\t&\t1.7\t\\\\\n$\\overline{3}$\t&\t3\t&\t0\t&\t0\t&\t0.149\t&\t0.150\t&\t0.01\t&\t34.9\t&\t30.0\t&\t4.9\t\\\\\n$\\overline{2}$\t&\t4\t&\t$\\overline{2}$\t&\t0\t&\t0.130\t&\t0.130\t&\t0.00\t&\t5.5\t&\t0.0\t&\t5.5\t\\\\\n2\t&\t$\\overline{4}$\t&\t2\t&\t0\t&\t0.130\t&\t0.130\t&\t0.00\t&\t178.5\t&\t180.0\t&\t1.5\t\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\\section{Introduction}\n\nSevere plastic deformation (SPD) comprises a suite of techniques which impose high plastic strain on a material in order to reduce the grain size to the micron or nanometer regime. Such grain refinement substantially increases the yield stress via the Hall-Petch effect, while simultaneously improving the ductility \\cite{Yamashita2001,DingChang2009,LiWei2011}, and can lead to high-strain rate superplasticity \\cite{Komura1998}. This desirable combination of properties has generated great interest in SPD techniques, and particularly in equal-channel angular extrusion (ECAE) and high-pressure torsion (HPT) which impose high strains with no net shape change. \n\nThe ultrafine grain sizes that develop during HPT are inherently thermodynamically unstable \\cite{WangIwahashi1996} and restriction of grain growth during and after processing is important in order to retain the improved mechanical properties. Grain growth occurs more rapidly and at lower temperatures than usual due to the high defect densities and the resulting ultra-fast atomic diffusion that occurs during SPD \\cite{Amouyal2007,Divinski2010,Ribbe2009a}. During HPT, dynamic grain growth occurs even without external heating and results in ``saturation'' where the rate of dynamic grain growth balances that of strain-induced recrystallisation and a further increase in strain will not lead to a reduction of grain size.\n\nOne approach used to restrict grain growth is the use of fine precipitates to pin grain boundaries. For example, fine-scale precipitation of Al-Sc particles has been shown to reduce the rate of grain growth of ECAE Al-Mg-Sc \\cite{Komura1998} leading to high-strain rate superplasticity. More recently it was demonstrated that grain sizes of approximately 140\\,nm could be achieved in HPT-deformed Mg-Zn \\cite{MengRosalie2014}. This is a much more refined grain size than in pure Mg, which reaches saturation at a grain size of  $\\sim1\\,\\mu$m \\cite{EdalatiHpt2011}. The difference in grain size was attributed to the presence of fine-scale zinc-rich grain boundary precipitates that were observed in high angle annular dark field (HAADF) STEM images after 20 revolutions of HPT deformation.\n\nIn this work we identify the precipitates formed during HPT of a Mg-Zn alloy using high resolution transmission electron microscopy (HRTEM), and quantify the precipitate size distribution using small angle X-ray scattering (SAXS). The complementary TEM and SAXS methodology has recently been shown to be a valuable tool for quantifying particle size and volume fraction in similar alloys \\cite{RosaliePauw2014} and here we demonstrate its applicability to studies of  precipitation in SPD materials.\n\n\\section{Experimental details}\n\nThe experiments were conducted on disc-shaped samples (10\\,mm diameter, $\\sim$0.85\\,mm height) cut in cross-section from an extruded rod of a Mg-3.4at.\\%Zn alloy (The alloy casting and prior heat treatment is described in detail by Rosalie and Pauw \\cite{RosaliePauw2014}). Discs were solution-treated at 300$^\\circ$C for 1\\,h in an Ar atmosphere and quenched in ambient temperature water. They were subsequently deformed at room temperature by HPT. Each disc was compressed with an applied pressure of 5\\,GPa. This load was maintained for 30\\,s before and during the application of rotational strain at a rotation speed of 1\\,RPM. Previous studies on a \\ce{Zr50(Cu , Al)50} alloy under similar deformation conditions found that frictional heating only  increased the anvil temperature adjacent to the sample to $\\sim$306\\,K \\cite{MengTsuchiya2013}. The temperature increase for the present material is expected to be significantly less and insufficient to facilitate precipitation during HPT. The height of the discs after HPT treatment was $\\sim$0.55\\,mm.\n\nTEM foils were prepared from two HPT discs, one of which was subjected to 3\\,HPT rotations (N=3), and the second subjected to 20 rotations (N=20). These conditions were selected as N$>$1 was required to give a homogeneous hardness distribution and microstructure \\cite{MengRosalie2014}, thus ensuring that the TEM foils are representative of the disc as a whole. The N=3 condition corresponds to the peak hardness condition, whereas the N=20 sample was in the strain-saturated condition \\cite{MengRosalie2014}. The samples were prepared for TEM by mechanical grinding and polishing, followed by dimple grinding and thinning to perforation by precision ion polishing. Foils were examined using JEOL 2100 and 2100F microscopes, operating at 200\\,kV. The precipitate phases were identified from Fast Fourier transforms of high-resolution images. A Hann filter was used to remove high frequency noise from the region of interest before performing the FFT.