diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzqdzz" "b/data_all_eng_slimpj/shuffled/split2/finalzzqdzz" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzqdzz" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{sec:1}\nEinstein's theory of General Relativity (GR) has experienced unprecedented success in its power to explain astrophysical phenomena ranging from Solar System tests to strong field gravitational wave physics \\cite{Misner:1974qy}. However, this theory of gravity has required important modifications due to observational realities which have arisen over the past few decades. In terms of the energy budget of the Universe, the first modification comes from observations of galaxies and their dynamical structure, which is only possible with the addition of approximately purely gravitational interacting particles, namely \\textit{dark matter}, which may potentially be beyond the standard model of particle physics \\cite{Rubin:1970zza,Navarro:1995iw}. The second and larger contribution to the modification of GR comes from the relatively recent observation of the accelerating expansion of the Universe \\cite{Riess:1998cb,Perlmutter:1998np} which is an observational fact, called \\textit{dark energy}. This can be accounted for in GR through the introduction of the cosmological constant however this poses its own problems \\cite{Weinberg:1988cp}. The $\\Lambda$CDM is the most successful model which can explain the current accelerated expansion and the evolution of the observable Universe at the level of background dynamics, involving the superposition between the dark matter fluid and the cosmological constant. On the other hand, the early period of the Universe also features several facets that need remedy. Most prominently, for $\\Lambda$CDM to correctly produce our current picture of the Universe, a period of cosmological inflation must of taken place \\cite{Guth:1980zm,Linde:1981mu} which would allow for a natural solution to the horizon problem. However, this may also necessitate further particles beyond the standard model \\cite{Lyth:1998xn}. Cosmologically, the time that should be best described by the $\\Lambda$CDM model is the present or late-time period of the Universe. However, recent releases by the Planck collaboration have revealed a growing tension in the local and global measurements of $H_0$ and $f\\sigma_8$ \\cite{Aghanim:2018eyx}. \\medskip\n\nNow, it may be the case that the fundamental and observational problems surrounding $\\Lambda$CDM may be resolved in the coming years, or it may be the case that $\\Lambda$CDM needs to be changed in some way. Over the previous decades there has been concerted efforts in extending GR to account for certain elements of these problems \\cite{Capozziello:2011et}. However, it may also be the case that a new paradigm is needed to confront the growing requirements of constructing a viable theory of gravity. One such treatment is the teleparallel gravity approach where the Levi-Civita connection is replaced with the Weitzenb\\\"{o}ck connection \\cite{AP}. The connection plays a crucial role in gravitational physics in that the expression of curvature, torsion, or nonmetricity is not a property of the manifold itself but of the connection which relates the elemental tangent spaces of the manifold \\cite{Heisenberg:2018vsk}. In this way, one can choose to consider gravitation in terms of the curvatureless Weitzenb\\\"{o}ck connection which also observes the metricity property. \\medskip\n\nIn teleparallel gravity, the gravitation is characterized by the torsion tensor, $\\udt{T}{\\rho}{\\mu\\nu}$, instead of the Riemann tensor in GR and its extensions. As in the GR framework, a Lagrangian can be constructed to represent the gravitational field. Of particular interest, in teleparallel gravity, is that a Lagrangian can be constructed such that it is equivalent to the Einstein-Hilbert Lagrangian up to a total divergence or boundary term, $B$, that is\n\\begin{equation}\\label{ricc_tor_sca_def}\n R=-T+B,\n\\end{equation}\nwhere $T$ is called the torsion scalar and contains only second order terms, while the boundary term, $B$, encapsulates the higher order contributions to the Ricci scalar, $R$. This is the so-called \\textit{teleparallel equivalent of general relativity} (TEGR) which is equivalent to GR at the level of the field equations \\cite{Cai:2015emx,Krssak:2018ywd}. The natural consequence of this realization is that every test of GR also becomes a test of TEGR with the difference that in TEGR gravity is again observed to act as a (Lorentz) force, and that the barrier with the quantum regime seems to have less tension \\cite{AP}. Moreover, due to the second order nature of the torsion tensor, even in the case where the theory is extended to an $f(T)$ Lagrangian, the resulting field equations remain second order which has important consequences for the gravitational wave polarization modes of the theory. In fact, $f(T)$ gravity continues to exhibit the equivalent polarization modes as in the GR and TEGR settings \\cite{Farrugia:2018gyz}. \\medskip\n\nGiven the decomposition of the Ricci scalar into the second and fourth order terms expressed in Eq.~(\\ref{ricc_tor_sca_def}), we consider the analysis of a tachyonic dark energy model nonminimally coupled to the aforementioned separate contributions through different functionals. In particular, we choose to study a tachyonic scalar field which has been shown to produce an inflationary epoch and late-time accelerating solutions that do not violate the strong energy condition \\cite{Quiros:2009mz,Aguirregabiria:2004xd}. These models are partially inspired by string theory \\cite{Bagla:2002yn,Fang:2010zze,Ortin:2015hya} and $k$-essence theory \\cite{Almeida:2016ixq}. In the teleparallel setting, scalar fields have been investigated to a moderate degree, with various extensions having been investigated in the cosmological context \\cite{Geng:2011aj,Xu:2012jf,Hohmann:2018rwf,PhysRevD.97.084008,Otalora:2013tba,Otalora:2014aoa}. Furthermore, the effects of the boundary couplings in scalar tensor theories have been discussed in various papers \\cite{Hohmann:2018ijr,Bahamonde:2018miw,Abedi:2018lkr,Hohmann:2018rwf, Bahamonde:2017wwk, Marciu:2017sji,Gecim:2017hmn, Bahamonde:2017ifa, Abedi:2017jqx, Fazlpour:2018jzz,Bahamonde:2016grb, Bahamonde:2016cul,MohseniSadjadi:2016ukp, Bahamonde:2015zma}. A recent review on various studies related to dynamical analysis in different cosmological constructions can be found in \\cite{BAHAMONDE20181}. In Ref.~\\cite{Otalora:2013dsa}, a tachyonic field is investigated for the modified teleparallel setting, late-time accelerating attractor solutions are found with a field equation of state that realistically tends to the current dark energy value. This approach was extended to more general models in Ref.~\\cite{Fazlpour:2014qaa} where the attractor solution context is further clarified. \\medskip\n\nThe work is divided as follows: in Sec.~\\ref{sec:2} the tachyonic approach to extended teleparallel theories of gravity is introduced with a focus on the cosmological consequences of the treatment. In Sec.~\\ref{sec:3}, the dynamical analysis of the system is undertaken for specific choices of tachyonic field. Finally, in Sec.~\\ref{sec:4} the conclusions are summarized and discussed. Unless stated otherwise geometric units are used throughout the paper. In addition, $e^a_\\mu$ and $E_a^\\mu$ represent the tetrads and the inverse of the tetrads respectively and the $(+---)$ metric signature is used.\n\n\n\\section{Generalized Tachyonic Teleparallel theories of gravity}\\label{sec:2}\n\\noindent In this paper, we present a new teleparallel tachyonic model based on the following action\n\\begin{equation}\nS = \\int\n\\left[\\frac{T}{2\\kappa^2}+\\frac{1}{2}f(\\phi) T+\\frac{1}{2}g(\\phi)B-V(\\phi)\\sqrt{1-\\frac{2X}{V(\\phi)}}+L_{\\rm m}\\right] e\\, d^4x \\,,\\label{1}\n\\end{equation}\nwhere $\\kappa^2=8\\pi G$, $L_{\\rm m}$ is a matter Lagrangian, $T$ is the scalar torsion, $B=(2\/e)\\partial_{\\mu}(eT^{\\mu})$ is the boundary term, $f(\\phi)$ and $g(\\phi)$ are scalar field dependent coupling functions, $V(\\phi)$ is the potential\nand\n\\begin{equation}\n X=\\frac{1}{2}(\\partial_\\mu \\phi)( \\partial^\\mu \\phi)\n\\end{equation}\nis the kinetic term. The torsion tensor is identified as\n\\begin{equation}\n \\udt{T}{a}{\\mu\\nu}=\\partial_{\\mu}e^a_{\\nu}-\\partial_{\\nu}e^a_{\\mu}+\\udt{\\omega}{a}{b\\mu}e^b_{\\nu}-\\udt{\\omega}{a}{b\\nu}e^b_{\\mu}\\,,\n\\end{equation}\nwhere $e^a_{\\nu}$ form a tetrad field of the gravitational system and represent coordinate transformations between the general manifold and the tangent space at any point, while $\\udt{\\omega}{a}{b\\mu}$ form the spin connection components which are purely inertial and sustain the local Lorentz invariance of the theory \\cite{Krssak:2015oua,Cai:2015emx}. The torsion scalar is then defined through the contraction\n\\begin{equation}\n T=\\udt{T}{a}{\\mu\\nu}\\dut{S}{a}{\\mu\\nu},\n\\end{equation}\nwhere the superpotential is defined as \n\\begin{equation}\n \\dut{S}{a}{\\mu\\nu}=\\frac{1}{2}\\left(\\dut{T}{a}{\\mu\\nu}+\\udt{T}{\\nu\\mu}{a}-\\udt{T}{\\mu\\nu}{a}\\right)-E^a_{\\nu}\\udt{T}{\\alpha\\mu}{\\alpha}+E^a_{\\mu}\\udt{T}{\\alpha\\nu}{\\alpha}\\,.\n\\end{equation}\nThis work considers an analogous generalization of other Tachyonic models studied in the literature \\cite{Otalora:2013dsa}. By taking, $f(\\phi)=-g(\\phi)$, one recovers a Tachyonic theory with a non-minimally coupling between the scalar field and the Ricci scalar $R$ due to the relation in Eq.~(\\ref{ricc_tor_sca_def}). By taking $g(\\phi)=0$, one recovers a teleparallel Tachyonic theory where the torsion scalar $T$ is non-minimally coupled with the scalar field. In \\cite{Banijamali:2012nx}, the authors found that this coupling allows the crossing of the phantom divide line. The new coupling between the scalar field and the boundary term is motivated from the scalar tensor theory studied in \\cite{Bahamonde:2015hza,Zubair:2016uhx}, where the authors found that, without fine-tunning, the system evolves to a late-time accelerating attractor solution.\n\n\\noindent By taking variation with respect to the tetrad, we find the corresponding gravitational field equations given by\n\\begin{eqnarray}\n2\\Big(\\frac{1}{\\kappa^2}+f(\\phi)\\Big)\\left[ e^{-1}\\partial_\\mu (e S_{a}{}^{\\mu\\nu})-E_{a}^{\\lambda}T^{\\rho}{}_{\\mu\\lambda}S_{\\rho}{}^{\\nu\\mu}-\\frac{1}{4}E^{\\nu}_{a}T\\right]-E_a^\\nu V(\\phi)\\sqrt{1-\\frac{2X}{V(\\phi)}}\n\\nonumber\\\\ -\\frac{1}{\\sqrt{1-\\frac{2X}{V(\\phi)}}}E_a^\\nu \\partial_\\mu \\phi \\partial^\\mu \\phi+ 2(\\partial_{\\mu}f(\\phi)+\\partial_{\\mu}g(\\phi)) E^\\rho_a S_{\\rho}{}^{\\mu\\nu}+E^{\\nu}_{a}\\Box g(\\phi)-E^\\mu_a \\nabla^{\\nu}\\nabla_{\\mu}g(\\phi)= \\mathcal{T}^\\nu_a\\,. \\label{2}\n\\end{eqnarray}\nHere, $\\nabla_\\mu$ is the covariant derivative with respect to the Levi-Civita connection and $\\Box=\\nabla^\\mu\\nabla_\\mu$. By taking variations with respect to the scalar field, one finds\n\\begin{align}\n \\frac{V'(\\phi)}{2V(\\phi)\\sqrt{1-\\frac{2X}{V(\\phi)}}}(\\partial_\\mu \\phi)( \\partial^\\mu \\phi) +\\partial_\\mu \\Big(\\frac{\\partial^\\mu \\phi}{\\sqrt{1-\\frac{2X}{V(\\phi)}}}\\Big)+V'(\\phi)\\sqrt{1-\\frac{2X}{V(\\phi)}}&=\\frac{1}{2}\\Big(f'(\\phi)T+g'(\\phi)B\\Big)\\,.\\label{KG}\n\\end{align}\n\nFor the flat FLRW metric in Cartesian coordinates, the metric is given by\n\\begin{equation}\n ds^2=dt^2-a(t)^2(dx^2+dy^2+dz^2)\\,,\n\\end{equation}\nwhere $a(t)$ is the cosmological scale factor. This can equivalently be described by the tetrad field\n\\begin{equation}\n e_\\mu^a=\\textrm{diag}(1,a(t),a(t),a(t))\\,,\n\\end{equation}\nwhich naturally induce vanishing spin connection components \\cite{Krssak:2015oua,Cai:2015emx}. The ensuing Friedmann equations then turn out to be represented by\n\\begin{eqnarray}\n3H^2&=&\\rho_{\\rm m}-3H^2 f(\\phi)+3 H \\dot{\\phi}g'(\\phi )+\\frac{V(\\phi )}{\\sqrt{1-\\frac{\\dot{\\phi}^2}{V(\\phi)}}}\\,,\\label{eq1}\\\\\n3 H^2+2 \\dot{H}&=&-p_{\\rm m}-3 H^2f(\\phi)-2 f(\\phi) \\dot{H}-2 H \\dot{\\phi}f'(\\phi)+\\dot{ \\phi}^2 g''(\\phi)+ \\ddot{ \\phi} g'(\\phi)+V(\\phi) \\sqrt{1-\\frac{\\dot{\\phi}^2}{V(\\phi)}}\\,,\\label{eq2}\n\\end{eqnarray}\nwhere $\\kappa=1$ was assumed, dots represent differentiation with respect to cosmic time and primes with respect to the scalar field. It should be noted that a standard perfect fluid with energy density $\\rho_{\\rm m}$ and pressure $p_{\\rm m}$ is being assumed. It is straightforward to show that the standard conservation of the energy-momentum tensor gives \n\\begin{align}\n \\dot{\\rho}_{\\rm m}+3H(\\rho_{\\rm m}+p_{\\rm m})=0\\,,\n\\end{align}\nwhich is the standard conservation equation for matter.\n\n\\noindent The scalar field relation in Eq.~(\\ref{KG}) takes the following form\n\\begin{eqnarray}\n\\ddot{\\phi}+3 H\\dot{\\phi} \\left(1-\\frac{\\dot{\\phi}^2}{V(\\phi )}\\right)+\\left(1-\\frac{3 \\dot{\\phi}^2}{2 V(\\phi)}\\right) V'(\\phi)&=&-\\left(1-\\frac{\\dot{\\phi}^2}{V(\\phi )}\\right)^{3\/2}\\left[3 H^2 f'(\\phi)+3 \\left(3 H^2+\\dot{H}\\right) g'(\\phi) \\right]\\,.\n\\end{eqnarray}\nIt can be shown that this equation can be also found directly from the modified FLRW equations \\eqref{eq1}--\\eqref{eq2}, therefore, it is not an extra constraint equation.\n\n\\noindent The Friedmann equations can also equivalently be rewritten as their GR analogue with an additional effective fluid component so that\n\\begin{eqnarray}\n3H^2=\\rho_{\\rm eff}\\,,\\quad 3 H^2+2 \\dot{H}=-p_{\\rm eff}\\,,\n\\end{eqnarray}\nwhere $\\rho_{\\rm eff}=\\rho_{\\rm m}+\\rho_{\\phi}$ and $p_{\\rm eff}=p_{\\rm m}+p_{\\phi}$ with the new fluid quantities defined as follows\n\\begin{align}\n \\rho_{\\phi}&=-3H^2 f(\\phi)+3 H \\dot{\\phi}g'(\\phi )+\\frac{V(\\phi )}{\\sqrt{1-\\frac{\\dot{\\phi}^2}{V(\\phi)}}}\\,,\\\\\n p_{\\phi}&=3 H^2f(\\phi)+2 f(\\phi) \\dot{H}+2 H \\dot{\\phi}f'(\\phi)-\\dot{ \\phi}^2 g''(\\phi)- \\ddot{ \\phi} g'(\\phi)-V(\\phi) \\sqrt{1-\\frac{\\dot{\\phi}^2}{V(\\phi)}}\\,.\n\\end{align}\nIt is then convenient to introduce the effective equation of state parameter\n\\begin{eqnarray}\nw_{\\rm eff}=\\frac{p_{\\rm eff}}{\\rho_{\\rm eff}}=\\frac{p_{\\rm m}+p_{\\phi}}{\\rho_{\\rm m}+\\rho_{\\phi}}\\,,\n\\end{eqnarray}\nwhich means that the density parameters of the contributing components take the form\n\\begin{align}\n \\Omega_{\\rm m}=\\frac{\\rho_{\\rm m}}{3H^2}\\,,\\quad \\Omega_{\\phi}=\\frac{\\rho_{\\phi}}{3H^2}\\,,\\label{energydensity}\n\\end{align}\nin such a way that $\\Omega_{\\rm m}+\\Omega_{\\phi}=1$ holds.\n\n\n\\section{Dynamical system of the model}\\label{sec:3}\nIn this section, we will study the dynamical system of our model which only has non-minimally couplings between the scalar field and the boundary terms, therefore we will assume that $f(\\phi)=0$, and $g(\\phi)\\neq0$. The dynamical system for the case $f(\\phi)\\neq 0$ and $g(\\phi)=0$ was studied previously in~\\cite{Otalora:2013dsa,Fazlpour:2014qaa}. Let us first introduce the following dimensionless variables\n\\begin{align}\n x=\\frac{\\dot{\\phi}}{\\sqrt{V(\\phi)}}\\,,\\quad y=\\frac{\\sqrt{V(\\phi)}}{\\sqrt{3}H}\\,,\\quad u=\\frac{1}{2}g'(\\phi)\\,,\\quad \\lambda=-\\frac{V'(\\phi)}{V(\\phi)}\\,,\n\\end{align}\nso that $y>0$. By using these variables in the first FLRW equation in Eq.~(\\ref{eq1}) and the definition of the energy density parameter given by Eq.~(\\ref{energydensity}), one gets the constraint\n\\begin{align}\n 0 \\leq \\Omega_{\\rm m}= 1-2 \\sqrt{3} u x y-\\frac{y^2}{\\sqrt{1-x^2}}\\leq 1\\,,\n\\end{align}\nwhich gives the phase space of the dynamical system. One needs to choose $g(\\phi)$ to write down the full dynamical system of the model. Therefore, in the next sections, we will study two different kind of couplings: power-law and exponential types. We will further assume standard a barotropic fluid given by $p_{\\rm m}=w_{\\rm m} \\rho_{\\rm m}$.\n\n\\subsection{Power-law coupling and exponential potential}\nAssuming a power-law coupling between the boundary term and the scalar field means setting\n\\begin{eqnarray}\ng(\\phi)=\\chi \\phi^p\\,, \n\\end{eqnarray}\nwhere $\\chi$ and $p$ are both constants. It is possible to write down the dynamical equations of the system as a 4-dimensional one with a generic potential. This can be done if one introduces the variable \n\\begin{align}\n \\Gamma=\\frac{V(\\phi) V''(\\phi)}{V'(\\phi ) V'(\\phi)}\\,.\n \\label{ecuatie_gamma}\n\\end{align}\nBy introducing the variable $N=\\log(a)$, the dynamical system for this kind of couplings can be written as\n\\begin{equation}\n\\frac{dx}{dN}=\\frac{\\ddot{\\phi}}{\\sqrt{3} H^2 y}+\\frac{1}{2} \\sqrt{3} \\lambda x^2 y, \\nonumber\n\\end{equation}\n\\begin{equation}\n \\frac{dy}{dN}=-\\frac{y \\dot{H}}{H^2}-\\frac{1}{2} \\sqrt{3} \\lambda x y^2, \\nonumber\n\\end{equation}\n\\begin{equation}\n\\frac{du}{dN}=\\sqrt{3} \\cdot 2^{\\frac{1}{1-p}} (p-1) p \\chi x y \\left(\\frac{u}{p \\chi }\\right)^{\\frac{p-2}{p-1}}, \\nonumber\n\\end{equation}\n\\begin{equation}\n\\frac{d\\lambda}{dN}=-\\sqrt{3} (\\Gamma-1) \\lambda^2 x y\\,,\n\\nonumber\n\\end{equation}\nwhere\n\\begin{equation}\n \\ddot{\\phi}=3 H^2 \\lambda \\left(1-\\frac{3 x^2}{2}\\right) y^2-3 \\sqrt{3} H^2 x \\left(1-x^2\\right) y-6 u \\left(1-x^2\\right)^{3\/2} \\left(3 H^2+\\dot{H}\\right),\\nonumber\n\\end{equation}\nand\n\\begin{multline}\n \\dot{H}=\\frac{3 H^2}{12 u^2 \\left(1-x^2\\right)^{3\/2}+2}\\cdot \\Bigg[ w_m \\left(2 \\sqrt{3} u x y+\\frac{y^2}{\\sqrt{1-x^2}}-1\\right)+2^{\\frac{p-2}{p-1}} (p-1) p \\chi x^2 y^2 \\left(\\frac{u}{p \\chi }\\right)^{\\frac{p-2}{p-1}}\n \\\\\n -12 u^2 \\left(1-x^2\\right)^{3\/2}-\\lambda u \\left(3 x^2-2\\right) y^2+2 \\sqrt{3} u x \\left(x^2-1\\right) y+\\sqrt{1-x^2} y^2-1 \\Bigg].\\label{eq:11}\n\\end{multline}\nThis dynamical system assumes that $p\\neq 1$. The case $p=1$ is a very special one which gives a 3-dimensional dynamical system of equations, but generically, does not give any interesting cosmological behaviour. One more realistic model is the one where $p=2$ which gives a coupling like $(1\/2)\\chi \\phi^2 B $ which was studied in \\cite{Bahamonde:2015hza}. In that case however, non-tachyonic scalar fields were considered. In what follows, we shall consider the case where we have a specific coupling, $g(\\phi)=\\chi \\phi^p$, with $p=2$ and an exponential potential $V(\\phi)=V_0 e^{- \\lambda \\phi}$, where $\\lambda$ is a positive constant. In this case, the dynamical evolution of the model can be described by a 3D autonomous system of differential equations\n\\begin{multline}\n\\label{equu1}\n \\frac{dx}{dN}=\\frac{1}{y \\left(6 u^2 \\left(x^2-1\\right)^2+\\sqrt{1-x^2}\\right)} \\Big( -18 u^2 x^5 y w_{\\rm m}+36 u^2 x^3 y w_{\\rm m}-18 u^2 x y w_{\\rm m}+3 \\sqrt{3} u x^4 w_{\\rm m}+3 u \\sqrt{3-3 x^2} x^2 y^2 w_{\\rm m}\n \\\\-3 u \\sqrt{3-3 x^2} y^2 w_{\\rm m}-6 \\sqrt{3} u x^2 w_{\\rm m}+3 \\sqrt{3} u w_{\\rm m}+3 \\sqrt{3} \\lambda u^2 x^6 y^2-6 \\sqrt{3} \\lambda u^2 x^4 y^2+3 \\sqrt{3} \\lambda u^2 x^2 y^2\n \\\\-6 \\sqrt{3} u \\chi x^6 y^2+12 \\sqrt{3} u \\chi x^4 y^2-3 \\sqrt{3} u x^4-6 \\sqrt{3} u \\chi x^2 y^2\n +6 \\sqrt{3} u \\sqrt{1-x^2} x^2 y^2-3 \\sqrt{3} u \\sqrt{1-x^2} y^2\n \\\\+6 \\sqrt{3} u x^2-3 \\sqrt{3} u \\sqrt{1-x^2} x^4 y^2-3 \\sqrt{3} u-\\sqrt{3} \\lambda \\sqrt{1-x^2} x^2 y^2+\\sqrt{3} \\lambda \\sqrt{1-x^2} y^2-3 \\sqrt{1-x^2} x y+3 \\sqrt{1-x^2} x^3 y\\Big) \\,,\n\\end{multline}\n\\begin{multline}\n \\frac{dy}{dN}=\\frac{1}{2 \\left(6 u^2 \\left(x^2-1\\right)^2+\\sqrt{1-x^2}\\right)} \\Big( -6 u \\sqrt{3-3 x^2} x y^2 w_{\\rm m}+3 \\sqrt{1-x^2} y w_{\\rm m}-3 y^3 w_{\\rm m}-6 \\sqrt{3} \\lambda u^2 x^5 y^2+36 u^2 x^4 y\n \\\\+12 \\sqrt{3} \\lambda u^2 x^3 y^2-72 u^2 x^2 y-6 \\sqrt{3} \\lambda u^2 x y^2+36 u^2 y+9 \\lambda u \\sqrt{1-x^2} x^2 y^3-6 \\lambda u \\sqrt{1-x^2} y^3\n \\\\+6 \\sqrt{3} u \\sqrt{1-x^2} x y^2-6 \\sqrt{3} u \\sqrt{1-x^2} x^3 y^2-6 \\chi \\sqrt{1-x^2} x^2 y^3+3 x^2 y^3-\\lambda \\sqrt{3-3 x^2} x y^2+3 \\sqrt{1-x^2} y-3 y^3 \n \\Big)\\,,\n\\end{multline}\n\\begin{equation}\n\\label{equu2}\n \\frac{du}{dN}=\\sqrt{3} \\chi x y.\n\\end{equation}\n\\par \n\\noindent In this scenario, the effective equation of state for the dark energy field can be written as:\n\\begin{multline}\n w_{\\rm eff}=\\frac{1}{6 u^2 \\left(x^2-1\\right)^2+\\sqrt{1-x^2}} \\Big( -2 u \\sqrt{3-3 x^2} x y w_{\\rm m}+\\sqrt{1-x^2} w_{\\rm m}-y^2 w_{\\rm m}+6 u^2 x^4-12 u^2 x^2+6 u^2+3 \\lambda u \\sqrt{1-x^2} x^2 y^2\n \\\\-2 \\lambda u \\sqrt{1-x^2} y^2+2 \\sqrt{3} u \\sqrt{1-x^2} x y-2 \\sqrt{3} u \\sqrt{1-x^2} x^3 y-2 \\chi \\sqrt{1-x^2} x^2 y^2+x^2 y^2-y^2 \\Big)\\,.\n\\end{multline}\nNext, the critical points are determined by considering that the RHS of Eqs.~(\\ref{equu1})--(\\ref{equu2}) are equal to zero, taking into account also the physical viability which requires: $0\\leq \\Omega_{m}=1-\\Omega_{\\phi}\\leq 1$, $y\\geq 0$, $1-x^2>0$ and the location of the corresponding critical points is the the real space. In the case of this specific power law coupling and for an exponential potential, the phase space structure has a reduced complexity, having only one critical point located at $O(x,y,u)=\\left(0, 1,\\frac{\\lambda }{6} \\right)$, with the following eigenvalues\n\\begin{equation}\n \\Xi_{O}=\\left[-\\frac{3 \\left(\\sqrt{\\left(\\lambda ^2+6\\right) \\left(\\lambda ^2-48 \\chi +6\\right)}+\\lambda ^2+6\\right)}{2 \\left(\\lambda ^2+6\\right)},\\frac{3 \\left(\\sqrt{\\left(\\lambda ^2+6\\right) \\left(\\lambda ^2-48 \\chi +6\\right)}-\\lambda ^2-6\\right)}{2 \\left(\\lambda ^2+6\\right)},-3 \\left(w_{\\rm m}+1\\right) \\right]\\,.\n\\end{equation}\nThis critical point corresponds to a de Sitter evolution, acting as a cosmological constant $ w_{\\rm eff}= -1 $, implying the full domination of the tachyonic dark energy field over the matter component, $\\Omega_m=1-\\Omega_{\\phi}=0$. For this critical point, we show in Fig.~\\ref{fig:unu}, various regions for the model's parameters which are connected to different cosmological scenarios, corresponding to a stable and stable spiral evolution, respectively.\n\\begin{figure}[H]\n \n \\begin{minipage}{0.45\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{O_stable_point.pdf}\n \\end{minipage}\\hfill\n \\begin{minipage}{0.45\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{O_stable_spiral_point.pdf}\n \\end{minipage}\n \\caption{(a) The case where the critical point $O$ represents a stable cosmological scenario (left panel); (b) The situation where $O$ corresponds to a stable spiral evolution, due to complex eigenvalues (right panel).}\n \\label{fig:unu}\n\\end{figure}\n\n\n\\subsection{Exponential coupling and inverse hyperbolic sine potential}\nHere, we investigate the case where the coupling functional is represented by an exponential, $g(\\phi)=g_0 e^{\\alpha \\phi}$, with $g_0$ and $\\alpha$ constants. The potential energy associated with the present tachyonic dark energy model is considered to be an inverse hyperbolic sine, $V(\\phi)= V_0 \\sinh^{-\\omega_1}(\\omega_2 \\phi)$, with $\\omega_{1,2}$ constants. This type of potential is beyond the usual exponential type considered in many dynamical analysis and is motivated by the recent work in Ref.~\\cite{Roy:2017uvr}, where the structure of the phase space for non--canonical fields with the potential energy beyond exponential type was investigated. The potential energy considered in this section, the inverse hyperbolic sine \\cite{UrenaLopez:2000aj,Sahni:1999gb} represents a possible parameterization for the dark energy field which is associated to a second order polynomial in the dynamical equation for the dimensionless variable $\\lambda$. As discussed in Ref.~\\cite{Roy:2017uvr}, if we denote $f(\\lambda)=\\lambda^2(\\Gamma-1)$ with $\\Gamma$ defined in the relation in Eq.~(\\ref{ecuatie_gamma}), then for this specific potential, the function $f(\\lambda)$ obeys a second order polynomial parameterization $f(\\lambda)=a \\lambda^2+b \\lambda + c$, with $a,b,c$ constant parameters. For the specific potential considered here, the inverse hyperbolic sine, we find that $f(\\lambda)=\\lambda^2\/\\omega_1-\\omega_1 \\omega_2^2$. \n\n\n\n\\begin{figure}[H]\n \n \\begin{minipage}{0.35\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{PointA_1.pdf}\n \\end{minipage}\\hfill\n \\begin{minipage}{0.35\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{PointA_2.pdf}\n \\end{minipage}\n \\caption{(a) The case where the critical line $A$ has a saddle cosmological behavior, considering $w_{\\rm m}=0$, $u=0$, $\\alpha=1$ (left panel); (b) The situation where $A$ corresponds to a saddle critical line, considering $w_{\\rm m}=0$, $\\omega_1=\\omega_2$, $\\alpha=1$ (right panel).}\n \\label{fig:aaa}\n\\end{figure}\n\n\\begin{figure}[th]\n \n \\begin{minipage}{0.35\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{PointBminusminus.pdf}\n \\end{minipage}\\hfill\n \\begin{minipage}{0.35\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{PointBplusplus.pdf}\n \\end{minipage}\n \\caption{(a) The case where the critical point $B_{-}^{-}$ has a stable cosmological behavior, considering $w_{\\rm m}=-0.001$, $\\alpha=2$ (left panel); (b) The situation where $B_{+}^{+}$ corresponds to a stable scenario, considering $w_{\\rm m}=-0.01$, $\\alpha=-1$ (right panel).}\n \\label{fig:bbbaaa}\n\\end{figure}\n\n\nIn this case, the evolution of the dynamical system can be written as the following system of differential equations\\begin{multline}\n\\frac{dx}{dN}=-\\frac{1}{y \\left(6 u^2 \\left(x^2-1\\right)^2+\\sqrt{1-x^2}\\right)}(x^2-1) \\Big[-3 u w_{\\rm m} \\left(-6 u x^3 y+6 u x y+\\sqrt{3-3 x^2} y^2+\\sqrt{3} x^2-\\sqrt{3}\\right)\n\\\\+3 \\sqrt{3} u^2 x^2 \\left(x^2-1\\right) y^2 (2 \\alpha -\\lambda )+3 \\sqrt{3} u \\left(x^2-1\\right) \\left(\\sqrt{1-x^2} y^2+1\\right)+\\sqrt{1-x^2} y \\left(\\sqrt{3} \\lambda y-3 x\\right) \\Big]\\,,\n\\label{eq:12}\n\\end{multline}\n\\begin{multline}\n\\frac{dy}{dN}=-\\frac{y}{2 \\left(6 u^2 \\left(x^2-1\\right)^2+\\sqrt{1-x^2}\\right)} \\Big[ 3 w_{\\rm m} \\left(2 u x \\sqrt{3-3 x^2} y-\\sqrt{1-x^2}+y^2\\right)+6 u^2 \\left(x^2-1\\right)^2 \\left(\\sqrt{3} \\lambda x y-6\\right)\n\\\\+3 u \\sqrt{1-x^2} y \\left(2 \\sqrt{3} x^3+x^2 y (2 \\alpha -3 \\lambda )-2 \\sqrt{3} x+2 \\lambda y\\right)-3 x^2 y^2+\\lambda x \\sqrt{3-3 x^2} y-3 \\sqrt{1-x^2}+3 y^2 \\Big]\\,,\n\\end{multline}\n\\begin{equation}\n\\frac{du}{dN}=\\sqrt{3} \\alpha u x y\\,,\n\\end{equation}\n\\newpage and finally the last equation for the dynamical system given by\n\\begin{equation}\n\\frac{d\\lambda}{dN}=-\\sqrt{3} x y \\left(\\frac{\\lambda ^2}{\\omega _1}-\\omega _1 \\omega _2^2\\right)\\,.\\label{eq:16}\n\\end{equation}\nThe total effective equation of state is:\n\\begin{align}\n w_{\\rm eff}&=\\frac{1}{6 u^2 \\left(x^2-1\\right)^2+\\sqrt{1-x^2}} \\Big( -u \\sqrt{1-x^2} y \\left(2 \\sqrt{3} x \\left(w_{\\rm m}-1\\right)+2 \\sqrt{3} x^3+x^2 y (2 \\alpha -3 \\lambda )+2 \\lambda y\\right)\n \\nonumber \\\\\n &+w_{\\rm m} \\left(\\sqrt{1-x^2}-y^2\\right)+6 u^2 \\left(x^2-1\\right)^2+\\left(x^2-1\\right) y^2\\Big)\\,.\n\\end{align}\n\nNext, as in the previous case, the critical points are obtained by considering that the RHS of the evolution equations in Eqs.~(\\ref{eq:12})--(\\ref{eq:16}) are equal to zero. For the potential energy of the inverse hyperbolic sine type, the critical points and the main physical properties are expressed in Table~\\ref{table:que}. It should be noted that for dust matter $w_{\\rm m}=0$, only the critical point $A$ exists in the phase space. As can be noted from this table, one can observe two main classes of critical points. The first class, denoted as $A$ represents a critical line which corresponds to a cosmological constant behavior, having an interrelation between the auxiliary variable $u$ associated to the coupling function $g(\\phi)$ and the strength of the potential energy, embedded into the non--constant $\\lambda$ variable. This epoch corresponds to a de Sitter universe, a critical line where the dark energy field dominates in terms of density parameters. Analyzing the corresponding eigenvalues, due to the presence of a zero eigenvalue, the linear stability method fails to provide a viable theoretical framework for determining the stability properties. Hence, for this critical line we can only argue on the specific cases where we have a saddle cosmological behavior, due to the presence of eigenvalues with both positive and negative real parts. Concerning this critical line, we display in Fig.~(\\ref{fig:aaa}) various regions for the model's parameters which correspond to a saddle cosmological behavior, due to the existence of at least one eigenvalue with negative real part, and at least an eigenvalue with positive real part.