\n\nSAXS analysis was performed on samples in four different states: one sample preceding the HPT treatment (i.e. solution-treated, ``ST''), one where the sample was compressed but not rotated (``Compressed'', or N=0), one sample after 1\\,HPT rotation (N=1), and one at N=20. The samples were thinned to approximately 0.1\\,mm, and measured for 21600\\,s each. SAXS measurements were performed on a Bruker Nanostar instrument with a chromium target producing 5.4\\,keV photons, and a sample-to-detector distance of 1.05m, resulting in a Q-range coverage of $0.05\\leq$Q$\\leq 1.3\\, \\mbox{nm}^{-1}$ (with $Q=4\\pi\/\\lambda \\sin(\\theta)$, where $2\\theta$ denotes the scattering angle, and $\\lambda$ the X-ray photon wavelength). A photon-counting wire array (delay line) detector was used to collect the scattered photons.\n\nCollected images were corrected for image distortion using Bruker's ``2D SAXS'' software supplied with the instrument. The following corrections were subsequently performed using an in-house developed data correction procedure (see \\cite{Pauw2013a} for the details of each correction step): data read-in corrections (DS), darkcurrent (DC), pixel masking (MK), flatfield (FF), time (TI), flux (FL), transmission (TR), sample self-absorption (SA), spherical distortion (SP), background (BG), and thickness (TH), followed by a data integration in logarithmically-spaced Q-bins. Uncertainty estimates, used as weights in the fitting procedure, have been determined as the largest of either the propagated Poisson uncertainties, 1\\% of the intensity in the bin, or the standard error in the bin. \n\nAfter correction, the data was subjected to a Monte Carlo-based size distribution determination procedure assuming spherical scatterers \\cite{Pauw2013}, whose size (radii) range is defined by the Q-range to between 2.5 and 60\\,nm. On average, the scattering patterns fit to within the uncertainty of the data over the entire Q-range (i.e. to $\\chi^2_r \\leq 1$). \n\n\\section{Results and discussion}\n\n\\subsection{TEM}\n\nTEM observations on samples deformed to N=3 show narrow regions of stronger atomic contrast at the grain boundaries. These darker regions were typically 5--20\\,nm in calliper diameter (see Fig.~\\ref{hrtem-15}, in agreement with SAXS results) with widths of 5-10\\,nm, no clearly defined facets, and extended along the grain boundary as a film. Figure~\\ref{hrtem-23-a} shows a HRTEM image of the grain boundary region in this condition.  The two Mg grains present ($A$, left) and  ($C$, right)  are separated by a darker region  ($B$) at the grain boundary. A FFT of region $B$ is shown in Fig~\\ref{hrtem-23-fft-Mg4Zn7} with reflections assigned to the  \\ce{Mg4Zn7} phase. This phase generally forms in the shape of high aspect ratio rods in isothermally aged Mg-Zn alloys \\cite{Gao2007,Singh2007}. \\ce{Mg4Zn7} has been reported at grain boundaries in conventionally cast Mg-Zn alloys  \\cite{GaoInter2007} where it is present in Mg--\\ce{Mg4Zn7} lamellae. The beam direction, $\\boldsymbol{B}$, is  parallel to the ${[}2\\overline{4}2\\overline{3}{]}$ zone axis in ($A$) and is close to the [010]  zone axis of \\ce{Mg4Zn7} in ($B$). This orientation relationship is not among those previously reported  for this phase in Mg \\cite{Singh2007} and lacks the usual alignment of the $[010]_\\ce{Mg4Zn7}$ ($d=0.52$\\,nm) parallel to $[002]_\\ce{Mg}$ ($d=0.26$\\,nm). The adjacent grain, (C), is not aligned along a rational, low-index direction. The insets show an enlargement of the matrix (left) and a simulated HRTEM image (right) for defocus=70\\,nm, thickness=70\\,nm.\nThe absence of  i) a favourable orientation relationship, ii) clear crystallographic facets and and iii) the usual rod-like morphology suggest that the precipitate nucleated at the grain boundary rather than intragranularly.  A full listing of the reflections used to index the phases in regions $(A, B)$ is set out in Tables~\\ref{tab-fft-hrtem-23a} and \\ref{tab-fft-hrtem-23b} in the supplementary information (SI).   \n\nMore extensive deformation results  in the grain-boundary precipitates coalescing into roughly equiaxed particles. A typical micrograph showing this condition is shown in Figure~\\ref{N20_10}. The larger, dark regions are Mg grains in a strongly diffracting condition. Precipitates are concentrated at the grain boundaries and triple points, with two examples indicated by arrows. \nIt was possible to obtain atomic resolution images of some of the precipitates, as shown in Figure~\\ref{hrtem-n20}.\nThe Mg matrix (lower right) is close to a two beam condition with $\\boldsymbol{g}=0002$. The precipitate (centre and left region)  has stronger atomic contrast.\nFigure~\\ref{hrtem-n20-fft} shows the FFT of the precipitate region which was assigned to the \\ce{MgZn2} phase, with beam direction $[0001]$. A full listing of the reflections used to index the phase is set out in Table~\\ref{tab-fft-_N20_6} in the SI.   \n\n\\begin{figure}\n\\begin{center}\n\t\\begin{center}\n\t\\makebox[2ex]{}\n\t\\hfill\n\tTEM\n\t\\hfill\\\n\t\\hfill\n\tHRTEM\n\t\\hfill\\\n\t\\hfill\n\tFFT\n\t\\hfill\\\n\t\\hfill\\\n\t\\end{center}\n\t\n\t\\raisebox{0.15\\textwidth}[0pt][0pt]{\n\t\t\\begin{rotate}{90}\n\t\t\tN=3\n\t\t\\end{rotate}}\n\t\\hfill\n       \\subfigure[\\label{hrtem-15}]{\\includegraphics[width=0.3\\textwidth]{Tem-2014-04-11_15}}\\hfill\n       \\subfigure[ \\label{hrtem-23-a}]{\\includegraphics[width=0.3\\textwidth]{Tem-2014-04-11_23b}}\n\t\\hfill\n\t\\subfigure[\\label{hrtem-23-fft-Mg4Zn7}]{\\includegraphics[width=0.3\\textwidth]{FFT_Tem-2014-04-11_23-center-assign}}\n\t\\hfill\\\n\n\t\\raisebox{0.15\\textwidth}[0pt][0pt]{\n\t\t\\begin{rotate}{90}\n\t\t\tN=20\n\t\t\\end{rotate}}\n\t\\hfill\n\\subfigure[\\label{N20_10}]{\\includegraphics[width=0.3\\textwidth]{MgZnN20_10b}}\n\t\\hfill\n\t\\subfigure[\\label{hrtem-n20}]{\\includegraphics[width=5cm]{N20_6a}}\n\t\\hfill\n\t\\subfigure[\\label{hrtem-n20-fft}]{\\includegraphics[width=5cm]{FFT_N20_6-prec}}\n\t\\hfill\\\n\n\t\\caption{Electron micrographs of grain boundary precipitates in HPT Mg-Zn. Figures (a)--(c) are from samples deformed to N=3 and (d)--(f) are for N=20. (a,d) show typical diffraction contrast images of grain boundary precipitation. High resolution images are presented in Figs.~(b,e) with FFTs of the precipitate-containing regions presented in (c) and (f), respectively. For N=3 (c) the FFT is indexed to the \\ce{Mg4Zn7} phase  (circles) and Mg matrix (squares). For N=20 (f) the FFT is indexed to the \\ce{MgZn2} phase. A Hann filter was used to remove high frequency noise from the regions of interest in (b,e) before performing the FFT.\n \\label{hrtem-23}}\n\\end{center}\n\\end{figure}\n\n\nThe TEM observations of N=3 samples suggest that the initial grain boundary precipitates are comprised of the \\ce{Mg4Zn7} phase. Further solute diffusion of Zn to the grain boundaries continues during HPT deformation, resulting in the formation  of the \\ce{MgZn2} phase. Although first principles calculations show \\ce{MgZn2}  has a lower formation enthalpy for Zn $>$66at.\\% \\cite{Xie2013} the difference is minimal and microdomains of the \\ce{Mg4Zn7} and \\ce{MgZn2} phases can co-exist within individual precipitates  \\cite{RosalieSingh2011,SinghRosalie2010,RosalieSomekawa2010} giving rise to a continuous spectrum of compositions. \nIt is probable that such mixed-phase precipitates are present in the HPT material but extensive analysis would be required to confirm this.\n\n\\subsection{SAXS}\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=\\textwidth]{SAXSyStuff}\n\t\t\\caption{Left: Small-angle scattering data and fits (points with uncertainties, and solid lines, respectively). Right: Size distributions of scatterers determined from fitting of the scattering data. Error bars indicate $\\pm$1\\,standard deviation (SD). \\label{fig-saxs}}\n\t\\end{center}\n\\end{figure}\n\nThe scattering behaviour of the ST alloy shows little evidence for any structure in the measurable size range, and only shows the $I\\propto Q^{-4}$ scattering from larger scattering structures (Fig.~\\ref{fig-saxs}). After compression (N=0), there are slight but measurable changes in the scattering. Major changes occur upon HPT rotation, with strong, almost identical shoulders appearing in the scattering patterns for the N=1 and N=20 samples. \n\nThe Monte Carlo method can retrieve a size distribution from the scattering patterns provided that a general scatterer shape is defined \\cite{Pauw2013}. Given the diversity of grain boundary precipitate morphologies observed in TEM, a generally applicable spherical shape was chosen for the characterisation of the precipitate dimensions. Application of this method shows that the shoulders in the scattering patterns for both HPT samples can be interpreted as scattering originating from a broad distribution of objects, with radii ranging from about 2.5--20\\,nm, and maximum volume-weighted radii of around 3--5\\,nm. This size is consistent with the particle sizes observed via TEM. Due to technical limitations, the volume fractions shown in Fig.~\\ref{fig-saxs} are not on absolute scale. They are, however, internally consistent and therefore allow for comparisons between samples  \\footnote{Analysis of simulated data indicates that the determined volume fraction remains largely unchanged provided the aspect ratio of globular scatterers is less than 1:5.}. The volume fraction and size of detected precipitates in the N=1 and N=20 samples exhibit striking similarity. \n\n\\section{Conclusions}\nIn conclusion, HRTEM and SAXS provided direct evidence of precipitation in an HPT-deformed Mg-3.4at.\\%Zn alloy without post-deformation heat treatment as well as the characterisation of the precipitates. Both techniques show fine-scale precipitates after deformation with dimensions between 2.5--20\\,nm, with the precipitate radii centred at around 3--5\\,nm. HRTEM observations indicate that precipitation initially takes the form of a grain boundary film consisting  of the \\ce{Mg4Zn7} phase.  \n\n\n\nSAXS shows little difference in the precipitate size and volume fraction in materials subjected to 20 rotations, where the alloy is in the the strain-saturated condition. However solute segregation to the grain boundaries continues during deformation, as evidenced by HRTEM showing a transition from a microstructure with a grain boundary \\ce{Mg4Zn7} film to one with more equiaxed particles of the \\ce{MgZn2} phase.\n\n\\section*{Author contributions}\nThe HPT samples were prepared by FM. JMR and FM performed the SEM and TEM investigations, with the analysis of the HRTEM micrographs being done by JMR. BRP performed the SAXS measurements, corrections and analysis. JMR and BRP wrote the majority of the manuscript, with contributions from FM and KT and with feedback and comments from all authors. HM and HK are responsible for the SAXS instrument used in the studies. KT supervised the investigations presented herein.