\n\nA second class of critical points displayed in Table~\\ref{table:que} is represented by the $B_{i}^{j}$, $i,j=\\left \\{ +,- \\right \\}$ critical points which have a scaling cosmological behavior, an epoch in which the dark energy field acts as a matter component and mimics a matter dominated epoch. This type of solutions can in principle solve the cosmic coincidence problem. For these solutions, we have shown in Table~\\ref{table:que} the locations in the phase space and the corresponding physical properties associated. From a physical point of view, the validity of these solutions implies that various existence conditions are satisfied. These corresponds to the requirement that the critical points belong to the real space, and the corresponding density parameters are physically viable, $\\Omega_{\\phi}=1-\\Omega_{m} \\in \\left[ 0,1 \\right]$. Moreover, due to the definition of the dimensionless variables within this section, one should add the requirement that the $y$ variable is real and positive and $x \\in \\left( -1,1 \\right)$ due to the form of the matter (dark energy) density parameter. From the expression of the $x$ variable presented in the table, it can be seen that the existence conditions imply that for the present tachyonic dark energy model $w_{\\rm m} \\in \\left[-1, 0\\right)$. Assuming that the matter component is embedded into the dark matter fluid, this implies that the pressure associated to the dark matter fluid is negative, an exotic situation which is not excluded by different cosmological observations~\\cite{Thomas:2016iav, Yang:2016dhx, PhysRevD.88.127301,PhysRevD.71.047302}. Moreover, for the critical points $B_{\\left[+,-\\right]}^{-}$, the existence conditions imply that the product $\\omega_{1}\\omega_2$ is negative, while for the solutions $B_{\\left[+,-\\right]}^{+}$ we have an inverted situation, $\\omega_{1}\\omega_2>0$. Hence, the current tachyonic dark energy model might contribute to a solution of the cosmic coincidence problem due to the scaling solutions since it can recover matter and de Sitter cosmological epochs. The stability of the points $B_{\\pm}^{\\pm}$ depend on the sign of the eigenvalues. In general, the stability conditions for each point are very cumbersome since they depend on the parameters $\\alpha, w_{\\rm m}$ and $\\omega_{1,2}$. Concerning the stability properties, we show in Fig.~\\ref{fig:bbbaaa} various cases which corresponds to the stability associated to the $B_{-}^{-}$ and $B_{+}^{+}$ critical points, determining possible values of the $\\omega_{1,2}$ parameters which result in stable scaling solutions. Point $A$ is a non-hyperbolic point and standard linear stability theory fails on describing any stability property of it. One can use other dynamical system techniques as centre manifold theory to study its stability (see~\\cite{BAHAMONDE20181, Leon:2015via}). However, due to limited physical effects associated to the $A$ critical point, we relied our analysis only on linear stability methods, exploring the specific conditions where the stability corresponds to a saddle dynamical behavior. The present discussion can be adapted also for various potential types beyond exponential which have been studied in Ref.~\\cite{Roy:2017uvr} with compatible results. \n\\begin{table}[t!]\n \n \\label{tab:table1}\n \\begin{tabular}{|c|c|c|c|c|c|c|c|} \n \\hline\n Point & x & y & u & $\\lambda$ & $\\Omega_{\\phi}$ & $w_{\\rm eff}$ & Eigenvalues\\\\ \n \\hline\\hline\n A& 0 & 1 & u & 6u & 1 & -1 & $0,-3 \\left(w_{\\rm m}+1\\right),-\\frac{3}{2} \\pm \\frac{\\sqrt{3} \\sqrt{\\left(6 u^2+1\\right) \\omega _1 \\left(\\omega _1 \\left(18 u^2-24 \\alpha u+3\\right)-144 u^2+4 \\omega _1^2 \\omega _2^2\\right)}}{2 \\left(6 u^2+1\\right) \\omega _1}$ \\\\ \n \\hline\n $B_{+}^{-}$& $\\sqrt{w_{\\rm m}+1}$ & $-\\frac{\\sqrt{3} \\sqrt{w_{\\rm m}+1}}{\\omega _1 \\omega _2}$ & 0 & $-\\omega _1 \\omega _2$ & $\\frac{3 \\left(w_{\\rm m}+1\\right)}{\\omega _1^2 \\omega _2^2 \\sqrt{-w_{\\rm m}}}$ & $w_{\\rm m}$ & $ -\\frac{3 \\alpha \\left(w_{\\rm m}+1\\right)}{\\omega _1 \\omega _2},-\\frac{6 \\left(w_{\\rm m}+1\\right)}{\\omega _1}, \\frac{3}{4} \\left( w_{\\rm m}-1\\pm \\Xi \\right)$ \\\\ \n \\hline\n $B_{-}^{+}$& $-\\sqrt{w_{\\rm m}+1}$ & $\\frac{\\sqrt{3} \\sqrt{w_{\\rm m}+1}}{\\omega _1 \\omega _2}$ & 0 & $-\\omega _1 \\omega _2$ & $\\frac{3 \\left(w_{\\rm m}+1\\right)}{\\omega _1^2 \\omega _2^2 \\sqrt{-w_{\\rm m}}}$ & $w_{\\rm m}$ & $-\\frac{3 \\alpha \\left(w_{\\rm m}+1\\right)}{\\omega _1 \\omega _2},-\\frac{6 \\left(w_{\\rm m}+1\\right)}{\\omega _1}, \\frac{3}{4}\\left(w_{\\rm m}-1 \\pm \\Xi \\right) $ \\\\ \n \\hline\n $B_{-}^{-}$& $-\\sqrt{w_{\\rm m}+1}$ & $-\\frac{\\sqrt{3} \\sqrt{w_{\\rm m}+1}}{\\omega _1 \\omega _2}$ & 0 & $\\omega _1 \\omega _2$ & $\\frac{3 \\left(w_{\\rm m}+1\\right)}{\\omega _1^2 \\omega _2^2 \\sqrt{-w_{\\rm m}}}$ & $w_{\\rm m}$ & $ \\frac{3 \\alpha \\left(w_{\\rm m}+1\\right)}{\\omega _1 \\omega _2},-\\frac{6 \\left(w_{\\rm m}+1\\right)}{\\omega _1}, \\frac{3}{4}\\left(w_{\\rm m}-1 \\pm \\Xi \\right)$ \\\\\n \\hline\n $B_{+}^{+}$& $\\sqrt{w_{\\rm m}+1}$ & $\\frac{\\sqrt{3} \\sqrt{w_{\\rm m}+1}}{\\omega _1 \\omega _2}$ & 0 & $\\omega _1 \\omega _2$ & $\\frac{3 \\left(w_{\\rm m}+1\\right)}{\\omega _1^2 \\omega _2^2 \\sqrt{-w_{\\rm m}}}$ & $w_{\\rm m}$ & $\\frac{3 \\alpha \\left(w_{\\rm m}+1\\right)}{\\omega _1 \\omega _2},-\\frac{6 \\left(w_{\\rm m}+1\\right)}{\\omega _1}, \\frac{3}{4}\\left(w_{\\rm m}-1 \\pm \\Xi \\right) $ \\\\ \n \\hline\n\\end{tabular}\n \n \\caption{The critical points in the case of exponential coupling and inverse hyperbolic sine potential. The auxiliary variable $\\Xi$ used in the description of the eigenvalues for the various critical points is equal to: $\\Xi=\\frac{\\sqrt{\\omega _1^4 \\omega _2^4 \\left(-w_{\\rm m}\\right) \\left(w_{\\rm m}+1\\right) \\left(17 \\omega _1^2 \\omega _2^2 w_{\\rm m}^2+14 \\omega _1^2 \\omega _2^2 w_{\\rm m}-96 \\left(-w_{\\rm m}\\right){}^{3\/2}+48 \\left(-w_{\\rm m}\\right){}^{5\/2}+48 \\sqrt{-w_{\\rm m}}+\\omega _1^2 \\omega _2^2\\right)}}{\\omega _1^3 \\omega _2^3 \\sqrt{-w_{\\rm m} \\left(w_{\\rm m}+1\\right)}}$}\n\\label{table:que}\n\\end{table}\n\n\n\n\\section{Conclusions}\\label{sec:4}\nIn this paper, we have considered a new dark energy model in the teleparallel equivalent of general relativity, based on modifications due to tachyonic fields which are non--minimally coupled with the torsion scalar and its boundary term. In this approach, the boundary term is related to the divergence of the torsion vector and the torsion scalar reproduces the same theory as GR at the level of the field equations. After finding the corresponding field equations for this tachyonic dark energy model, we analyzed the effects of the non--minimal coupling by employing the linear stability theory. In the first scenario, we considered the case where the coupling function is represented by a power law dependence, and the potential energy term corresponds to an exponential. In this case, we observed that the structure of the phase space has a reduced complexity. The critical points which are present corresponds to a cosmological constant behavior. Hence, in this case one notices that the evolution of the dynamical system can explain the current accelerated expansion of the present Universe due to the cosmological constant behavior of the system at the critical points. However, due to the reduced complexity of the phase space and without the presence of the scaling solutions, the cosmic coincidence problem cannot be alleviated. \n\\par \nA second cosmological scenario was also considered by taking into account that the non--minimal coupling functional has an exponential type parameterization. Concerning the potential energy term in the action, the study considered that the potential is beyond the usual exponential type found in many dynamical constructions in scalar tensor theories. As can be noted from the previous section, the dynamical equation associated to the potential term involves a second order polynomial parameterization. In this specific case in the analysis, the potential corresponds to an inverse hyperbolic sine. For the second cosmological scenario the phase space have a 4-dimensional structure and a richer complexity. As can be noted from the previous section, we have shown that the second cosmological scenario analyzed can reproduce the known evolution of the Universe and solve the cosmic coincidence problem due to the existence of scaling solutions. In these critical points, the dark energy field mimics a matter era due to the specific form the corresponding effective equation of state. However, the existence conditions associated to the scaling solutions imply the existence of an exotic warm dark matter fluid having a limited negative pressure, a comportment not ruled out by present astrophysical observations \\cite{Thomas:2016iav,Aghanim:2018eyx}. Furthermore, the rest of the critical points in the phase space corresponds to a cosmological constant behavior and can explain the current accelerated expansion of the Universe with a constant equation of state.\n\\par \nOne can then say that the current dark energy model constructed in the teleparallel equivalent of general relativity modified by a tachyonic field non--minimally coupled with a boundary term represents a potentially realistic model in scalar tensor theory which might solve the cosmic coincidence problem and the nature of the dark energy phenomenon, a feasible tachyonic prototype. This offers one possible alternative avenue to tachyonic fields to compliment the curvature-based work in the literature \\cite{Quiros:2009mz,Aguirregabiria:2004xd,Bagla:2002yn,Fang:2010zze,Otalora:2013tba,Otalora:2014aoa}.\n\n\n\\begin{acknowledgments}\nThe JLS and SB would like to acknowledge networking support by the COST Action GWverse CA16104. This article is based upon work from CANTATA COST (European Cooperation in Science and Technology) action CA15117, EU Framework Programme Horizon 2020. SB is supported by Mobilitas Pluss N$^\\circ$ MOBJD423 by the Estonian government. M. Marciu acknowledge partial support by the project 29\/2016 ELI-RO from the Institute of Atomic Physics, Bucharest--Magurele.\n\\end{acknowledgments}\n\n\n\\bibliographystyle{Style}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{INTRODUCTION AND EXPERIMENTAL SUMMARY}\n\nThe general study of phase transitions has been of continuing\n interest to condensed matter chemistry\nand physics for many years. Particular efforts have been made to\ncharacterize and understand second-order (or continuous)\nphase transitions and associated critical phenomena~\\cite{goldenfeld}.\n Good\nexamples of such continuous phase transitions are found in certain\natomic and molecular solids when they undergo order to disorder\ntransitions~\\cite{disorder}. These order-disorder transitions are\nprobed experimentally by observing anomalies in thermodynamic properties\nlike heat capacities and various susceptibilities.\n Some examples of systems that are known\nto exhibit disordering phenomena are ${\\rm NH}_4{\\rm Cl}_{\\rm (s)}$, $\\beta$-brass, and\n ${\\rm KCN}_{({\\rm s})}$.\n\n\nIn this study we focus on the order-disorder phase transition\nin ${\\rm NH}_4{\\rm Cl}_{\\rm (s)}$, an interesting case that has received considerable\nprevious\ntheoretical and experimental\nattention~\\cite{disorder}. A helpful\nreview of the experimental data up to 1978\nis given\nby Parsonage and Staveley~\\cite{disorder}.\nAs discussed in Reference 2 ammonium chloride has three\nphases.\nThe two phases generally referred to as $\\delta$ and $\\gamma$\nshare the CsCl lattice type\nshown in Figure 1, whereas the $\\alpha$ phase has a NaCl lattice\ntype. First measured and reported by Simon in 1922~\\cite{Simon}\nand subsequently by other workers\\cite{ber,TrappVander,VorGar,Schwartz}, the\n$\\delta \\rightarrow \\gamma$ transition has been used as a textbook example\nof an order-disorder phase transition~\\cite{experpchem}.\n{}From experiment it is known that in the low-temperature $\\delta$ phase,\nmost of the ${\\rm NH}_4^+$ molecules are in the same orientation, (\nthe ammonium ions orient with their hydrogen atoms pointing\ntowards the ``same\" crystallographic axes), and the $\\delta$ phase\nis said to be ``orientationally ordered.'' In\nthe $\\gamma$ phase, the orientations with respect to the chloride ions\nhave no long range order~\\cite{disorder}. This model of the\n$\\delta \\rightarrow \\gamma$ phase transition\nhas been supported by various spectroscopic\nmeasurements, including neutron scattering experiments and NMR\nstudies~\\cite{WagHorn,LevyPet,ManTrapp,TopRichtSpring,BornHohlEck,Mukhetal}.\n\nThe origin of the orientational order in NH$_4$Cl can be understood in terms\nof the intermolecular forces of the crystal. There are attractive\ninteractions between the hydrogen atoms in an ammonium ion and the\nadjacent cage of chloride ions (see Figure 1). Any isolated ammonium\nion in a chloride cage tends to align with either of two\nequivalent sets of four chloride ions. In Figure 1 the hydrogen atoms\npoint toward the set of chloride ions that has been shaded.\n In\naddition to the interactions of ammonium ions with the near-neighbor\nchloride cage, the ammonium ions interact with each other\nelectrostatically, with the lowest-order, non-zero contribution to the\ninteraction arising\n from the\noctapole-octapole interactions characteristic of tetrahedral charge\ndistributions. The octapole-octapole interaction has an asymptotic decay\nof $R^{-7}$ ($R$ is the distance between the nitrogen centers on any two\nammonium ions). There is an absolute potential energy minimum if two\ninteracting ammonium ions are aligned,\nand there is a higher energy local minimum if two ammonium ions have\nopposite alignments.\nWe can\nthink of the two potential minima for the ammonium ion in a chloride\ncage as two available states. These two available states can be mapped\nonto a two-state Ising model by identifying one orientation with\nan up spin and the other possible orientation with a down\nspin\\cite{expspin}.\nThe\norder-disorder transition has often been interpreted~\\cite{isingf}\n in terms of such a two\nstate Ising model with a positive ferromagnetic coupling constant between near\nneighbor spins. We shall investigate the energetics of this Ising\npicture within a simple pair site potential model in Section\n\\ref{s:model}\nof this paper.\n\nAlthough the interpretation of the transition in terms of the two-state\n Ising model has been insightful, it is not a perfect description\nof the transition.\nAt atmospheric pressure, a measurement of the heat capacity $C_p$ as a function\nof\ntemperature shows a distinctly $\\lambda$-like shape with a peak at\n243K\\cite{Schwartz}.\n While this $\\lambda$ character has been used to support\n the interpretation\nof this transition in terms of the Ising model, it is known that\nthere is a discontinuous change in the specific volume at\nthis temperature~\\cite{Fredericks,NisPoy,WeinGarl}, as well as a latent heat of\ntransition~\\cite{TrappVander,VorGar}. This transition\nat atmospheric pressure is, at least,\nweakly first order, not withstanding the qualitative similarity the system\nhas to an Ising lattice. The first-order character of the transition\nhas led to descriptions of the system in terms of a compressible Ising\nmodel\\cite{isingf,rgar,baker,baker1,slichter}.\n\nIt has also been established that there is hysteresis associated with\nthe $\\delta \\rightarrow \\gamma$ phase transition in measurements of\nthe specific volume~\\cite{Fredericks,NisPoy},\nthe spin-lattice relaxation time~\\cite{ManTrapp,TopRichtSpring}, and neutron\nscattering measurements of the [111] Bragg\nintensity~\\cite{TopRichtSpring}. Pressure-dependent\nstudies of this system~\\cite{TrappVander,ManTrapp,WeinGarl}\nshow that the nature of the transition is a function of pressure.