\n\n\\section*{Acknowledgements}\nThe authors would like to thank Dr. H. Somekawa for supplying the extruded material used in the investigations. This work was supported in part by a Grant-in-Aid for Scientific Research on Innovative Area, \"Bulk Nanostructured Metals\", through MEXT, Japan (contract no. 22102004). \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\nMarkov Chain Monte Carlo algorithms (MCMC) enjoy great success in simulating \nmany theories from the Ising model up to Lattice QCD. \nAlbeit the potential, MCMC has a hard time whenever the action \nbecomes complex-valued due to the associated Boltzmann weight loosing its \ninterpretability as probability distribution. \n\nUsing MCMC, we focus on the Hubbard model capturing electronic properties \nof systems with strongly interacting electrons propagating on a fixed \nspatial lattice of ions.\nExamples for such systems are carbon nano structures like Graphene and \nFullerene $C_n$.\nIn the Hubbard model the sign problem is observed at finite chemical potential as well as on \nnon-bipartite lattices\\footnote{ \n    Bipartite describes lattice geometries at which the sites can be two \n    coloured such that no neighbouring sites have the same colour.\n    For example, the square is bipartite while the triangle is non-bipartite.\n}.\nReweighting can treat the complex-valued Boltzmann weight though, at the \nsame time, introducing an exponentially hard to estimate normalization.\n\nDeforming the region of integration onto Manifolds with an almost constant imaginary \naction showed great promise in reducing the sign problem substantially~\\cite{\nKashiwa:2018vxr, alexandru2020complex, DetmoldPathIntegral, Detmold:2021ulb}.\nPractically, this deformation requires numerical integration of differential \nequations which becomes infeasible for larger systems. \nWe aim to identify efficient Neural Network architectures to learn such  \nbeneficial deformations.\nThis removes the cost of numerically integrating configurations and enables \nsimulations of large systems with a sign problem beyond the standard reweighting\napproach.\n\nIn this proceedings, we collect material from our earlier publications~\\cite{leveragingML,rodekampMitigating} and a master thesis~\\cite{christophThesis}.\nThe manuscript is organized in the following way.\nIn section~\\ref{sec:Formalism} a brief introduction to the Hubbard model and \nthe tested system is presented along a short discussion of the sign problem.\nThis discussion is then followed by the definition of the neural network \narchitectures as published in~\\cite{leveragingML,rodekampMitigating}.\nFurther, in section~\\ref{sec:NumericalResults} correlator results are presented \nand we discuss the obtained charge density of one of the larger systems.\n %\n\\label{sec:Introduction}\n\n\\section{Formalism}\\label{sec:Formalism}\nThe Hubbard model~\\cite{Hubbard1963} describes a fixed spatial lattice $X$ of ions on which electrons can move and interact.\nIts Hamiltonian, in particle-hole basis, is\n\\begin{equation}\n    \\nonumber\n    \\mathcal{H}\\left[K, V, \\mu\\right]\n    =\n           - \\sum_{x,y\\in X} \\left( p_x^\\dagger K^{xy} p_y - h_x^\\dagger K^{xy} h_y \\right)\n         + \\frac{1}{2} \\sum_{x,y\\in X} \\rho_x V^{xy} \\rho_y\n            + \\mu \\sum_{x\\in X} \\rho_x,\n    \\label{eq:hubbard-hamiltonian}\n\\end{equation}\nwhere the amplitudes in $K$ encode the hopping of fermionic particles $p$ \nand holes $h$, the potential $V$ describes the interactions between charges\n\\begin{equation}\n    \\rho_x = p^{\\dagger}_x p_x - h^{\\dagger}_x h_x\n    \\label{eq:net-charge}\n\\end{equation}\nand the chemical potential $\\mu$ incentivizes charge.\nBy adjusting the hopping and lattice symmetry $K$ as well as the interaction $V$ \nthis model can describe a wide variety of physical systems.\nIn the following investigation, five systems are considered as displayed in figure~\\ref{fig:systems}.\nThe 2 site system describes one unit cell of the honeycomb structure used \nfor Graphene type models which we successively built up with the 8 and 18 site ones.\nSecondly, we present preliminary results for fullerenes $C_{20}$ and $C_{60}$ at zero chemical potential.\nIn all cases $K$ encodes nearest-neighbor hopping\nand we assume an on-site interaction,\n\\begin{align}\n    K &= \\kappa \\delta_{\\langle xy \\rangle}\n    &\n    V &= U \\delta_{xy}.\n\\end{align}\nIn figure~\\ref{fig:systems} the sites, i.e.\\@ ions and their nearest neighbor connections are depicted. \nLines stretching out display periodic boundary of the spatial lattice (suppress in the 18 site case).\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=1] {figure\/systemPlots.pdf}\n    \\caption{\n        Showing the different geometries considered in this proceedings. \n        Each vertex represents an ion and each (dashed) line depicts the \n        nearest-neighbor hopping that is allowed by the Hubbard model. \n        Dashed lines indicate periodic boundary condition where possible.