\nAt high pressures ( greater than 1400 atmospheres),\nthe transition changes from first order to continuous. This\ntransformation in behavior is characteristic of a\ntricritical point~\\cite{goldenfeld}. The location of the tricritical\npoint on the phase diagram is known to shift to lower pressures when the\nammonium ions are deuterated\\cite{NisPoy,WeinGarl},\nimplying the importance of\nquantum effects.\nTo date no complete explanation of the microscopic\norigin of the tricritical point in ammonium chloride and its\ndeuterated counterparts is available.\n\nThe purpose of the work presented here is to provide information leading\nto a\nmicroscopic description of the order-disorder phenomena in ammonium\nchloride that is more complete than the traditional Ising picture.\nOur motivation for the study comes from recent\ninvestigations\\cite{berry}\ninto the analogues of phase transition phenomena in small\nclusters. Many small clusters have thermodynamic properties as a\nfunction of temperature that have been interpreted as analogues of bulk\nphase transition behavior. As examples of\nfirst-order transition behavior, small clusters of rare gas\natoms exhibit heat capacity anomalies\\cite{ffd}, and rapid increases in\ndiffusivities\\cite{berry} that are characteristic of bulk melting.\nRecently, the study of phase transition analogues in small clusters has\nbeen extended to cases that are similar to second-order transitions.\nLopez and Freeman~\\cite{lopez} observed\nheat capacity\nanomalies in model\nPd-Ni alloy clusters,\nthat are reminiscent of the order-disorder transitions known to occur in\nsome bulk alloy materials like $\\beta$-brass\\cite{kittel}. Studies of\nanalogues to second-order phase transitions in other kinds of materials\ncontinue to be of interest. Given this background, it is our\nintention ultimately\nto investigate\nthe possibility of orientational disordering transitions in ammonium\nchloride clusters. To do justice to such a study\na detailed simulation of the analogous bulk\ntransition using the same model potential is important. Providing bulk\ninformation for future work on the cluster systems is an important\nmotivation of the work we report.\n\nA review of some previous computational studies of\ntransitions in molecular and\nionic crystals has been given by Klein~\\cite{kleinrev}.\nThere have also been studies related to the present\nwork by Smith~\\cite{smith}\nand by H\\\"{u}ller and Kane~\\cite{huller}\n who focused on the orientational motion of the\nammonium ions and by\nO'Shea~\\cite{shea} who studied related octapolar solids. More pertinent\nto the present discussion is the work\nof Klein, McDonald and Ozaki~\\cite{KleinMacOz}\n who studied NH$_4$Br, KClO$_4$ and\nLi$_2$SO$_4$ using molecular dynamics methods. Klein {\\em et\nal.}~\\cite{KleinMacOz}\nintroduced a suitable order parameter to monitor the disordering\ntransition in ammonium bromide. We shall make use of the same order\nparameter in the work we report.\n\nThe contents and organization of the remainder of this paper are as\nfollows. In the next section the model potential is discussed along\nwith the basic structural features of the ammonium chloride lattice\nimplied by the model. In Section III we discuss the computational\ndetails including how we resolve quasiergodicity problems in the\nsimulations. We present the computed thermodynamic properties in\nSection IV including a discussion of the chosen order parameter. Of the\nproperties given in Section IV, we show that the model predicts no peak\nin the isothermal compressibility at the disordering transition for the\nsimulation sample sizes used in the current work. In Section V we\nidentify the lack of peak in the compressibility with finite size\neffects by presenting results of thermodynamic properties predicted in a\ncompressible Ising calculation as a function of sample size. We\nsummarize our findings and discuss their significance in Section VI.\n\n\\section{Theoretical Model} \\label{s:model}\n\n\\subsection{Model Potential}\n\nIn the present study we use classical Monte Carlo methods to compute\nthermodynamic\nproperties for ${\\rm NH}_4{\\rm Cl}_{\\rm (s)}$ as it undergoes an order to disorder transition. The\napplication of classical mechanics to\npair models of ionic solids has been explored by Klein and coworkers\n{}~\\cite{kleinrev,KleinMacOz},\nwho have\nused molecular dynamics methods to study dynamical processes in\nseveral ionic crystals. Their work gives us\ngood reason\nto expect that useful information about the order to disorder transition\nin ammonium chloride can be obtained using a simple pair model potential\ndominated by Coulombic contributions.\nWe independently evaluate the success of this approach\nby using the model potential to predict such bulk properties\nas the lattice constant\nand the barrier to rotation of a single ammonium ion\n between the two local potential\nminima. We shall confirm that the model potential yields values for\nthese observables that are in reasonable\nagreement with experiment, and we can expect that the potential is\nsufficiently accurate\nfor our purposes.\nThe approach we use to validate the effective pair potential has been\nadvocated elsewhere~\\cite{AllenTild}.\n\nThe specification of the potential surface is somewhat simplified by the\ndynamical\nmodel we assume. In all calculations reported here, we treat the ${\\rm NH}_4^+$\nmolecular\n ions\nas rigidly rotating bodies, with the center of mass of each ${\\rm NH}_4^+$ fixed at\nthe\ncenter of\neach cubic cell. This dynamical model makes it unnecessary to specify an\ninternal vibrational potential\nfor the ${\\rm NH}_4^+$ molecule. We need only set ${\\rm NH}_4^+$ - ${\\rm NH}_4^+$,\n${\\rm NH}_4^+$ - ${\\rm Cl}^-$, and ${\\rm Cl}^-$ - ${\\rm Cl}^-$ interactions to\nspecify a useful effective interaction potential.\nEach of these interactions is\nfurther\n decomposed\ninto a total of six pair atom-atom interaction terms. Letting {\\bf r}\n designate the\ncoordinates of all interacting particles in the system under consideration,\nwe assume that the total interaction potential $U({\\bf r})$ is given by\n\n\\begin{equation}\nU({\\bf r}) = \\sum_i \\sum_{j > i}\nu_{2} (r_{ij})\n\\end {equation}\nwhere $r_{ij}$ is the distance between particle $i$ and particle $j$, and\n$u_{2}$ is\na pair potential of the form\n\n\\begin {equation}\nu_2 (r_{ij}) = A_{ij} \\exp(- \\alpha_{ij} r_{ij})\n + { {D_{ij} }\\over {r_{ij}^{12}} }\n + {{q_i q_j }\\over {r_{ij}} }\n - { {C_{ij} \\over {r_{ij}^6} } } .\\label{eq:pair}\n\\end{equation}\nMost of the parameters used to define the potential are presented in\nTable 1. In addition as used elsewhere~\\cite{KleinMacOz,jorg1} we set\n$q_H=.35$\nand $q_N=-.40$.\n We also set $q_{Cl}=-1.00$~\\cite{jorg}. The charges used on the\nnitrogen and hydrogen atoms of the ammonium ion were confirmed by\nindependent {\\em\nab initio} calculations~\\cite{ferg}.\nIn Table 1\nwe have assumed transferability of most of the atom-atom interaction\nparameters from other, similar\nsystems studied previously by Klein {\\em et al.}~\\cite{KleinMacOz}\n and Pettit and Rossky~\\cite{pr}. For\nthe parameters not available from previous work, we use\nstandard combination rules, i.e.\nrelations of the form\n\\begin{equation}\nA_{ij} = \\sqrt{A_{ii}A_{jj}} ,\n\\end{equation}\n\\begin{equation}\nC_{ij} = \\sqrt{C_{ii}C_{jj}} ,\n\\end{equation}\nand\n\\begin{equation}\n\\alpha_{ij} = { {\\alpha_{ii}+\\alpha_{jj}}\\over{2} } .\n\\end{equation}\nThe specific sources for the parameters used in Eq.(\\ref{eq:pair}) are\ngiven as footnotes in Table 1.\n\nAs in any calculation,\nto generate the results that follow both in this and subsequent sections, a\nfinite representation of the lattice was used. To reduce the\nimportance of edge\neffects, we also used standard minimum image periodic boundary\nconditions~\\cite{AllenTild}.\nIn the main (or central) sample cell we included either 8\nion pairs (48 atoms) corresponding to a 2x2x2 lattice, or 27 ion pairs\n(162 atoms)\ncorresponding to a 3x3x3 lattice. To evaluate efficiently the sums over\nthe periodic images we also used standard Ewald methods with vacuum\nboundary conditions~\\cite{AllenTild}.\nWe found the Ewald calculations to be converged by\nsetting the decay parameter of the error functions to 5.55 in units of\nthe inverse length of a side of the central sample cell~\\cite{heyes}\nand including\n125 reciprocal lattice vectors in the reciprocal lattice sum.\n\n\\subsection{Properties of the lattice within the model potential}\n\\label{s:prop}\n\nA primary indication of the validity of the parameters listed\nin Table 1 is given by comparing the lattice constant and cohesive energy\nat 0K predicted\nby the model potential with experiment. Using the parameters in Table\n1, we have minimized the energy of ammonium chloride using the cesium\nchloride phase. We have assumed all ammonium ions in the lattice are\noriented in the same direction with respect to the crystallographic axes\nin the manner shown in Figure 1. The number of ion pairs in the\ncentral simulation cell was taken to be 27, and the Ewald sums have been\nevaluated as discussed above. The resulting lattice constant\n$a_0$ (the\ndistance between the nitrogen atom on an ammonium ion and an adjacent\nchloride ion) is calculated to be 3.789 \\AA \\ with a cohesive energy of\n-759.7 kJ mol$^{-1}$. The agreement with the experimental lattice\nconstant (3.868 \\ \\AA)~\\cite{lbj}\n and the experimental cohesive energy at 298K (-697 kj\nmol$^{-1}$)~\\cite{wilson}\nis within acceptable limits. The differences\nbetween the calculated and experimental values are a result of finite size\nand thermal effects as well as the details of the model potential.\n\nAnother indication of the accuracy of the model potential is given in\nFigure 2 where the potential energy $U$ of the lattice with 27 ion pairs\nin the\ncentral lattice is plotted as a function of the rotation angle $\\phi$ of the\ncentral ammonium ion [see Figure 1]. At an angle of $\\phi=0$\n(where we set the zero of energy in this figure)\nall the ammonium ions in\nthe lattice are oriented with respect to a set of crytallographically\nequivalent chloride ions. The central ammonium ion is then rotated as\nin Figure 1 until an angle of $\\phi=\\pi \/2$ where the central ammonium\nion is oriented with respect to the alternate set of chloride ions in\nthe lattice. At this angle the ammonium ion finds a local potential\nminimum higher in energy than the absolute minimum at $\\phi=0$. The\n$\\sim$18 kJ mol$^{-1}$ barrier to rotation evident in Figure 2 is in\ngood agreement with experimental estimates based on NMR\nand other data~\\cite{disorder}. This\nagreement is another indication that the potential model can be expected to\nbe at least\nsufficiently accurate to account\nqualitatively for the order to disorder phenomenology\nof the system.\n\nIn addition to calculating the barrier to rotation for the ammonium\nions, we investigated the effective range of the ammonium-ammonium\ninteractions in the lattice. As an initial geometry we considered a\nlattice with all ammonium ions oriented in the same direction with\nrespect to a set of crytallographically equivalent set of chloride ions\nexcept for the central ammonium ion in the simulation cell.\nThis central ammonium ion was oriented to the other set of chloride ions (i.e.\nat an angle of $\\phi =\\pi \/2$ in Figure 2).\nWe then compared the energy of the lattice with a single ammonium ion\nout of orientation, with the energy obtained by\nrotating another ammonium ion in the lattice at a distance $R$ from the\ncentral ammonium ion so that the central ion and the additional ion were\noriented in the same direction.\nIt is important to recognize that the energy calculations were performed\nwithout any nearest neighbor assumptions about the range of the interactions.\nThe change in potential energy $\\Delta U$\n obtained from this\nprocess is given in Figure 3 as a function of $R$.\nAt distances beyond the first ammonium ion cage, $\\Delta U$ becomes\nconstant,\nand\nas is evident from Figure\n3, the interaction between ammonium ions can be well represented by\na nearest neighbor model.\n\n\\subsection{Implications}\n\nThe simple pair model expressed in Eq.(\\ref{eq:pair}) gives rise to a stable\nCsCl lattice with the ammonium ions oriented in the same direction in\nits lowest energy 0K structure. The orientational ordering is a\nconsequence of the attractive octapole-octapole interaction between the\nammonium ions. For ammonium chloride the octapole-octapole\ncoupling is sufficiently weak that interactions beyond the near neighbors\ncan be neglected.\n\nThese preliminary indications lend support to representing the\nsystem\nby a simple Ising model with a positive ferromagnetic coupling\nconstant. However,\nas indicated in the Introduction the simple Ising picture is\nnot of sufficient complexity to explain the phenomenology of the order\nto disorder transition. The transition from first order to second order\nbehavior, characteristic of a tricritical point, provides justification\nfor investigating the transition with more detail than the Ising\npicture. In the next section we develop the necessary Monte Carlo tools\nto investigate the system within the model potential of\nEq.(\\ref{eq:pair}). By performing simulations in the\nisothermal-isobaric ensemble, coupling between lattice motions and\nrotational motions of the ammonium ions will be included.\n\n\\section{Simulation method}\n\nThe details of the Metropolis Monte Carlo method~\\cite{mrrtt}\nboth in the canonical and in the isothermal-isobaric ensemble\nbeen discussed in many references~\\cite{AllenTild,KalWhit,Mac}.\nIn this section we explain some of the\ndetails specific to the simulations performed in the current work, and\nthe approach we used to insure that the simulations were done in\nan ergodic fashion.\n\nSince the Ewald sums\nconstituted the dominant fraction of the computational effort in this\nwork, we examined the consequences of decreasing the number of reciprocal\nlattice\nvectors. Although the absolute value of the cohesive energy was\nsensitive to including fewer reciprocal lattice vectors,\nwe found the lattice constant\nand the barrier to rotation of the ammonium ions (See Section\n\\ref{s:prop})\nto change by $\\sim$0.3 percent when only 8 reciprocal lattice vectors\n(rather than the converged 125) were included.\n Our interest is in the order to disorder\ntransition, and the transition can be expected to be sensitive to the\nbarriers and not to the absolute value of the cohesive energy of the\ncrystal. We tested this assumption by comparing fluctuation\nquantities\n(e.g. heat capacities, compressibilities, etc.)\nsensitive to the location of the transition in the canonical\nensemble with 8 and 125 reciprocal lattice vectors included. We observed no\nsignificant changes in the computed properties. Most of the Monte Carlo\nresults\nwere determined with the smaller set of\nreciprocal lattice vectors.