\n    }\n    \\label{fig:systems}\n\\end{figure}\n %\n\\subsection{Simulation Setup}\nCalculating observables follows the standard procedure~\\cite{browerHybridMonteCarlo2012} \nof evaluating the thermal trace.\nAfter trotterizing into $\\ensuremath{\\mathrm{N}_{\\mathrm{t}}}$ time slices, inserting Grassmannian resolutions of\nthe identity and linearizing the interaction via the Hubbard-Stratonovich \ntransformation~\\cite{Hubbard1959} the Hamiltonian is transformed into the action\n\\begin{equation}\n    S\\left[\\Phi \\,\\vert \\, K, V, \\mu \\right]\n       =\n            -   \\log\\det{ M\\left[\\Phi\\,\\vert\\, K,\\mu\\right] \\cdot M\\left[-\\Phi\\,\\vert\\, -K,-\\mu\\right] }\n         +   \\frac{1}{2} \\sum_{t=0}^{\\ensuremath{\\mathrm{N}_{\\mathrm{t}}}-1}\\sum_{x,y\\in X} \\Phi_{tx} \\delta V^{-1\\,xy} \\Phi_{ty},\n    \\label{eq:hubbard-action}\n\\end{equation}\nwhere $\\Phi \\in \\ensuremath{\\mathbb{R}}^{\\abs{\\Lambda}}$ is an auxiliary field on the \nspacetime lattice $\\Lambda = [0, N_t-1]\\otimes X$ and $\\delta=\\beta\/N_t$ \nis the (temporal) lattice spacing controlled by the inverse temperature $\\beta$.\nThe fermion matrix is not uniquely defined on the lattice, we choose the exponential discretization~\\cite{Wynen:2018ryx}\n\\begin{equation}\n    M\\left[\\Phi\\,\\vert\\, K,\\mu\\right]_{x't';xt} \n    =   \n        \\delta_{x'x}\\delta_{t't}\n    -  \\left( e^{\\delta(K + \\mu)} \\right)_{x'x} e^{+ i \\Phi_{xt}} \\mathcal{B}_{t'}\\delta_{t'(t+1)}\n\\end{equation}\nwhere $\\mathcal{B}$ encodes the anti-periodic boundary conditions in time.\nFor bipartite systems we may replace $-K$ with $+K$ in the holes' fermion \nmatrix~\\cite{browerHybridMonteCarlo2012}.\nThe thermal trace for this is expressed as path integral\n\\begin{equation}\n    \\expval{\\ensuremath{\\mathcal{O}}}\n    =\n    \\frac{1}{\\ensuremath{\\mathcal{Z}}} \\int  \\DD{\\Phi} e^{- S\\left[\\Phi\\right]} \\ensuremath{\\mathcal{O}}\\left[\\Phi\\right]\n    =\n    \\int  \\DD{\\Phi} p_S\\left[\\Phi\\right] \\ensuremath{\\mathcal{O}}\\left[\\Phi\\right]\n    \\label{eq:true-expectation-value}\n\\end{equation}\nFor cases of real action we can apply MCMC to generate $\\ensuremath{\\mathrm{N}_{\\mathrm{conf}}}$ configurations according to the Boltzmann distribution $p_S\\left[\\Phi\\right] $\nand estimate observables \\eqref{eq:true-expectation-value} by an unweighted expectation value.\nIf the action is complex valued we apply reweighting\n\\begin{align}\n        \\expval{\\ensuremath{\\mathcal{O}}}\n        &   = \\frac{\\expval{e^{-i\\Im{S}}\\ensuremath{\\mathcal{O}}}_{\\Re{S}}}{\\expval{e^{-i \\Im{S}}}_{\\Re{S}}}\n            = \\frac{1}{\\Sigma} \\expval{ e^{-i \\Im{S}} \\ensuremath{\\mathcal{O}}}_{\\Re{S}}.\n        \\label{eq:reweighting}\n\\end{align}\nand sample configurations according to the Boltzmann distribution under the real part of the action.\nIt has been shown~\\cite{berger2021complex} that an effective number of configurations\n\\begin{equation}\n    \\ensuremath{\\mathrm{N}_{\\mathrm{conf}}}^{\\textrm{eff}} = \\abs{\\Sigma}^2 \\cdot \\ensuremath{\\mathrm{N}_{\\mathrm{conf}}} \\label{eq:effective-Nconf}\n\\end{equation}\ncontrols the scaling of statistical errors $\\sim \\left(\\ensuremath{\\mathrm{N}_{\\mathrm{conf}}}^{\\textrm{eff}}\\right)^{-1\/2}$.\nThis translates the sign problem to the ability of calculating the denominator $\\Sigma$, \ni.e. the statistical power, reliably~\\cite{berger2021complex,leveragingML,PhysRevD.93.014504,mori2018lefschetz}.\n\\\\\n \nA promising approach to mitigate, or even remove, the sign problem is to \ndeform the region of integration $\\Phi \\in \\ensuremath{\\mathcal{M}}_{\\ensuremath{\\mathbb{R}}} = \\ensuremath{\\mathbb{R}}^{\\abs{\\Lambda}}$ onto a manifold \non which the imaginary part of the action is (nearly\\footnote{\n    If $\\Im{S}\\approx const$, the statistical power $\\abs{\\Sigma} = \n    \\abs{\\expval{e^{i\\Im{\\ensuremath{S_{\\mathrm{eff}}}}}}} \\approx \\abs{\\expval{e^{i \\cdot const}}} \\approx 1$\n    yielding nearly no reduction in effective number of configurations \n    $\\ensuremath{\\mathrm{N}_{\\mathrm{conf}}}^{\\mathrm{eff}} \\approx \\ensuremath{\\mathrm{N}_{\\mathrm{conf}}}$.\n}) constant~\\cite{PhysRevD.86.074506,alexandru2020complex}\n, $\\ensuremath{\\mathcal{M}} = \\left\\{ \\Phi \\in \\ensuremath{\\mathbb{C}}^\\abs{\\Lambda} \\, \\vert \\, \\Im {S\\left[\\Phi\\right]} = const. \\right\\}$.\nIf $\\ensuremath{\\mathcal{M}}$ is in the same homology class as $\\ensuremath{\\mathcal{M}}_{\\ensuremath{\\mathbb{R}}}$\nan analogue of the Cauchy integral theorem ensures that the observables are unchanged.\nParametrizing fields $\\Phi(\\phi)\\in\\ensuremath{\\mathcal{M}}$ then adds a Jacobian defining the \nused effective action\n\\begin{equation}\n    \\ensuremath{S_{\\mathrm{eff}}}\\left[\\phi\\right] = S[\\Phi\\left(\\phi\\right)] - \\log\\det{J\\left[\\phi\\right]}, \\quad J_{ij} = \\pdv{\\Phi(\\phi)_i}{\\phi_j}\n    \\label{eq:effective-action}\n\\end{equation}\n\nThere are many strategies for picking target manifolds $\\ensuremath{\\mathcal{M}}$~\\cite{Tanizaki:2017yow}.\nOne choice is to try to approximate the Lefschetz thimbles~\\cite{PhysRevD.86.074506}.\nEach thimble contains a critical point $\\Phi_{cr}$ that is a fixed point \nof the holomorphic flow\n\\begin{equation}\n    \\frac{d\\Phi(\\tau)}{d\\tau} = \\pm\\left(\\frac{\\partial S\\left[ \\Phi(\\tau) \\right]}{\\partial \\Phi(\\tau)}\\right)^*, \\quad \\Phi(0) = \\phi\n    \\label{eq:holomorphic-flow}\n\\end{equation}\nintroducing the fictitious flow time $\\tau$.