\n\nThe calculations in the canonical ensemble were performed using central\ncell sizes of both 8 ion pairs and 27 ion pairs. For each central cell\nsize, the lattice parameters were adjusted to give the minimum energy\nfor the orientation having\n all the ammonium ions in the same direction.\nAbout each ammonium ion in the lattice we constructed a set of\northogonal Cartesian axes.\nEach\nMonte Carlo point in the canonical simulations consisted of a rotation\nof a randomly chosen ammonium ion about\none Cartesian axis randomly chosen~\\cite{watt}.\n The\nmaximum allowed rotation angle was adjusted so that about fifty percent\nof the moves were accepted. As is typical in Monte Carlo simulations,\nthis maximum allowed rotation angle was a function of temperature and\nwas found by performing short run experiments at each temperature. In\naddition to these normal Metropolis moves, we found it necessary to\ninclude moves with larger maximum displacements ten percent of the\ntime. The purpose of the moves with magnified displacements (magwalking)\nwas to insure that the ammonium ions were given the opportunity to\novercome the potential barrier separating the two minima in the\npotential surface (See Figure 2). Consequently, the maximum\ndisplacement in the magwalking moves was taken to be $\\pi \/2$.\n\nEvidence that the magwalking scheme discussed in the previous paragraph\nprovides an ergodic distribution is given in Figure 4 where the average\npotential\nenergy of the 8 ion pair lattice is plotted as a function of\ntemperature. The data used in generating both curves of Figure 4 was\ninitiated with configurations having\n the ammonium ions in random orientations. Quasiergodicity\ndifficulties~\\cite{ffd,vw}\ncan be anticipated at low temperatures where all the\nammonium ions are expected to have the same orientation with respect to\nthe chloride ions. The data in the upper curve of Figure 4 was\ngenerated using Metropolis Monte Carlo methods with a fixed step size\nfor all moves. The unstable behavior at low temperatures is evident.\nThe instability is a result of rapid quenching of the initial random\norientations of\nthe ammonium ions into\nlocal, high-energy minima. This disordered trapping causes the\nquasiergodicity in the sampling of the rotations.\nIn the lower curve of Figure 4, the magwalking scheme described in\nthe previous paragraph was used. The sensible behavior at low\ntemperatures is clear, and the simple magwalking scheme can be expected\nto solve the quasiergodicity problems in this system.\n\nIn the isothermal-isobaric simulations the Monte Carlo sampling is with\nrespect to the distribution\n\\begin{equation}\n\\rho({\\bf r})= \\Delta ^{-1} \\exp(-\\beta U ({\\bf r}) -\\beta pV )\n\\end{equation}\nwhere $U$ is the system potential energy, $\\beta = 1\/k_B T$ where $T$ is\nthe temperature and $k_B$ is the Boltzmann constant, $p$ is the\npressure,\n $V$ is the volume\nand $\\Delta^{-1}$ is a normalization.\n To perform the isothermal-isobaric simulations\n in addition to the rotational\nmoves used in the canonical study, the lattice parameter (and consequently\nthe system volume) was varied.\n{}From numerical experiments we found that including volume fluctuations\nabout forty percent of the time gave good convergence of the computed\nproperties.\nThe volume fluctuations were included with Metropolis Monte Carlo moves\nagain with a maximum displacement chosen so that about fifty percent of\nthe attempted fluctuations were accepted. Unlike the rotational moves,\nwe found no evidence of quasiergodicity in the volume fluctuations.\nWhen a mixture of maximum step sizes were used in the volume\nmoves, we found no\nstatistically significant changes in the computed thermodynamic\nproperties.\n\nThe volume fluctuations included in this work correspond to\na single vibrational breathing mode for the lattice. Of course, the\nreal lattice dynamics in NH$_4$Cl is more complex than this simple\npicture. However, at least we have been able to include important\ncontributions to the coupling between the rotational and vibrational\nmodes of the lattice.\n\nIn the calculations that follow, we have calculated several fluctuation\nquantities in addition to the energy $E$ and enthalpy $H=E+pV$\n of the crystal.\nThese fluctuation quantities include the constant volume and constant\npressure heat capacities\n\\begin{equation}\nC_V\/k_B = \\frac {3 N }{2} + \\beta ^2 [ -^2] \\label{eq:cv}\n\\end{equation}\nand\n\\begin{equation}\nC_p\/k_B = \\beta ^2[-^2] \\label{eq:cp}\n\\end{equation}\nthe isobaric coefficient of thermal expansion\n\\begin{equation}\n\\alpha = \\frac {1}{V} \\left ( \\frac {\\partial V}{\\partial T} \\right\n)_{N,p}\n\\end{equation}\n\\begin{equation}\n = (k_B T^2 V)^{-1} [-] \\label{eq:alpha}\n\\end{equation}\nand the isothermal compressibility\n\\begin{equation}\n\\kappa=-\\frac {1}{V} \\left ( \\frac {\\partial V}{\\partial p} \\right\n)_{N,T}\n\\end{equation}\n\\begin{equation}\n = (k_B T V)^{-1} [-^2] \\label{eq:kappa}\n\\end{equation}\nIn Eq.(\\ref{eq:cv}) the notation $<>$ represents averages in the\ncanonical ensemble, and in Eqs.(\\ref{eq:cp}),(\\ref{eq:alpha})\nand (\\ref{eq:kappa}) the notation\n$<>$ represents averages in the isothermal-isobaric ensemble. In the\nresults we report in the next section, $C_V, C_p, \\alpha$ and $\\kappa$\nwere calculated directly from these fluctuation expressions.\n\nIn the calculations that follow, error bars were estimated at the double\nstandard deviation level by ``binning'' the data and estimating the\nvariance of the bin averages about the total walker average. In the\nlimit of large numbers, this method is known to be appropriate for the\ncorrelated data generated by a Metropolis walk. However, for finite\nMonte Carlo samples the suitability of the binning parameters should be\nchecked for a given model system to ensure the reliability of the error\nestimates. We performed these checks by comparing the computed error\nestimates to those obtained through covariance calculations of the\nerror~\\cite{stam,top}.\nWe observed the two methods to give similar results, indicating\nthat suitable binning parameters were chosen for this study.\n\n\\section{Results}\n\nIn this section we provide results of the Monte Carlo studies of the\nproperties of the NH$_4$Cl lattice as a function of temperature. As\nexpected we shall find features in the thermodynamic properties as a\nfunction of temperature that can be associated with a transition from\nrotational order to disorder. To monitor the degree of\n orientational order in the\nlattice, we use an order parameter introduced by Klein {\\em et al}~\\cite\n{KleinMacOz}.\nTo\ndefine the order parameter we place the origin of a set of Cartesian axes\non the nitrogen atom of each ammonium ion, and we orient the Cartesian\naxes so that the $z$-axis is orthogonal to\na square face of chloride\nions in the chloride cage. The $x$ and $y$ axes are then oriented along\nthe edges of the lattice as shown by the coordinates displayed in Figure\n1.\n For each ammonium ion we define\n\\begin{equation}\nM_j=\\frac {3 \\sqrt{2}}{4} \\sum_{i=1}^4 x_j^i y_j^i z_j^i \\label{eq:mj}\n\\end{equation}\nwhere $x_j^i$ is the $x$-component of the coordinate of\na unit vector pointing from the origin toward\nthe $i$'th\nhydrogen atom on ammonium ion $j$ and the summation runs over the 4\nhydrogen atoms on the ammonium ion. $M_j$ is defined so that\n$M_j=1$\nwhen\nammonium ion $j$ is oriented exactly to one set of chloride ions, and\n$M_j=-1$ when it is oriented exactly to the alternate set of chloride\nions (see Figure 1).\n Of course $M_j$ is a continuous function of the coordinates, so\nthat it takes on values of $\\pm$ 1 only when the hydrogen atoms point\nexactly to a set of chloride ions. This definition of $M_j$ enables a\nmapping of the orientations of the ammonium ions onto a spin variable.\nWhen $M_j$ is positive we can think of ammonium ion $j$ as having a\npositive or ``up'' spin and when $M_j$ is negative, we can think of\nammonium ion $j$ as having a ``down'' or negative spin. To clarify the\nmapping,\nin Figure 5 we display a particular configuration of the ammonium chloride\nlattice\ntaken from a canonical simulation with 8 ion pairs in the central\ncell at 300K. We have placed arrows on each nitrogen atom in the\nlattice to carry information about the algebraic sign of $M_j$.\nPositive $M_j$ is represented by an up arrow and can be interpreted as\nan up spin. Similarly, negative $M_j$ is represented by a down arrow and\ncan be interpreted as a down spin.\nTo simplify the\ndiscussion often we shall describe the orientations of the ammonium ions in\nterms of these spin variables. In\nanalogy with the Ising model, we can also define the {\\em magnetization per\nsite}\n$M$ of the lattice by\n\\begin{equation}\nM=\\frac {1}{N} \\sum_{j=1}^N M_j\n\\end{equation}\nwhere the summation on $j$ is over the $N$ ammonium ions in the lattice.\nAssociated with the magnetization is a susceptibility per site $\\chi$\ndefined by\n\\begin{equation}\n\\chi = N(-^2) \\label{eq:chi}\n\\end{equation}\nwhere the notation $<>$ in Eq.(\\ref{eq:chi}) represents an ensemble\naverage. We use $M$ as the order parameter for the orientational order to\ndisorder\ntransition in the system.\n\n\\subsection{Canonical simulations}\n\nIn the canonical simulations for each temperature we included 50000\npasses\nwithout the accumulation of data followed by 200000 passes with data\naccumulation. Each pass consisted of cycling through the ammonium ions\nin the central simulation cell and attempting to rotate each ion once.\n\nWe begin by examining the changes in computed thermodynamic properties\nthat accompany alterations in the size of the central sample cell. In\nFigure \\ref{f:cv} we present the constant volume heat capacity $C_V$\nper ion pair in\nunits of the Boltzmann constant as a function of temperature. The upper\ncurve gives results when the central cell consists of 8 ion pairs and the\nlower curve gives results for a 27 ion pair central cell. By increasing\nthe size of the central cell the width of the heat capacity maximum\nnarrows as expected.\n\nBy examining configuration files it is possible to verify that the\nmaxima in the heat capacity seen in Figure \\ref{f:cv} are a result of\nthe rotational disordering of the ammonium chloride lattice. We can\nconfirm this interpretation by monitoring the magnetization as a\nfunction of temperature. In Figure \\ref{f:m2} the curve connected by\ndiamonds are computed values of $$ as a function of temperature for\nthe 27 ion pair lattice. We have chosen to present $$ rather than\n$$ because in any finite lattice $=0$ at any finite\ntemperature. In a completely\noriented configuration as the temperature approaches zero, $M^2$ is\nalways 1 whereas $M$ can be either 1 or -1. Then at low temperatures\n$$ must approach unity, and at high temperatures $$ must\napproach zero. This behavior is evident in Figure \\ref{f:m2}. At the\ntransition temperature of $\\sim$210K $$ changes rapidly. This\ntransition temperature matches the peak of the maximum in $C_V$ as a\nfunction of temperature. Also presented in Figure \\ref{f:m2} is\nthe result of mapping the\nmagnetization onto a spin model. The data for this spin model are\nplotted in Figure \\ref{f:m2} as points represented by triangles. In the\nspin model we set\n\\begin{equation}\nM_s=\\frac{1}{N} \\sum_{j=1}^{N} \\mbox{sign} (M_j)\n\\end{equation}\nwhere sign($M_j$) is the algebraic sign of $M_j$ and $M_s$ represents\nthe magnetization of the system described by the spin variables.\n\n The difference\nbetween the two curves given in Figure \\ref{f:m2} clarifies the extent\nto which the decrease in the magnetization can be attributed to\nlibrational modes. Since $M$ is a continuous function of the\ncoordinates of each ammonium ion, the magnetization will decrease as a\nfunction of temperature even if all the ions remain ordered. In\ncontrast, $M_s$ will change only if an ammonium\nion in a particular configuration\nchanges its ``spin;''\ni.e. overcomes the barrier between the two minima in the rotational\npotential surface. By comparing the decay of the order parameter to the decay\nof the\nIsing-like spin parameter in Figure \\ref{f:m2}, we observe\nthat the initial decay of the order parameter is a result of the onset of\nlibrational motions. At the transition temperature, the order parameter\nrapidly decays to zero because of the loss of long range order of the\nNH$_4^+$ orientations.\n\nAlthough in a finite system $$=0,\n in an actual Monte Carlo simulation $$ may differ from\nzero, because both inverted configurations may be reachable only in\nMetropolis walks that are sufficiently long. The length of the walk required\nto actually calculate a zero value for $$ will grow both with\ndecreasing temperature and increasing system size. The effect of the\nfinite walks can be made apparent by examining the susceptibility $\\chi$\nas a function of temperature. Such a graph is given in Figure\n\\ref{f:chi} for the lattice consisting of 27 ion pairs in the central\ncell. The maximum of $\\chi$ occurs at the same temperature as the\ntemperature at which $$ changes rapidly. The fluctuations in\n$\\chi$ at this temperature are also large. Of course in the limit of an\ninfinite system $ \\neq 0$, and the susceptibility we calculate should\napproach the infinite system result with increasing system size.\n\n\\subsection{Isothermal-isobaric simulations}\n\nCanonical simulations do not have sufficient flexibility to incorporate\ncoupling of the rotational modes of the ammonium ions with the\nvibrational modes of the lattice. To obtain preliminary understanding\nof the effects of such couplings, we have performed Monte Carlo\nsimulations of the thermodynamic properties of the system in the\nisothermal-isobaric ensemble. By fixing the pressure of the system,\nthe isothermal-isobaric simulations more closely match the experimental\nsituation than the canonical studies. In exchange for this closer\nconnection with experimental data, calculations in the\nisothermal-isobaric ensemble are computationally more demanding than\ncalculations in the canonical ensemble. The range of system sizes that\ncan be studied while achieving acceptable levels of convergence is\nrestricted.\n\nThe simulations in the isothermal-isobaric ensemble were performed on a\n27 ion pair representation of the ammonium chloride lattice in the\ncentral cell. At each temperature 300000 Monte Carlo passes were\nperformed without data accumulation followed by about one million passes\nwith data accumulation. Each pass consisted of sequential attempted\nrotations of each ammonium ion in the same manner as in the canonical\nsimulations. Additionally, volume fluctuations were attempted in forty\npercent of the passes. Ten percent of the rotational moves used a\nmaximum rotation angle of $\\pi \/2$ using the magwalking scheme found to\nbe successful in the canonical simulations. The remaining ninety per\ncent of rotational moves used a maximum rotational displacement\ndetermined so that about fifty percent of the attempted moves were\naccepted.