\nA thimble is the set of complexified configurations that flow to a critical \npoint under downward flow, i.e. $-$ in~\\eqref{eq:holomorphic-flow}.\nAn integrator for~\\eqref{eq:holomorphic-flow} will always be computationally \nexpensive~\\cite{alexandru2020complex,rodekampMitigating,leveragingML}.\nHowever, the non-interacting solution $\\phi = 0$ for~\\eqref{eq:holomorphic-flow}\nassuming a constant field $\\Phi_{t,x} = \\Phi_{t',x'}$ defines \na (hyper-) plane parallel to the real plane $\\ensuremath{\\mathcal{M}}_\\ensuremath{\\mathbb{R}}$ that goes through the main critical point $i \\Phi_c^0$.\nThis so called tangent plane $\\ensuremath{\\mathcal{M}}_{T} = \\{\\phi + i \\Phi_c^0 \\, \\vert \\, \\forall \\phi \\in \\ensuremath{\\mathcal{M}}_{\\ensuremath{\\mathbb{R}}}\\}$\nshowed promise in sufficiently mitigating the sign problem, at least for smaller systems~\\cite{PhysRevD.93.014504,Alexandru:2018ddf, Warrington:2019kzf, leveragingML, rodekampMitigating}.\n\n \n\\subsubsection{Neural Network Architectures}\nTo improve beyond the tangent plane it seems plausible to \nidentify a transformation that transforms a given configuration $\\phi\\in\\ensuremath{\\mathcal{M}}_\\ensuremath{\\mathbb{R}}$ \nto a target manifold $\\tilde{\\ensuremath{\\mathcal{M}}}$ that may be closer to $\\ensuremath{\\mathcal{M}}$ than the tangent plane.\nSuch a transformation may be parametrized by a neural network \n\\begin{equation}\n    \\ensuremath{\\mathrm{SHIFT}}: \\ensuremath{\\mathcal{M}}_\\ensuremath{\\mathbb{R}} \\to \\tilde{\\ensuremath{\\mathcal{M}}}, \\, \\phi \\mapsto \\phi + i\\left(\\Phi_c^0 + NN\\left( \\phi \\right)\\right).\n    \\label{eq:SHIFT}\n\\end{equation}\nFor the neural network part $NN$ we pick two linear layers $\\omega \\phi + b$ with real trainable weights $\\omega$ and biases $b$\nwhich are separated by a leacky-ReLU. \nAs the effective action~\\eqref{eq:effective-action} suggests the defining \ntransformations Jacobian needs to be computed \n\\begin{equation}\n    \\log\\det{J_{\\ensuremath{\\mathrm{SHIFT}}}[\\phi]} = \\log\\det{\\mathds{1} + i \\pdv{NN\\left(\\phi\\right)}{\\phi} } \n\\end{equation}\nwhich requires an $\\order{V^3}$ algorithm for the determinant calculation.\nThis scaling is not feasible for large scale systems but still much cheaper \nthan applying a Runge-Kutta --- or similar algorithms --- to integrate the \nholomorphic flow equations.\n\nTo improve the scaling, we identify a neural network that has a cheaper determinant. \nOne such architectures is given by Affine Coupling Layers~\\cite{albergo2021introduction,foreman2021hmc}\nthat approximate the integrator $\\Phi(\\phi) \\approx \\ensuremath{\\mathcal{NN}}(\\phi) $\n\\begin{equation}\n    \\ensuremath{\\mathcal{NN}}_\\ell(\\Phi) =\n    \\begin{cases}\n        c_\\ell\\left[ \\Phi_A, \\, \\Phi_B \\right] & A_\\ell \\text{ components} \\\\\n        \\Phi_B & B_\\ell \\text{ components}\n    \\end{cases}\n    \\label{eq:ACL-def}\n\\end{equation}\nHere $A$ and $B$ are layer-specific partitions of the input vector $\\Phi$ of equal cardinality $\\nicefrac{1}{2}\\abs{\\Lambda}$. $\\Phi_{A,B}$ are the components of the input belonging to the indicated partition.\nWe apply the affine coupling~\\cite{albergo2021introduction}\n\\begin{equation}\n    c_\\ell\\left[\\Phi_A, \\Phi_B \\right] = e^{m_\\ell\\left(\\Phi_B\\right)} \\odot \\Phi_A + a_\\ell\\left(\\Phi_B\\right).\n    \\label{eq:affine-coupling}\n\\end{equation}\nThe coupling functions \n$m_\\ell,a_\\ell: \\mathbb{C}^{\\nicefrac{\\abs{\\Lambda}}{2}} \\to \\mathbb{C}^{\\nicefrac{\\abs{\\Lambda}}{2}}$ \nare again two linear layers separated by the non-linear Softsign function.\nTo ensure that the neural network produces a complex configuration as is required by the holomorphic flow, \nthe weights and biases need to be complex valued which is discussed in more detail in~\\cite{rodekampMitigating}.\nA single layer just transforms half of the configuration, we thus pair it up with a second layer that is set up in the same way but with the roles of $A$ and $B$ interchanged.\nThis setup allows to express the Jacobian, with $L\/2$ of these pairs, in the form\n\\begin{equation}\n    \\log{\\det{J_{\\ensuremath{\\mathcal{NN}}}(\\phi)}} = \\sum_{\\ell = 1}^{L} \\sum_{i =0}^{\\abs{A}-1} m_\\ell\\left( \\Phi_{\\ell-1}(\\phi)_B\\right)_i.\n    \\label{eq:logDetJ-NN}\n\\end{equation}\nwhich adds only an $\\order{V}$ cost to the application of the transformation.\n\nFor any of these architectures we perform Molecular Dynamics on $\\ensuremath{\\mathcal{M}}_\\ensuremath{\\mathbb{R}}$ using \na standard leapfrog algorithm and then apply the network to move onto $\\tilde{\\ensuremath{\\mathcal{M}}}$.\nAccept\/Reject is then performed according to the effective action~\\eqref{eq:effective-action}\nusing the transformed configuration from the Network and the Jacobian defined by the \nnetwork.\nThis machine learning enhanced Hybrid Monte Carlo is referred to as ML HMC.\n\n \n\\subsection{Observables}\nWe are interested in the electronic properties of a given system.