\n\nIn Figure \\ref{f:m2npt} we present\n$$ as a function of temperature calculated both using the\ndefinition of the order parameter given in Eq.(\\ref{eq:mj}) (the points\nrepresented by diamonds) and the projection of the order parameter onto\nspin variables (the points represented by triangles).\nThe projected spin order parameter is the same as that used in the\ncanonical simulations and discussed previously. The region of rapid\nchange in the order parameter is at a temperature of about 250K. The\nonset of the transition is more clearly seen in Figure \\ref{f:chinpt}\nwhere we plot the susceptibility as a function of temperature. The\nmaximum occurs at a temperature of about 250K, a temperature where the\nfluctuations in $\\chi$ are large.\n\nWe measure the average volume of the crystal in terms of the lattice\nparameter $a_0$. The average volume is\nof interest because we would expect to observe a rapid volume change\nnear any first order transition. In Figure \\ref{f:a0npt} the lattice\nparameter in \\AA \\ is plotted as a function of temperature. Although no\nrapid change in the lattice parameter is seen, there is a clear\nchange of slope at the transition temperature.\n\nIn Figure \\ref{f:cpnpt} we give the constant pressure heat capacity\n$C_p$ per ion pair,\nthe isothermal compressibility $\\kappa$ and the isobaric\ncoefficient of thermal expansion $\\alpha$ as a function of temperature.\nThe onset of the maxima in $C_p$ and $\\alpha$ match the onset of the\norder to disorder transition as identified by the order parameter. From\nbasic considerations~\\cite{whee}\na maximum is expected in $\\kappa$ at the transition\ntemperature as well. As is evident from Figure \\ref{f:cpnpt}, $\\kappa$\nis found to be a monatomic increasing function of the temperature with\nno apparent peak.\nTo test the sensitivity of the behavior of $\\kappa$ to the number of\nreciprocal lattice vectors included in the Ewald sums, we increased the\nset from 8 to 125 reciprocal lattice vectors. No significant change in the\ncompressibility as a function of temperature was observed.\nWe believe that the lack of peak is a consequence of\nthe finite size of the central simulation cell, and we give evidence for\nthis in the next section.\n\n\\section{A Compressible Ising Model}\n\nIn the previous section we showed that in the\nisothermal-isobaric ensemble at the disordering transition,\npeaks were observed in the constant\npressure heat capacity and in the isobaric coefficient of thermal\nexpansion. The location of the peak maxima were in good agreement with\nthe temperature at which there were rapid changes in the order\nparameter. The identification of the peak maxima with the disordering\nphenomena seems well justified.\n\nIn contrast to the properties discussed in the previous paragraph, no\nmaximum was observed in the isothermal compressibility.\nSince the compressibility should have a specific heat-like divergence at the\ntransition temperature in the infinite system~\\cite{whee},\nthe lack of observation is a concern. In this section we\nshow that similar behavior is found in finite representations of a\ncompressible two-dimensional Ising model\non a square lattice, and the lack of peak in the\nisothermal\ncompressibility we observed for NH$_4$Cl may be attributable to finite size\neffects.\n\nThe compressible model we use is based on the two-dimensional Hamiltonian\n\\begin{equation}\nH=-\\frac{a}{R^7} \\sum_{} S_i S_j + 4 \\epsilon \\left [ \\left ( \\frac\n{\\sigma}{R} \\right )^{12}\n- \\left ( \\frac {\\sigma}{R} \\right ) ^6 \\right ] \\label{eq:iham}\n\\end{equation}\nwhere $S_i$ is the spin on site $i$ and can take on values of + or - 1,\nthe notation $$ on the sum represents summation over nearest\nneighbors only, $a$, $\\epsilon$ and $\\sigma$ are parameters and $R$ is\nthe lattice constant.\nThe first term in the Hamiltonian is the standard spin-spin interaction\nin the Ising model with a lattice-parameter dependent coupling constant.\nWe choose the $R$-dependence of the coupling constant to decay with\ndistance like the octapole-octapole interaction in ammonium chloride\n($R^{-7}$). The second term in the Hamiltonian is in the form of a\nusual Lennard-Jones potential and provides a balance for the volume\nfluctuations so that the lattice will relax to a physical nearest\nneighbor distance.\nIn the Ising simulations that follow we have\nattempted to choose parameters\nfor Eq. (\\ref{eq:iham})\nthat mimic the ammonium chloride results\nof the previous section.\nFor the bulk Lennard-Jones $\\sigma$ parameter, we take a value that can\nmake the equilibrium lattice distance near that of the real crystal. We\nthen take $\\sigma=R_e\/2^{1\/6}$ where $R_e=7.31$ Bohr. We take $\\epsilon$\nto match the ammonium chloride lattice energy of 8380K.\nSince the barrier to changing the orientation\nof a single ammonium ion in a completely ordered ammonium chloride\nlattice is about 1000K, we take $a=2000 R_e^{7}\/N_{\\mbox{spin}}$, where\n$N_{\\mbox{spin}}$ is the number of spins in the primary cell.\n\nWe determined the properties of the two-dimensional compressible Ising\nlattice defined by Eq.(\\ref{eq:iham}) using Monte Carlo simulations\nin the isothermal-isobaric ensemble. The\nsimulations consisted of two-million moves without data accumulation\nfollowed by 10 million Monte Carlo moves with the accumulation of data\nat each temperature. Changes in the $R$-parameter were made forty per\ncent of the time and the two-dimensional pressure was arbitrarily set to\n1 atomic unit of pressure. Simulations were performed on 4x4, 8x8,\n16x16 and 32x32 lattices with periodic boundary conditions included.\n\nShown in Figure \\ref{f:cpi} is the constant pressure heat capacity of the\ncompressible Ising\nmodel for 4x4, 8x8, 16x16 and 32x32 lattices from top to bottom in\nthe figure. The peak at 240K in the heat capacity occurs at the same\ntemperature as rapid changes in the magnetization of the model.\nAs the number of spins in the simulation is increased the width\nof the peak narrows as expected. Given in Figure \\ref{f:ki} is the isothermal\ncompressibility as a function of temperature. Again from top to bottom\nis $\\kappa$ for a 4x4, 8x8, 16x16 and 32x32 lattice. For the\n4x4 lattice, no peak appears in the compressibility. As the sample size\nis increased, a peak develops in the compressibility at the transition\ntemperature. Evidently,\nthe existence of a peak in\nthe compressibility is more sensitive to finite\nsize effects than the heat capacity. Although not shown here, the\nisobaric coefficient of thermal expansion shows a peak at the transition\ntemperature\nfor all lattice\nsizes. These Ising results lend support to our assumption that the lack\nof peak in $\\kappa$ in the isothermal-isobaric simulations of ammonium\nchloride is a result of finite size effects.\n\n\\section{Summary and discussion}\n\nIn this work we have applied Monte Carlo methods in the canonical and\nisothermal-isobaric ensembles to calculate the thermodynamic properties\nof ${\\rm NH}_4{\\rm Cl}_{\\rm (s)}$ modeled by simple point charge pair interaction potentials.\nIn both ensembles the model potential predicts an order to disorder\ntransition associated with the rotational orientations of the ammonium\nions in the lattice. In the isothermal-isobaric ensemble at 1\natmosphere pressure, the transition temperature is about 250K,\nas determined from the rapid growth of the susceptibility.\n Although\nwe have made no effort to determine the transition temperature with any\nprecision, the agreement between the approximately determined\ntemperature and the experimental result (243K) is satisfying.\n\n\nThe central sample cell used in the simulations contained a maximum of\n27 ion pairs. Extensions to larger central cell sizes were inhibited by\nthe large portion of the computer time needed to evaluate the Ewald\ncorrections. Apparently this size limitation produced unreliable values\nfor the isothermal compressibility, a quantity that compressible Ising\ncalculations imply is sensitive to the size of the central cell.\n\nThere are several outstanding questions about the behavior of ammonium\nchloride that require further attention. We found no evidence in our\nsimulations for a rapid change in the molar volume at the transition\ntemperature. Since the transition is known to be first order at\natmospheric pressure\\cite{WeinGarl},\nit is worth examining some of the physical effects\nnot included in the simulations. A true discontinuity in the lattice\nconstant would require an infinite central simulation cell. The unseen\nrapid change in volume may be another example of a finite size effect.\nHowever, in the simulations presented here we have included only those\nlow frequency vibrational modes where the overall crystal symmetry is\nunaltered. Calculations that include more general variations in the\nlocations of the ionic mass centers would clearly be of interest.\nInclusion of the internal vibrations on the ammonium ions can be\nexpected to be of less importance owing to the high frequencies of those\nmotions.\n\nAnother approximation in the results has been the application of classical\nmechanics. Classical calculations can be interpreted as the infinite\nmass limit of a corresponding quantum calculation.\nIf quantum effects were not important, our simulations would be equally\napplicable to the phase diagram of ${\\rm ND}_4{\\rm Cl}_{\\rm (s)}$.\nExperimentally,\na\nsignificant shift of the tricritical point to lower pressures is\nknown to occur when the\nammonium ions are deuterated~\\cite{NisPoy,WeinGarl}.\nIt is possible that the classical system at\natmospheric pressure has a\ncontinuous disordering transition,\n and the location (or existence) of a tricritical point is a consequence of\nquantum effects. A quantum path integral study of ammonium chloride\nwithin the same model potential would clearly be of interest.\n\n\n\\section*{Acknowledgements}\n\nWe would like to thank Professors P. Nightingale, R. Stratt and J. Doll\nfor helpful discussions. RQT would like to thank Dr. Michael New\nand the members of the Ohio Supercomputer Center\n Computational Chemistry Electronic Forum (chemistry@osc.edu)\n for\nhelpful e-mail conversations regarding simulation methods.\nAcknowledgement is made to the Donors of the Petroleum Research Fund of\nthe American Chemical Society for support of this work. The\ncomputational work reported here was supported by an equipment grant\nfrom the National Science Foundation(CHE-9203498).\n\n\\newpage\n\\begin{table} \\centering\n\\caption{The parameters used in the model potential$^d$}\n\n\\vspace{.15in}\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{tabular}{*{5}{c}}\nPair & $A_{ij}$ & $\\alpha _{ij}$ & $C_6$ & $D_{12}$ \\\\\n\\hline\nH-H$^a$ & 1.0162 & 1.9950 & 2.9973& 0 \\\\\nN-N$^b$ & 104.74 & 1.5611 & 25.393 & 0 \\\\\nCl-Cl$^b$ & 125.55 & 1.7489 & 113.68 & 0 \\\\\nH-N$^a$ & 10.318 & 1.7780 & 8.7229 & 0 \\\\\nH-Cl$^c$ & 0 & 0 & 10.033 & 43884.0 \\\\\nN-Cl$^a$ & 114.22 & 1.6550 & 53.736 & 0 \\\\\n\\end{tabular}\n\\end{table}\n\\parbox[b]{10in}{$^a$ Combining rules \\\\\n$^b$ Reference \\onlinecite{KleinMacOz} \\\\ $^c$ Reference \\onlinecite{pr}\n\\\\$^d$ Units of energy in Hartree and units of length in Bohr}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nSince the seminal work of Anderson \\cite{And58}, the investigation of the localization for noninteracting quasi-particles in random media has attracted great attention in physics and mathematics community.\n\nIn mathematics the first rigorous proof of the localization for random operators was due to Goldsheid-Molchanov-Pastur \\cite{GMP77}. They obtained the pure point spectrum for a class of $1D$ continuous random Schr\\\"odinger operators. In higher dimensions, Fr\\\"ohlich-Spencer \\cite{FS83} proved, either at high disorder or low energy, the absence of diffusion for some random Schr\\\"odinger operators by developing the celebrated multi-scale analysis (MSA) method.\nBased on the MSA method of Fr\\\"ohlich-Spencer, \\cite{FMSS85,DLS85,SW86} finally obtained the Anderson localization at either high disorder or extreme energy. We should remark that the method of \\cite{FS83} was simplified and extended by von Dreifus-Klein \\cite{vDK89} via introducing a scaling argument.\nLater, the method of \\cite{vDK89} was generalized by Klein \\cite{Kle94} to prove the Anderson localization for random operators with the \\textit{exponential} long-range hopping. Finally, we want to mention that the MSA method has been strengthened to establish the Anderson localization for random Schr\\\"odinger operators with Bernoulli potentials \\cite{CKM87,BK05,DS19} \\footnote{Very recently, Jitomirskaya-Zhu \\cite{JZ19} provided a delicate proof of the Anderson localization for the $1D$ random Schr\\\"odinger operators with Bernoulli potentials using ideas from \\cite{Jit99}.\n}.\n\n\n\n\n\n\nAn alternative method for the proof of the localization for random operators, known as the fractional moment method (FMM), was developed by Aizenman-Molchanov \\cite{AM93}. This remarkable method also has numerous applications in localization problems \\cite{AWB}. As one of its main applications, the FMM was enhanced to prove the first dynamical localization \\cite{Aiz94} for random operators on $\\Z^d$.\nAnother key application of the FMM in \\cite{AM93} is a proof of the \\textit{power-law} localization for random operators (on $\\Z^d$) with polynomially decaying long-range hopping. Later, the \\textit{power-law} localization for $1D$ polynomial long-range hopping random operators was also established in \\cite{JM99} via the method of trace class perturbations. However, there is simply no MSA proof of the \\textit{power-law} localization for {polynomial} long-range hopping random operators, as far as I know. This is the main motivation of the present work. We remark that the Green's functions estimate in the FMM requires a mild condition (such as the H\\\"older continuity) on the probability distribution of the potential, while it does not apply to purely singular potentials, such as the Bernoulli ones. In order to deduce localization using the FMM, the \\textit{absolute continuity} of the measure was even needed in \\cite{AM93} so that the Simon-Wolff criterion \\cite{SW86} can work.\n\n\nIn this paper we develop a MSA scheme to handle random operators with the \\textit{polynomial} long-range hopping and H\\\"older continuous distributed potentials (including some \\textit{singular continuous} ones). Although the Bernoulli potentials are not considered in the paper, the suggested method is a promising candidate to be applicable in handling operators with Bernoulli potentials and the \\textit{polynomial} long-range hopping. In addition, we think our formulations in this paper may have applications in localization problems for other models. In fact, in a forthcoming paper \\cite{Shifc} we develop a MSA scheme to study some quasi-periodic operators with \\textit{polynomial} long-range hopping.\n\nOur proof is based essentially on Fr\\\"ohlich-Spencer type MSA method \\cite{FS83}. In particular, it employs heavily the simplified MSA method of von Dreifus-Klein \\cite{vDK89} (see also \\cite{K08}). However, one of the key ingredients in our proof is different from that of \\cite{FS83,vDK89} in which the \\textit{geometric resolvent identity} was iterated to obtain the \\textit{exponentially} decaying of off-diagonal elements of Green's functions. Instead, we directly estimate the \\textit{left} inverses of the truncated matrix via information on small scales Green's functions and a \\textit{prior} $\\ell^2$ norm bound of the inverse itself. This method was initiated by Kriecherbauer \\cite{Kri98} to deal with matrices with \\textit{sub-exponentially} decaying (even more general cases) off-diagonal elements, and largely extended by Berti-Bolle \\cite{BB13} to study matrices with \\textit{polynomially} decaying off-diagonal elements in the context of nonlinear PDEs.\n\n Once the Green's functions estimate was established, the proof of the \\textit{power-law} localization can be accomplished with the Shnol's Theorem.\n\n\n\n\n\nThe structure of the paper is as follows. The \\S2 contains our main results on Green's functions estimate (Theorem \\ref{msa}) and the \\textit{power-law} localization (Theorem \\ref{mthm}). The proof of Theorem \\ref{msa} is given in \\S3. In \\S4, the verification of the assumptions (\\textbf{P1}) and (\\textbf{P2}) in Theorem \\ref{msa} is presented. Moreover, the whole MSA argument on Green's functions is also proved there. In \\S5, the proof of Theorem \\ref{mthm} is finished. Some useful estimates are included in the appendix.\n\n\n\n \n\n\\section{Main Results}\n Here is the set-up for our main results.\n \\subsection{Random Operators with the Polynomial Long-range Hopping}\n Define on $\\Z^d$ the {polynomial} long-range hopping $\\mathcal{T}$ as\n\\begin{align}\\label{sp2}\n \n \n \n\\mathcal{T}(m,n)=\\left\\{\\begin{aligned}\n&|m-n|^{-r},\\ {\\rm for}\\ m\\neq n\\ {\\rm with}\\ m,n\\in{\\Z}^d,\\\\\n&0,\\ {\\rm for\\ } m=n\\in{\\Z}^d,\n\\end{aligned}\\right.\n\\end{align}\nwhere $|n|=\\max\\limits_{1\\leq i\\leq d}|n_i|$ and $r>0$.\n\nLet $\\{V_\\omega(n)\\}_{n\\in\\mathbb{Z}^d}$ be independent identically distributed (\\textit{i.i.d}) random variables (with the common probability distribution $\\mu$) on some probability space $(\\Omega,\\mathcal{F}, \\mathbb{P})$ ($\\mathcal{F}$ a $\\sigma$-algebra\non $\\Omega$ and $\\mathbb{P}$ a probability measure on $(\\Omega,\\mathcal{F})$).\n\n\n\nLet $\\mathrm{supp}(\\mu)=\\{x:\\ \\mu(x-\\varepsilon,x+\\varepsilon)>0\\ \\mathrm{for}\\ \\mathrm{any}\\ \\varepsilon>0\\}$ be the support of the common distribution $\\mu.$ Throughout this paper we assume $dd$)\n\\begin{align*}\n \\|\\mathcal{T}\\|\\leq \\sup_{m\\in{\\Z}^d}\\sum_{n\\neq m}|m-n|^{-r}\\leq \\sum_{n\\in{\\Z}^d\\setminus\\{0\\}}|n|^{-r}<\\infty,\n \\end{align*}\nwhere $\\|\\cdot\\|$ is the standard operator norm on $\\ell^2({\\Z}^d)$.} and\n \\begin{itemize}\n \\item[$\\bullet$] ${\\rm supp}(\\mu)$ contains at least two points.\n \\item[$\\bullet$] ${\\rm supp}(\\mu)$ is \\textit{compact}: We have $\\mathrm{supp}(\\mu)\\subset[-{M},{M}]$ for some $M>0$ \\footnote{\nFrom \\cite{K08}, we have for $\\mathbb{P}$ almost all $\\omega$,\n\\begin{align*}\n\\sup_{n\\in{\\Z}^d}|V_\\omega(n)|\\leq M.\n\\end{align*}\nThus we can assume $\\sup\\limits_{n\\in{\\Z}^d}|V_\\omega(n)|\\leq M$ for all $\\omega\\in\\Omega$.\n}.\n\\end{itemize}\n\nIn this paper we study the $dD$ random operators with the {polynomial} long-range hoppin\n\\begin{align}\\label{qps}\n{H}_{\\omega}=\\lambda^{-1}\\mathcal{T} + V_\\omega(n)\\delta_{nn'},\\ \\lambda\\geq1\n\\end{align}\nwhere $\\lambda$ is the coupling constant for describing the effect of disorder.\n\nUnder the above assumptions, $H_\\omega$ is a \\textit{bounded self-adjoint} operator on $\\ell^2({\\Z}^d)$ for each $\\omega\\in\\Omega$.\nDenote by $\\sigma(H_\\omega)$ the spectrum of $H_\\omega$. A well-known result due to Pastur \\cite{Pas80} can imply that there exists a set $\\Sigma$ (\\textit{compact} and \\textit{non-random}) such that for $\\mathbb{P}$ almost all $\\omega$, $\\sigma(H_\\omega)=\\Sigma$.\n\n\n\n\\subsection{Sobolev Norms of a Matrix}\n Since we are dealing with matrices with polynomially decaying off-diagonal elements, the Sobolev norms introduced by Berti-Bolle \\cite{BB13} are useful.\n\nFix $s_0>d\/2$ (s.t. the Sobolev embedding works).\n\nLet $\\langle k\\rangle=\\max\\{1,|k|\\}$ if $k\\in{\\Z}^d$. Define for $u=\\{ u(k)\\}\\in{\\C}^{{\\Z}^d}$ and $s>0$ the Sobolev norm\n\\begin{align}\\label{us}\n\\|u\\|_s^2=C_0(s_0)\\sum_{k\\in{\\Z}^d}|{u}(k)|^2{\\langle k\\rangle}^{2s},\n\\end{align}\nwhere $C_0(s_0)>0$ is fixed so that (for $s\\geq s_0$)\n\\begin{align*}\n\\|u_1u_2\\|_s\\leq \\frac{1}{2}\\|u_1\\|_{s_0}\\|u_2\\|_s+C(s)\\|u_1\\|_s\\|u_2\\|_{s_0}\n\\end{align*}\nwith $C(s)>0$, $C(s_0)=1\/2$ and $(u_1u_2)(k)=\\sum\\limits_{k'\\in{\\Z}^d}{u}_1{(k-k')}{u}_2(k')$.\n\n\n\nLet $X_1,X_2\\subset{\\Z}^d$ be finite sets. Define $$\\mathbf{M}^{X_1}_{X_2}=\\left\\{\\mathcal{M}=(\\mathcal{M}(k,k')\\in \\C)_{k\\in X_1,k'\\in X_2}\\right\\}$$ to be the set of all complex matrices with row indexes in $X_1$ and column indexes in $X_2$. If $Y_1\\subset X_1,Y_2\\subset X_2$, we write $\\mathcal{M}^{Y_1}_{Y_2}=(\\mathcal{M}(k,k'))_{k\\in Y_1,k'\\in Y_2}$ for any $\\mathcal{M}\\in\\mathbf{M}^{X_1}_{X_2}$.\n\n\n\\begin{defn}\\label{snorm}\nLet $\\mathcal{M}\\in\\mathbf{M}^{X_1}_{X_2}$. Define for $s\\geq s_0$ the Sobolev norm of $\\mathcal{M}$ as\n\\begin{align*}\n\\|\\mathcal{M}\\|_s^2=C_0(s_0)\\sum_{k\\in X_1-X_2}\\left(\\sup_{k_1-k_2=k}|\\mathcal{M}(k_1,k_2)|\\right)^2\\langle k\\rangle^{2s},\n\\end{align*}\nwhere $C_0(s_0)>0$ is defined in \\eqref{us}.\n\\end{defn}\n\n\\begin{rem}\n From this definition, we have $\\|\\mathcal{T}\\|_{r_1}<\\infty$ if $r_10$, define the cube $\\Lambda_L(n)=\\{k\\in{\\Z}^d:\\ |k-n|\\leq L\\}$. Moreover, write $\\Lambda_L=\\Lambda_L(0)$. The volume of a finite set $\\Lambda\\subset{\\Z}^d$\nis defined to be $|\\Lambda|=\\# \\Lambda$. We have $|\\Lambda_L(n)|=(2L+1)^d$ ($L\\in\\N$) for example.\n\n\nIf $\\Lambda\\subset \\Z^d$, denote ${H}_{\\Lambda}=R_{\\Lambda}{{H}_{\\omega}}R_{\\Lambda}$, where $R_{\\Lambda}$ is the restriction operator. Define the Green's function (if it exists) as\n\\begin{align*}\nG_{\\Lambda}(E)=({{H}}_{\\Lambda}-E)^{-1},\\ E\\in\\R.\n\\end{align*}\n\n Let us introduce \\textbf{good} cubes in ${\\Z}^d$.\n\\begin{defn}\nFix $\\tau'>0$, $\\delta\\in (0,1)$ and $d\/20$ and $|n'-n''|\\geq L\/2$,\n\\begin{align}\\label{gdec}\n|G_{\\Lambda_L(n)}(E)(n',n'')\n&\\leq |n'-n''|^{-(1-\\zeta)r_1}.\n\\end{align}\n\\end{rem}\n\n\nAssume the following inequalities hold true\\footnote{These inequalities will be explained in the proof of the \\textbf{Coupling Lemma} in the following.}:\n\\begin{align}\\label{para}\n\\left\\{\n\\begin{aligned}\n&-(1-\\delta)r_1+\\tau'+2s_0<0,\\\\\n&-\\xi r_1+\\tau'+\\alpha\\tau+(3+\\delta+4\\xi)s_0<0,\\\\\n&\\alpha^{-1}(2\\tau'+2\\alpha\\tau+(5+4\\xi+2\\delta)s_0)+s_0<\\tau',\\\\\n\\end{aligned}\n\\right.\n\\end{align}\nwhere $\\alpha,\\tau,\\tau',r_1>1, \\xi>0, s_0>d\/2$ and $\\delta\\in(0,1)$.\n\nDenote by $[x]$ the integer part of some $x\\in\\R$. In what follows let $E$ be in an interval $I$ satisfying $|I|\\leq 1$ and $I\\cap[-\\|\\mathcal{T}\\|-M,\\|\\mathcal{T}\\|+M]\\neq\\emptyset$. The main result on Green's functions estimate is as follows.\n\\begin{thm}[]\\label{msa}\nSuppose that $1+\\xi<\\alpha$ and \\eqref{para} holds true. Fix $p>\\alpha d, J\\in2\\mathbb{N}$ and let $L=[l^\\alpha]\\in \\N$ with $l\\in \\N$.\nThen there exists $$\\underline{l}_0=\\underline{l}_0(\\|\\mathcal{T}\\|_{r_1},M,J,\\alpha,\\tau,\\xi,\\tau',\\delta, p, r_1,s_0,d)>0$$ such that, if $l\\geq \\underline{l}_0$,\n\\begin{itemize}\n \\item[(\\textbf{P1})]\n$\\mathbb{P}({\\exists}\\ E\\in I \\ {\\rm s.t.}\\ {\\rm both}\\ \\Lambda_{l}(m)\\ {\\rm and}\\ \\Lambda_{l}(n)\\ {\\rm are\\ } (E,\\delta)-{\\bf bad})\n \\leq l^{-2p}$\nfor all $|m-n|>2l,$\n\n\n\n \\item[(\\textbf{P2})] $\\mathbb{P}({\\rm dist}(\\sigma(H_{\\Lambda_{L}(m)}),\\sigma(H_{\\Lambda_{L}(n)}))\\leq 2L^{-\\tau})\\leq L^{-2p}\/2$ for all $|m-n|>2L$,\n\\end{itemize}\nthen we have\n$$\\mathbb{P}({\\exists}\\ E\\in I \\ {\\rm s.t.}\\ {\\rm both}\\ \\Lambda_{L}(m)\\ {\\rm and}\\ \\Lambda_{L}(n)\\ {\\rm are\\ }(E,\\frac{1+\\xi}{\\alpha})-{\\bf bad})\n \\leq C(d)L^{-J(\\alpha^{-1}p-d)}+L^{-2p}\/2$$\nfor all $|m-n|>2L$.\n\n\\end{thm}\n\\begin{rem}\nIn this theorem no regularity assumption on $\\mu$ is needed.\nMoreover, if we assume further in this theorem $(1+\\xi)\/\\alpha\\leq \\delta$ and $p>\\alpha d+2\\alpha p\/J$, then the ``propagation of smallness'' for the probability occurs (see Theorem \\ref{kthm} in the following for details).\n\\end{rem}\n\n\\subsection{Power-law Localization}\nA sufficient condition for the validity of (\\textbf{P1}) and (\\textbf{P2}) in Theorem \\ref{msa} can be derived from some regularity assumption on $\\mu$.\n\nLet us recall the H\\\"older continuity of a distribution defined in \\cite{CKM87}.\n\n\\begin{defn}[\\cite{CKM87}]\nWe will say a probability measure $\\mu$ is H\\\"older continuous of order $\\rho>0$ if\n\\begin{align}\n\\frac{1}{\\mathcal{K}_\\rho(\\mu)}=\\inf_{\\kappa>0}\\sup_{0<|a-b|\\leq \\kappa}|a-b|^{-\\rho}\\mu([a,b])<\\infty.\n\\end{align}\nIn this case will call $\\mathcal{K}_\\rho(\\mu)>0$ the order of $\\mu$.\n\\end{defn}\n\\begin{rem}\n\\begin{itemize}\n\\item Let $\\mu$ be H\\\"older continuous of order $\\rho$ (i.e., $\\mathcal{K}_\\rho(\\mu)>0$). Then for any $0<\\kappa<\\mathcal{K}_\\rho(\\mu)$, there is some\n$\\kappa_0=\\kappa_0(\\kappa,\\mu)>0$ so that\n\\begin{align}\\label{mu}\n\\mu([a,b])\\leq \\kappa^{-1}|a-b|^\\rho\\ \\mathrm{for}\\ 0\\leq b-a\\leq \\kappa_0.\n\\end{align}\n\n\\item If $\\mu$ is \\textit{absolutely continuous} with a density in $L^q$ with $10$ \\cite{BH80}.\n\\end{itemize}\n\n\\end{rem}\n\nNow we can state the second main result on the \\textit{power-law} localization.\n\\begin{thm}\\label{mthm}\nLet ${H}_{\\omega}$ be defined by $(\\ref{qps})$ with the common distribution $\\mu$ being H\\\"older continuous of order $\\rho>0$, i.e., $\\mathcal{K}_\\rho(\\mu)>0$. Let $r\\geq\\max\\{\\frac{100d+23\\rho d}{\\rho}, 331d\\}$. Fix any $0<\\kappa<\\mathcal{K}_\\rho(\\mu)$. Then there exists $\\lambda_0=\\lambda_0(\\kappa,\\mu,\\rho,{M},r,d)>0$ such that for $\\lambda\\geq\\lambda_0$, $H_\\omega$\nhas pure point spectrum for $\\mathbb{P}$ almost all $\\omega\\in\\Omega$. Moreover, for $\\mathbb{P}$ almost all $\\omega\\in\\Omega$, there exists a complete system of eigenfunctions $\\psi_\\omega=\\{\\psi_\\omega(n)\\}_{n\\in{\\Z}^d}$ satisfying $|\\psi_\\omega(n)|\\leq |n|^{-r\/600}$ for $|n|\\gg1$.\n\\end{thm}\n\\begin{rem}\nOne may replace $\\max\\{\\frac{100d+23\\rho d}{\\rho}, 331d\\}$ with a smaller one. Actually, if $\\mu$ is \\textit{absolutely continuous}, it has been proven by Aizenman-Molchanov \\cite{AM93} that the \\textit{power-law} localization holds for $r>d$ by using the FMM.\n\\end{rem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Proof of Theorem \\ref{msa}}\n\n\n\n\n\\begin{proof}[\\bf Proof of Theorem \\ref{msa}]\nThe proof consists of a deterministic and a probabilistic part.\n\n\nWe begin with the following definition.\n\n\\begin{defn}[]\nWe call a site $n\\in\\Lambda\\subset{\\Z}^d$ is $(l,E,\\delta)$-\\textbf{good} with respect to (\\textit{w.r.t}) $\\Lambda$ if there exists some $\\Lambda_{l}(m)\\subset\\Lambda$ such that $\\Lambda_{l}(m)$ is $(E,\\delta)$-\\textbf{good} and $n\\in \\Lambda_{l}(m)$ with ${\\rm dist}(n,\\Lambda\\setminus\\Lambda_{l}(m))\\geq l\/2$. Otherwise, we call $n\\in\\Lambda\\subset{\\Z}^d$ is $(l,E,\\delta)$-\\textbf{bad} \\textit{w.r.t} $\\Lambda$.\n\\end{defn}\n\nWe then prove a key \\textbf{Coupling Lemma}. Recall that $E\\in I$ with $|I|\\leq 1$ and $I\\cap \\sigma(H_\\omega)\\neq\\emptyset.$\n\\begin{lem}[\\textbf{Coupling Lemma}]\\label{klem}\nLet $L=[l^\\alpha]$. Assume that\n\\begin{itemize}\n\\item We have \\eqref{para} holds true and $1+\\xi<\\alpha$.\n\\item We can decompose $\\Lambda_L(n)$ into two disjoint subsets $\\Lambda_L(n)=B\\cup G$ with the following properties: We have\n\\begin{align*}\nB=\\bigcup_{1\\leq j<\\infty}\\Omega_{j},\n\\end{align*}\nwhere for each $j$, ${\\rm diam}(\\Omega_j)\\leq C_\\star l^{1+\\xi}$ ($C_\\star>1$), and for $j\\neq j'$, ${\\rm dist}(\\Omega_j,\\Omega_{j'})\\geq l^{1+\\xi}$. For each $k\\in G$, $k$ is $(l,E,\\delta)$-{\\bf good} w.r.t $\\Lambda_{L}(n)$.\n\n\\item\n $$\\|G_{\\Lambda_L(n)}(E)\\|\\leq L^{\\tau}.$$\n\\end{itemize}\nThen for $$l\\geq \\underline{l}_0(\\|\\mathcal{T}\\|_{r_1},M,C_\\star,\\alpha,\\tau,\\xi,\\tau',\\delta, r_1,s_0,d)>0,$$ we have $\\Lambda_L(n)$ is $(E,\\frac{1+\\xi}{\\alpha})$-{\\bf good}.\n\\end{lem}\n\\begin{rem}\\label{clrem}\nThe main scheme of the proof is definitely from Berti-Bolle \\cite{BB13} in dealing with nonlinear PDEs.\nSince we are also interested in improving the lower bound on $r$, we have to figure out the dependence relations (i.e., \\eqref{para}) among various parameters in the iterations. Then the Coupling Lemma (i.e., Proposition 4.1 in \\cite{BB13}) of Berti-Bolle may not be used directly here and it needs some small modifications on the proof of Berti-Bolle \\cite{BB13}. It is a key feature that in random operators case the number of disjoint \\textbf{bad} cubes of smaller size contained in a larger cube is fixed and independent of the iteration scales. This permits us to get \\textit{separation} distance of $l^{1+\\xi}$ ($\\xi>0$) without increasing the diameter order (of order also $l^{1+\\xi}$) of \\textbf{bad} cubes clusters (see Lemma \\ref{jlem} in the following for details). As a result, it provides a possible way for improving the lower bound on $r$.\n\\end{rem}\n\n\\begin{proof}[{\\bf Proof of Lemma \\ref{klem}}]\nThe Sobolev norms introduced in \\cite{BB13} are convenient to the proof. Below, we collect some useful properties of Sobolev norms for matrices (see \\cite{BB13} for details):\n\\begin{itemize}\n\\item ({\\bf Interpolation property}): Let $B,C,D$ be finite subsets of ${\\Z}^d$ and let $\\mathcal{M}_1\\in \\mathbf{M}^{C}_D,\\mathcal{M}_2\\in \\mathbf{M}_C^B$. Then for any $s\\geq s_0$,\n\\begin{align}\\label{ip1}\n\\|\\mathcal{M}_1\\mathcal{M}_2\\|_s\\leq (1\/2)\\|\\mathcal{M}_1\\|_{s_0}\\|\\mathcal{M}_2\\|_s+(C(s)\/2)\\|\\mathcal{M}_1\\|_s\\|\\mathcal{M}_2\\|_{s_0},\n\\end{align}\nand\n\\begin{align}\n\\label{ip2}\\|\\mathcal{M}_1\\mathcal{M}_2\\|_{s_0}&\\leq \\|\\mathcal{M}_1\\|_{s_0}\\|\\mathcal{M}_2\\|_{s_0},\\\\\n\\label{ip3}\\|\\mathcal{M}_1\\mathcal{M}_2\\|_s&\\leq C(s)\\|\\mathcal{M}_1\\|_s\\|\\mathcal{M}_2\\|_s,\n\\end{align}\nwhere $C(s)\\geq 1$ and $C(s_0)=1$. In particular, if $\\mathcal{M}\\in \\mathbf{M}_B^B$ and $n\\geq 1$, then\n\\begin{align}\n\\label{ip4}\\|\\mathcal{M}^n\\|_{s_0}&\\leq \\|\\mathcal{M}\\|_{s_0}^n, \\|\\mathcal{M}\\|\\leq \\|\\mathcal{M}\\|_{s_0},\\\\\n\\label{ip5}\\|\\mathcal{M}^n\\|_s&\\leq C(s)\\|\\mathcal{M}\\|_{s_0}^{n-1}\\|\\mathcal{M}\\|_s.