\nEuclidean correlation functions of a single particle or a single hole %\n\\begin{equation}\n    C_{xy}^{\\tiny\\substack{p\\\\h}}(t) = \\expval{ p_x^\\dagger(t) p_y(0) } = \\expval{ M[\\pm\\Phi | \\pm K, \\pm\\mu ]^{-1}_{xt;y0} },\n\\end{equation}\nmomentum projected and averaged we obtain $C_{sp}(t)$~\\cite{rodekampMitigating}.\nIn the future we aim to match the parameters $\\nicefrac{U}{\\kappa},\\mu$ to real-world systems\nand extract the band-gap $C_{xy}(t) \\sim e^{-t\\cdot \\Delta E} $.\nFurther, the charge density is defined by\n\\begin{equation}\n    \\rho(\\mu) = \\frac{1}{\\abs{X}} \\sum_{x\\in X} \\expval{\\rho_x} \n            = \\frac{1}{\\abs{X}} \\sum_{x\\in X} C_{x,x}^p(0) - C_{x,x}^h(0).\n            \\label{eq:charge-density}\n\\end{equation}\nIt is point symmetric around the electric neutral half-filling point, $\\mu = 0$, due to \nparticle-hole symmetry.\nFor very large $\\mu\\to \\pm \\infty$ the charges (+) or holes (-) are favoured\nyielding a charge density of $\\pm 1$.\nQualitatively, it is expected that the system's charge \nequals integer multiples of the electric charge $n\\cdot e^{-}$ with \n$n \\in [-\\ensuremath{\\mathrm{N}_{\\mathrm{x}}},\\ensuremath{\\mathrm{N}_{\\mathrm{x}}}]$, i.e.\\@ $\\rho(\\mu) = \\nicefrac{n}{\\ensuremath{\\mathrm{N}_{\\mathrm{x}}}}$.\n\n \n\\section{Numerical Results}\\label{sec:NumericalResults}\nWe experimented with different training setups. \nForemost, supervised training using ADAM to minimize the $L1-$Loss.\nThe training data consists out of $\\order{\\num{10000}}$ pairs $(\\phi,\\Phi(\\phi))$\nobtained by a Runge-Kutta of \\nth{4} order.\nFor further details consider~\\cite{leveragingML,rodekampMitigating}.\n\n\n %\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\linewidth,height = 0.4\\textheight]{figure\/Correlators}\n\\caption{\n    Each row in this figure shows the correlators\n    of a system with 2 (upper row) and 8 (lower row) sites~\\cite{rodekampMitigating}.\n    The different columns correspond to a number of configurations, \n    $\\ensuremath{\\mathrm{N}_{\\mathrm{conf}}}\\in\\{\\num{1000},\\num{100000}\\}$, used to estimate the correlators.\n    Three methods --- ML HMC with coupling layer $\\ensuremath{\\mathcal{NN}}$ (blue), HMC on the Tangent Plane (red) and exact diagonalization (black) --- \n    are compared to show the effectiveness and correctness of the introduced \n    machine learning enhanced method. \n    The sign problem manifests as a loss of signal, \n    i.e. small number of effective configurations $\\ensuremath{\\mathrm{N}_{\\mathrm{conf}}}^{\\mathrm{eff}}$~\\eqref{eq:effective-Nconf}, \n    and greatly increases as the number of sites expands.\n    It can be seen that the ML HMC has a substantially reduced sign problem.\n}\n\\label{fig:SmallSystemCorrelators}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\linewidth,height = 0.3\\textheight]{figure\/18Site_Correlators.pdf}\n\\caption{The correlators of Graphene with 18 ions are shown~\\cite{rodekampMitigating}.\n    $\\num{100000}$ measurements are taken.\n    With this larger lattice direct diagonalization as in figure~\\ref{fig:SmallSystemCorrelators} \n    is not tractable any more hence only the two statistical methods ML HMC using the coupling network $\\ensuremath{\\mathcal{NN}}$ (blue)\n    and HMC on the tangent plane (red) are compared. \n    As expected the ML HMC improves the signal drastically.\n}\n\\label{fig:18SiteCorrelator}\n\\end{figure}\n\\begin{figure}\n    \\centering\n\\includegraphics[width = 0.9\\linewidth,height = 0.4\\textheight]{figure\/ChargeDensity.pdf}\n\\caption{\n    We computed the charge density for several values of the chemical potential $\\mu\\in \\left[0,\\,5.2\\right]$\n    for an 18 site Graphene sheet~\\cite{christophThesis}.\n    For most smaller and larger values of $\\mu$ the sign problem is small enough\n    that estimation with HMC on the tangent plane (red) is sufficient. \n    However, in the region $\\mu \\in \\left[2,3\\right]$ an ML HMC (blue) is used \n    to narrow particular values for which the sign problem becomes untraceable.\n    The features at $\\mu = 0$ and $\\mu \\to \\infty$ are captured as expected.\n    Finding the charge plateaus at $\\rho(\\mu)\\sim n$, however, is yet unavailable \n    due to the small $\\beta$.\n    The dashed line at $\\rho(\\mu)=\\nicefrac{4}{18}$ \n    indicate an expected plateau which may be surmised around $\\mu \\approx 1$.\n}\n\\label{fig:ChargeDensityScan}\n\\end{figure}\n\\begin{figure}\n    \\centering\n\\includegraphics[width = 0.9\\linewidth,height = 0.3\\textheight]{figure\/Correlators_buckyballs.pdf}\n\\caption{\n    The correlators of Fullerene $C_{20}$ (upper row) and $C_{60}$ (lower row) are shown.\n    A real plane (red) and a tangent plane (blue) HMC at \n    inverse temperature $\\beta = 8$ and $\\ensuremath{\\mathrm{N}_{\\mathrm{t}}} = 32$ time slices are compared.\n    We consider an on-site interaction $U = 3$ and zero chemical potential.\n    The, here negligible, sign problem solely stems from the non-bipartiteness \n    of the system due to the pentagonal rings.  \n    Already, at small number of configurations $\\ensuremath{\\mathrm{N}_{\\mathrm{conf}}} \\leq \\num{10000}$ the \n    signals are very good.\n}\\label{fig:buckyball-correlators}\n\\end{figure}\n %\nIn figure~\\ref{fig:SmallSystemCorrelators} correlators for systems with \n2 and 8 sites are compared using the different algorithms HMC (red) --- on the tangent plane --- \nML HMC (blue), applying the coupling network $\\ensuremath{\\mathcal{NN}}$, and exact diagonalization (black)~\\cite{rodekampMitigating}.\nWe use $\\ensuremath{\\mathrm{N}_{\\mathrm{conf}}}\\in\\{\\num{1000},\\num{100000}\\}$ to portray the effect of the \nstatistical power on the effective number of configurations. \nCorresponding statistical powers $\\abs{\\Sigma}$ can be found in~\\cite{rodekampMitigating}.\nThe system parameters --- $\\ensuremath{\\mathrm{N}_{\\mathrm{t}}} = 32$, $\\beta = 4$, $U = 4$ and $\\mu = 3$ --- are kept fix.\nThe ML HMC outperforms the HMC in terms of signal.\nThe 8 site system has a stronger sign problem to an extend that HMC retrieves no signal.\nIf a signal is obtained, both algorithms agree with the exact diagonalization.\nIn figure~\\ref{fig:18SiteCorrelator} the correlators for the system with 18 sites are displayed~\\cite{rodekampMitigating}.\nThe system is computed at $\\ensuremath{\\mathrm{N}_{\\mathrm{t}}} = 32$, $\\beta = 4$, $U=3$ and $\\mu = 3$.\nThe sign problem is much stronger than in the previous cases due to the larger volume.\nNevertheless the ML HMC extracts a good signal for the correlators. \nSimilar to the 8 site case HMC can't keep up.\nDue to the number of sites exact diagonalization is not feasible. \n\n %\nContinuing the 18 site model --- with $U = 2$, $\\beta = 5$, $\\ensuremath{\\mathrm{N}_{\\mathrm{t}}}=32$ ---\nwe want to study the charge density~\\eqref{eq:charge-density} subjected to the chemical potential. \nThis can be seen in figure~\\ref{fig:ChargeDensityScan}~\\cite{christophThesis}.\nWe compare tangent plane HMC (red) and ML HMC (blue) using \nthe SHIFT network.\nVarying the chemical potential has shown that for small and large values \nthe sign problem is mild (on the tangent plane).\nHowever, in the intermediate range of $\\mu \\in [2,3]$ the tangent plane is \nnot sufficient for a reasonable estimate, where we apply ML HMC with the SHIFT network.\nThe point at $\\mu = 2.5$ requires more attention and we plan to address it\nwith the coupling network in the future expecting much better results.\nThe expected behaviour of the charge density, $\\rho(\\mu=0) = 0$ and $\\rho(\\mu\\to\\infty) \\to 1$, \nis found numerically.\nThe dashed line exemplarily indicates an expected plateau at $\\expval{\\rho(\\mu)} = \\nicefrac{4}{18}$.\nAs it can be seen, this plateau is not fully deducible but may be surmised around $\\mu \\approx \\num{1}$.\nStudies of smaller systems, see~\\cite{christophThesis}, indicate increasing $\\beta$\nmakes these plateaus more pronounced. %\n\n \n\nTo probe our method in physically more relevant systems than the 18 Site Graphene sheet, \nwe started an investigation of Fullerene $C_{20}$ and $C_{60}$.\nThe correlators at $\\ensuremath{\\mathrm{N}_{\\mathrm{t}}} = 32$, $\\beta = 8$, $U = 3$ and zero chemical potential \nare displayed in figure~\\ref{fig:buckyball-correlators}.\nThe mild sign problem is solely due to the non-bipartiteness of the lattice structure.\nWe compare standard HMC (red) with tangent plane HMC (blue) to show \nthat the tangent plane obtains a good signal already at small \nnumber of configurations $\\ensuremath{\\mathrm{N}_{\\mathrm{conf}}} = 1000$ in both systems.\nFor $C_{60}$ the sign problem is negligible and the real plane HMC gives a\ngood signal too.\nFor finite chemical potential the sign problem\naggravates as it imposes a second source. \nWe are currently working on this particular lattice geometry. \n\n \n\\section{Conclusions}\nSimulating systems with strongly correlated electrons is a rather challenging \ntask due to the innate sign problem for doped systems.\nCurrent methods, like deformation onto Lefschetz thimbles, suffer from a very difficult \nscaling in computational cost.\nWe overcome this issue by identifying efficient Neural Network architectures and incorporating them\nin a HMC algorithm.\nWe present first studies of doped Graphene sheets using this enhanced HMC and \ndemonstrate a substantial improvement of the signal, effectively mitigating the sign problem.\nConsidering systems with increasing volume illustrates some stability of this \nmethod for larger volumes.\nFurther, preliminary simulation of Fullerene $C_{20}$ and $C_{60}$ at \nvanishing chemical potential are shown.\nIn the near future we will apply the neural network enhanced HMC also\nat finite chemical potential.\n\n \n\\begin{acknowledgments}\nWe thank the original authors of our recent papers for many helpful discussions and hard work.\nThis work was funded in part by the NSFC and the Deutsche Forschungsgemeinschaft (DFG, German Research\nFoundation) through the funds provided to the Sino-German Collaborative\nResearch Center ``Symmetries and the Emergence of Structure in QCD''\n(NSFC Grant No.~12070131001, DFG Project-ID 196253076 -- TRR110)\nas well as the STFC Consolidated Grant ST\/T000988\/1.  \nMR was supported under the RWTH Exploratory Research Space (ERS) grant PF-JARA-SDS005.\nWe gratefully acknowledge the computing time granted by the JARA Vergabegremium and provided on the JARA Partition part of the supercomputer JURECA at Forschungszentrum J\u00fclich.\n\\end{acknowledgments}\n \n\\bibliographystyle{apsrev4-1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}