\n\\end{align}\n\n\\item ({\\bf Smoothing property}): Let $\\mathcal{M}\\in \\mathbf{M}^B_C$. Then for $s\\geq s'\\geq 0$,\n\\begin{align}\n\\label{sp1}\\mathcal{M}(k,k')=0\\ {\\rm for}\\ |k-k'|0$,\n\\begin{align}\\label{sp2}\n\\mathcal{M}(k,k')=0\\ {\\rm for}\\ |k-k'|>N\\Rightarrow \\left\\{\\begin{aligned}\n&\\|\\mathcal{M}\\|_{s}\\leq N^{s-s'}\\|\\mathcal{M}\\|_{s'},\\\\\n&\\|\\mathcal{M}\\|_{s}\\leq N^{s+s_0}\\|\\mathcal{M}\\|.\n\\end{aligned}\\right.\n\\end{align}\n\\item ({\\bf Columns estimate}): Let $\\mathcal{\\mathcal{M}}\\in \\mathbf{M}^B_C$. Then for $s\\geq0$,\n\\begin{align}\\label{cl1}\n\\|\\mathcal{M}\\|_s\\leq C(s_0,d)\\max_{k\\in C}\\|\\mathcal{M}_{\\{k\\}}\\|_{s+s_0},\n\\end{align}\nwhere $\\mathcal{M}_{\\{k\\}}:=(\\mathcal{M}(k_1,k))_{k_1\\in B}\\in \\mathbf{M}^B_{\\{k\\}}$ is a column sub-matrix of $\\mathcal{M}$.\n\n\\item ({\\bf Perturbation argument}): If $\\mathcal{M}\\in \\mathbf{M}^B_C$ has a left inverse $\\mathcal{N}\\in \\mathbf{M}^C_B$ (i.e., $\\mathcal{N}\\mathcal{M}=\\mathcal{I}$, where $\\mathcal{I}$ the identity matrix), then for all $\\mathcal{P}\\in \\mathbf{M}_C^B$ with\n$\\|\\mathcal{P}\\|_{s_0}\\|\\mathcal{N}\\|_{s_0}\\leq 1\/2,$\nthe matrix $\\mathcal{M}+\\mathcal{P}$ has a left inverse $\\mathcal{N}_\\mathcal{P}$ that satisfies\n\\begin{align}\n\\label{pl2}\\|\\mathcal{N}_\\mathcal{P}\\|_{s_0}&\\leq 2\\|\\mathcal{N}\\|_{s_0},\\\\\n\\label{pl3}\\|\\mathcal{N}_\\mathcal{P}\\|_s&\\leq C(s)(\\|\\mathcal{N}\\|_s+\\|\\mathcal{N}\\|_{s_0}^2\\|\\mathcal{P}\\|_s)\\ {\\rm for\\ } s\\geq {s_0}.\n\\end{align}\nMoreover, if\n$\\|\\mathcal{P}\\|\\cdot\\|\\mathcal{N}\\|\\leq 1\/2,$\nthen\n\\begin{align}\\label{pl5}\n\\|\\mathcal{N}_\\mathcal{P}\\|\\leq 2\\|\\mathcal{N}\\|.\n\\end{align}\n\\end{itemize}\n\nWe then turn to the proof of the \\textbf{Coupling Lemma}.\n\nWrite $\\mathcal{A}={H}_{X}-E$ with $X=\\Lambda_L(n)$. Let $\\mathcal{T}_X=R_X\\mathcal{T}R_X$. For $u\\in \\mathbb{C}^X$ with $X=B\\cup G$, define $u_{G}=R_Gu\\in \\mathbb{C}^G$ and $u_B=R_Bu\\in \\mathbb{C}^B$. Let $h$ be an arbitrary fixed vector in $\\ell^2(X)$ and consider the equation\n\\begin{align}\\label{auh}\n\\mathcal{A}u=h.\n\\end{align}\n\n\n\nFollowing \\cite{BB13}, we have three steps:\n\n\n{\\bf Step 1: Reduction on \\textbf{good} sites}\n\\begin{lem}[]\nLet $l\\geq l_0(\\tau', \\delta, {r_1} , s_0, d)>0$. Then there exist $\\mathcal{M}\\in{\\bf M}_G^X$ and $\\mathcal{N}\\in{\\bf M}_G^B$ satisfying the following:\n\\begin{align}\\label{mnd}\n\\|\\mathcal{M}\\|_{s_0}\\leq C(s_0,d)l^{\\tau'+(1+\\delta)s_0},\\ \\|\\mathcal{N}\\|_{s_0}\\leq C({r_1},s_0,d)\\|\\mathcal{T}_X\\|_{r_1}l^{-(1-\\delta){r_1}+\\tau'+2s_0}\\leq 1\/2,\n\\end{align}\n and for all $s>{s_0}$:\n\\begin{align}\n\\label{ms}&\\|\\mathcal{M}\\|_s\\leq C(s,s_0,d) l^{2\\tau'+(1+2\\delta) s_0}(l^{s}\\|\\mathcal{T}_X\\|_{s_0}+\\|\\mathcal{T}_X\\|_{s+s_0}),\\\\\n\\label{ns}&\\|\\mathcal{N}\\|_s\\leq C(s,s_0,d)l^{\\tau'+\\delta s_0}(l^{s}\\|\\mathcal{T}_X\\|_{s_0}+\\|\\mathcal{T}_X\\|_{s+s_0}),\n\\end{align}\nsuch that\n\\begin{align}\\label{ugub}\nu_G=\\mathcal{N}u_B+\\mathcal{M}h.\n\\end{align}\n\\end{lem}\n\\begin{proof}\nFix $k\\in G$. Then there exists some $l$-cube $F_k=\\Lambda_l(k_1)$ such that $k\\in F_k$, ${\\rm dist}(k,X\\setminus F_k)\\geq l\/2$ and $F_k$ is $(E,\\delta)$-\\textbf{good}. Define $\\mathcal{Q}_k=\\lambda^{-1} G_{F_k}(E)\\mathcal{T}_{F_k}^{X\\setminus F_k}\\in \\mathbf{M}_{F_k}^{X\\setminus F_k}$. Since $F_k$ is $(E,\\delta)$-\\textbf{good} and using the \\textbf{Interpolation property} \\eqref{ip3}, we obtain (since $\\lambda\\geq 1$)\n\\begin{align}\\label{qr}\n\\|\\mathcal{Q}_k\\|_{r_1}\\leq C(r_1)\\|G_{F_k}(E)\\|_{r_1}\\|\\mathcal{T}_X\\|_{r_1}\\leq C({r_1})\\|\\mathcal{T}_X\\|_{r_1}l^{\\tau'+\\delta {r_1}}.\n\\end{align}\nBy the \\textbf{Interpolation property} \\eqref{ip1} and the \\textbf{Smoothing property} \\eqref{sp2}, for $s\\geq s_0$ we have (if $|k-k'|>2l$, then $G_{F_k}(E)(k',k)=0$)\n\\begin{align}\n\\nonumber\\|\\mathcal{Q}_k\\|_{s+s_0}&\\leq C(s)(\\|G_{F_k}(E)\\|_{s+s_0}\\|\\mathcal{T}_X\\|_{s_0}+\\|G_{F_k}(E)\\|_{s_0}\\|\\mathcal{T}_X\\|_{s+s_0})\\\\\n\\nonumber&\\leq C(s)((2l)^{s}\\|G_{F_k}(E)\\|_{s_0}\\|\\mathcal{T}_X\\|_{s_0}+l^{\\tau'+\\delta s_0}\\|\\mathcal{T}_X\\|_{s+s_0})\\\\\n\\label{qd}&\\leq C(s,d)l^{\\tau'+\\delta s_0}(l^{s}\\|\\mathcal{T}_X\\|_{s_0}+\\|\\mathcal{T}_X\\|_{s+s_0}).\n\\end{align}\n\n\nWe now vary $k\\in G$. Define the following operators\\footnote{Both $\\Gamma$ and $\\mathcal{L}$ are globally well-defined since we define the operators ``{column by column}''. More precisely, we have shown $G\\subset\\bigcup\\limits_{k\\in G}F_k$, and $k\\in F_k$ for each $k\\in G$. From the definition, we have {$\\Gamma(\\cdot,k)$ and $\\mathcal{L}(\\cdot,k)$ come from that of ${G}_{F_k}(\\cdot,k)$}. One may argue that it is likely that there exists $(m,m')$ such that $m'\\in F_{k}\\cap F_{k'}\\neq \\emptyset$ for some $k'\\neq k$. As a result, it is likely that ${G}_{F_{k}}(m,m')\\neq {G}_{F_{k'}}(m,m')$ and $\\mathcal{L}(m,m')$ (or $\\Gamma(m,m')$) is not uniquely defined! In fact, this is not the case since our definition of $\\mathcal{L}(m,m')$ (or $\\Gamma(m,m')$) comes from the column ${G}_{F_{m'}}(\\cdot,m')$ rather than that ${G}_{F_k}(\\cdot,m')$ or ${G}_{F_{k'}}(\\cdot,m')$.}:\n\\begin{equation*}\n\\Gamma (k',k)=\\left\\{\\begin{aligned}\n&0,\\ {\\rm for\\ } k'\\in F_k,\\\\\n&\\mathcal{Q}_k(k',k), \\ \\mathrm{for}\\ k'\\in X\\setminus F_k,\n\\end{aligned}\\right.\n\\end{equation*}\nand\n\\begin{equation*}\n\\mathcal{L} (k',k)=\\left\\{\\begin{aligned}\n&G_{F_k}(E)(k',k),\\ {\\rm for\\ } k'\\in F_k,\\\\\n&0, \\ \\mathrm{for}\\ k'\\in X\\setminus F_k.\n\\end{aligned}\\right.\n\\end{equation*}\nFrom \\eqref{auh}, we have\n\\begin{align}\\label{ug}\nu_G+\\Gamma u=\\mathcal{L}h.\n\\end{align}\nWe estimate $\\Gamma\\in \\mathbf{M}_G^X$. Fix $k\\in G$. Note that if $k'\\in X\\setminus F_k$, then $|k-k'|\\geq l\/2$.\nThis implies $\\Gamma_{\\{k\\}}(k',k)=0$\nfor $|k'-k|2l$, then $k'\\notin F_k$. This implies $\\mathcal{L}_{\\{k\\}}(k',k)=0$ for $|k'-k|>2l$.\nBy the \\textbf{Columns estimate} \\eqref{cl1} and the \\textbf{Smoothing property} \\eqref{sp2},\nwe have for $s\\geq 0$,\n\\begin{align}\n\\nonumber\\|\\mathcal{L}\\|_{s+s_0}&\\leq C(s_0,d)\\sup_{k\\in G}\\|\\mathcal{L}_{\\{k\\}}\\|_{s+2s_0}\\\\\n\\nonumber&\\leq C(s_0,d) \\sup_{k\\in G}(2l)^{s+s_0}\\|\\mathcal{L}_{\\{k\\}}\\|_{s_0}\\\\\n\\nonumber&\\leq C(s,s_0,d) \\sup_{k\\in G}l^{s+s_0}\\|G_{F_k}(E)\\|_{s_0}\\\\\n\\label{lsd}&\\leq C(s,s_0,d)l^{s+\\tau'+(1+\\delta)s_0}.\n\\end{align}\n\nNotice that we have ${-(1-\\delta){r_1}+\\tau'+2s_0<0}$. Thus for $${l\\geq l_0(\\tau',\\delta,{r_1},s_0,d)>0},$$\n we have since \\eqref{gmd} $\\|\\Gamma\\|_{s_0}\\leq 1\/2$. Recalling the \\textbf{Perturbation argument} \\eqref{pl2}--\\eqref{pl3}, we have that $\\mathcal{I}+\\Gamma_G^G$ is invertible and satisfies\n\\begin{align}\n\\label{ggd}\\|(\\mathcal{I}+\\Gamma_G^G)^{-1}\\|_{s_0}&\\leq 2,\\\\\n\\label{ggs}\\|(\\mathcal{I}+\\Gamma_G^G)^{-1}\\|_s&\\leq C(s) \\|\\Gamma\\|_s\\ {\\rm for}\\ s\\geq s_0.\n\\end{align}\nFrom \\eqref{ug}, we have\n\\begin{align*}\nu_G=-(\\mathcal{I}+\\Gamma_G^G)^{-1}\\Gamma_G^{B}u_B+(\\mathcal{I}+\\Gamma_G^G)^{-1}\\mathcal{L}h\n\\end{align*}\nand then\n\\begin{align}\\label{mnl}\n\\mathcal{N}=-(\\mathcal{I}+\\Gamma_G^G)^{-1}\\Gamma_G^{B}, \\ \\mathcal{M}=(\\mathcal{I}+\\Gamma_G^G)^{-1}\\mathcal{L}.\n\\end{align}\nRecalling the \\textbf{Interpolation property} \\eqref{ip1} and since \\eqref{ggd}--\\eqref{mnl}, we have\n\\begin{align*}\n\\|\\mathcal{N}\\|_{s_0}&\\leq \\|(\\mathcal{I}+\\Gamma_G^G)^{-1}\\|_{s_0}\\|\\Gamma\\|_{s_0}\\leq C({r_1},s_0,d)\\|\\mathcal{T}_X\\|_{r_1}l^{-(1-\\delta){r_1}+\\tau'+2s_0},\\\\\n\\|\\mathcal{M}\\|_{s_0}&\\leq \\|(\\mathcal{I}+\\Gamma_G^G)^{-1}\\|_{s_0} \\|\\mathcal{L}\\|_{s_0}\\leq C(s_0,d)l^{\\tau'+(1+\\delta)s_0},\n\\end{align*}\nand for $s\\geq s_0$,\n\\begin{align*}\n\\nonumber\\|\\mathcal{N}\\|_s&\\leq C(s)(\\|(\\mathcal{I}+\\Gamma_G^G)^{-1}\\|_s\\|\\Gamma\\|_{s_0}+\\|(\\mathcal{I}+\\Gamma_G^G)^{-1}\\|_{s_0}\\|\\Gamma\\|_s)\\\\\n\\nonumber&\\leq C(s)\\|\\Gamma\\|_s\\\\\n\\nonumber&\\leq C(s,s_0,d)l^{\\tau'+\\delta s_0}(l^{s}\\|\\mathcal{T}_X\\|_{s_0}+\\|\\mathcal{T}_X\\|_{s+s_0})\\ ({\\rm since} \\ \\eqref{gms}),\\\\\n\\|\\mathcal{M}\\|_s& \\leq C(s)(\\|(\\mathcal{I}+\\Gamma_G^G)^{-1}\\|_s\\|\\mathcal{L}\\|_{s_0}+\\|(\\mathcal{I}+\\Gamma_G^G)^{-1}\\|_{s_0}\\|\\mathcal{L}\\|_s)\\\\\n&\\leq C(s)({\\|\\Gamma\\|_s\\|\\mathcal{L}\\|_{s_0}}+\\|\\Gamma\\|_{s_0}\\|\\mathcal{L}\\|_s) \\\\\n&\\leq C(s,s_0,d) l^{2\\tau'+(1+2\\delta) s_0}(l^{s}\\|\\mathcal{T}_X\\|_{s_0}+\\|\\mathcal{T}_X\\|_{s+s_0})\\ ({\\rm since} \\ \\eqref{gmd}-\\eqref{lsd}).\n\\end{align*}\n\\end{proof}\n\n\n\n\n{\\bf Step 2: Reduction on \\textbf{bad} sites}\n\\begin{lem} Let $l\\geq l_0(\\tau',\\delta,{r_1},s_0,d)>0$. We have\n\\begin{align}\\label{apub}\n\\mathcal{A}'u_B=\\mathcal{Z}h,\n\\end{align}\nwhere\n\\begin{align*}\n\\mathcal{A}'=\\mathcal{A}_X^B+\\mathcal{A}_X^G\\mathcal{N}\\in\\mathbf{M}^B_X,\\ \\mathcal{Z}={\\mathcal{I}}-\\mathcal{A}_X^G\\mathcal{M}\\in\\mathbf{M}^X_X\n\\end{align*}\nsatisfy for $s\\geq s_0$,\n\\begin{align}\n\\label{apzd1}\\|\\mathcal{A}'\\|_{s_0}&\\leq C(M)(1+\\|\\mathcal{T}_X\\|_{s_0}),\\\\\n\\label{apzd2} \\|\\mathcal{Z}\\|_{s_0}&\\leq C(M,s_0,d)(1+\\|\\mathcal{T}_X\\|_{s_0})l^{\\tau'+(1+\\delta)s_0},\\\\\n\\label{aps}\\|\\mathcal{A}'\\|_s&\\leq C(M,s,s_0,d)(1+\\|\\mathcal{T}_X\\|_{s})l^{\\tau'+\\delta s_0}(l^{s}\\|\\mathcal{T}_X\\|_{s_0}+\\|\\mathcal{T}_X\\|_{s+s_0}),\\\\\n\\label{zs}\\|\\mathcal{Z}\\|_s&\\leq C(M,s,s_0,d)(1+\\|\\mathcal{T}_X\\|_{s})l^{2\\tau'+(1+2\\delta) s_0}(l^{s}\\|\\mathcal{T}_X\\|_{s_0}+\\|\\mathcal{T}_X\\|_{s+s_0}).\n\\end{align}\nMoreover, $(\\mathcal{A}^{-1})_B^X$ is a left inverse of $\\mathcal{A}'$.\n\\end{lem}\n\n\\begin{proof}\nSince $I\\cap[-\\|\\mathcal{T}\\|-M,\\|\\mathcal{T}\\|+M]\\neq\\emptyset$, $\\sup_{\\omega,n}{|V_\\omega(n)|}\\leq M$, $|I|\\leq 1$ and $\\lambda\\geq1$, we have for all $E\\in I$ and $n\\in{\\Z}^d$,\n\\begin{align*}\n|V_\\omega(n)-E|\\leq \\|\\mathcal{T}\\|+2M+1.\n\\end{align*}\nThus for any $s\\geq 0$, we obtain\n\\begin{align}\\label{as}\n\\|\\mathcal{A}\\|_s=\\|{H}_X-E\\|_s\\leq \\|\\lambda^{-1}\\mathcal{T}_X\\|_s+\\|\\mathcal{T}\\|+2M+1\\leq 2(1+\\|\\mathcal{T}_X\\|_s+\\|\\mathcal{T}\\|+M).\n\\end{align}\nFrom \\eqref{mnd}, \\eqref{as} and the \\textbf{Interpolation property} \\eqref{ip1}--\\eqref{ip2}, we have\n\\begin{align*}\n&\\|\\mathcal{A}'\\|_{s_0}\\leq \\|\\mathcal{A}\\|_{s_0}+\\|\\mathcal{A}\\|_{s_0}\\|\\mathcal{N}\\|_{s_0}\\leq C(M)(1+\\|\\mathcal{T}_X\\|_{s_0})\\ ({\\rm since}\\ \\|\\mathcal{T}_X\\|\\leq \\|\\mathcal{T}_X\\|_{s_0} ),\\\\\n&\\|\\mathcal{Z}\\|_{s_0}\\leq 1+\\|\\mathcal{A}\\|_{s_0}\\|\\mathcal{M}\\|_{s_0}\\leq C(M,s_0,d)(1+\\|\\mathcal{T}_X\\|_{s_0})l^{\\tau'+(1+\\delta)s_0},\n\\end{align*}\nand for $s\\geq s_0$,\n\\begin{align*}\n\\|\\mathcal{A}'\\|_s&\\leq \\|\\mathcal{A}\\|_s+C(s)(\\|\\mathcal{A}\\|_s\\|\\mathcal{N}\\|_{s_0}+\\|\\mathcal{A}\\|_{s_0}\\|\\mathcal{N}\\|_s)\\\\\n&\\leq C(M,s,s_0,d)(1+\\|\\mathcal{T}_X\\|_{s})l^{\\tau'+\\delta s_0}(l^{s}\\|\\mathcal{T}_X\\|_{s_0}+\\|\\mathcal{T}_X\\|_{s+s_0}),\\\\\n\\|\\mathcal{Z}\\|_s&\\leq 1+C(s)(\\|\\mathcal{A}\\|_s\\|\\mathcal{M}\\|_{s_0}+\\|\\mathcal{A}\\|_{s_0}\\|\\mathcal{M}\\|_s)\\\\\n&\\leq C(M,s,s_0,d)(1+\\|\\mathcal{T}_X\\|_{s})l^{2\\tau'+(1+2\\delta) s_0}(l^{s}\\|\\mathcal{T}_X\\|_{s_0}+\\|\\mathcal{T}_X\\|_{s+s_0}).\n\\end{align*}\nIt is easy to see $(\\mathcal{A}^{-1})_B^X$ is a left inverse of $\\mathcal{A}'$.\n\\end{proof}\n\n\n\n\\begin{lem}[Left inverse of $\\mathcal{A}'$] Let $l\\geq l_0(\\|\\mathcal{T}_X\\|_{r_1}, M, C_\\star, \\tau, \\xi, \\tau',\\delta, {r_1}, s_0, d)>0$. Then $\\mathcal{A}'$ has a left inverse $\\mathcal{V}$ satisfying for $s\\geq d$,\n\\begin{align}\\label{vs0}\n& \\|\\mathcal{V}\\|_{s_0}\\leq C(C_\\star,s_0,d)l^{\\alpha\\tau+(2+2\\xi)s_0},\n\\end{align}\nand for $s>s_0$,\n\\begin{align}\\label{vs}\n&\\|\\mathcal{V}\\|_s\\leq C(M,C_\\star,s,s_0,d)(1+\\|\\mathcal{T}_X\\|_{s})l^{\\tau'+2\\alpha\\tau+(4+4\\xi+\\delta)s_0}(l^{(1+\\xi)s}\\|\\mathcal{T}_X\\|_{s_0}+\\|\\mathcal{T}_X\\|_{s+s_0}).\n\\end{align}\n\\end{lem}\n\n\\begin{proof}\nThe proof is based on the perturbation of left inverses as in \\cite{BB13}. Let $\\widetilde \\Omega_j$ be the $l^{1+\\xi}\/4$-neighborhood of $\\Omega_j$, i.e., $\\widetilde \\Omega_j=\\{k\\in\\Z^d:\\ {\\rm dist}(k,\\Omega_j)\\leq l^{1+\\xi}\/4\\}$.\nLet $\\mathcal{D}\\in \\mathbf{M}^B_X$ satisfy the following:\n\\begin{equation*}\n\\mathcal{D} (k,k')=\\left\\{\\begin{aligned}\n&\\mathcal{A}'(k,k'),\\ {\\rm for\\ } (k,k')\\in \\bigcup_{j}(\\Omega_j\\times\\widetilde\\Omega_j),\\\\\n&0, \\ \\mathrm{for}\\ (k,k')\\notin \\bigcup_{j}(\\Omega_j\\times\\widetilde\\Omega_j).\n\\end{aligned}\\right.\n\\end{equation*}\n\nWe claim that $\\mathcal{D}$ has a left inverse $\\mathcal{W}$ satisfying $\\|\\mathcal{W}\\|\\leq 2L^{\\tau}$. Let $|k-k'|0$}.\nIt follows from the \\textbf{Perturbation argument} \\eqref{pl5} that $\\mathcal{D}$ has a left inverse $\\mathcal{W}$ satisfying $\\|\\mathcal{W}\\|\\leq 2\\|\\mathcal{A}^{-1}\\|\\leq 2L^{\\tau}$.\n\n\nFrom \\cite{BB13}, we know that\n\\begin{equation*}\n\\mathcal{W}_0 (k,k')=\\left\\{\\begin{aligned}\n&\\mathcal{W}(k,k'),\\ {\\rm for\\ } (k,k')\\in \\bigcup_{j}(\\Omega_j\\times\\widetilde\\Omega_j),\\\\\n&0, \\ \\mathrm{for}\\ (k,k')\\notin \\bigcup_{j}(\\Omega_j\\times\\widetilde\\Omega_j)\n\\end{aligned}\\right.\n\\end{equation*}\nis a left inverse of $\\mathcal{D}$. We then estimate $\\|\\mathcal{W}_0\\|_s$. Since ${\\rm diam}(\\widetilde\\Omega_j)\\leq 2C_\\star l^{1+\\xi}$, we have $\\mathcal{W}_0(k,k')=0$ if $|k-k'|> 2C_\\star l^{1+\\xi}$. Using the \\textbf{Smoothing property }\\eqref{sp2} yields for $s\\geq 0$,\n\\begin{align}\n\\label{w0s}&\\|\\mathcal{W}_0\\|_s\\leq C(C_\\star,s,s_0,d)l^{(1+\\xi)(s+s_0)}\\|\\mathcal{W}\\|\\leq C(C_\\star,s,s_0,d)l^{(1+\\xi)(s+s_0)+\\alpha\\tau}.\n\\end{align}\n\nFinally, recall that $\\mathcal{A}'=\\mathcal{D}+\\mathcal{R}$ and $\\mathcal{W}_0$ is a left inverse of $\\mathcal{D}$. We have by \\eqref{rd} and \\eqref{w0s},\n\\begin{align*}\n\\|\\mathcal{R}\\|_{s_0}\\|\\mathcal{W}_0\\|_{s_0}\\leq C(M,C_\\star,{r_1},s_0,d)l^{-\\xi {r_1}+\\tau'+\\alpha\\tau+(3+\\delta+4\\xi)s_0}\\leq 1\/2\n\\end{align*}\nsince {$-\\xi {r_1}+\\tau'+\\alpha\\tau+(3+\\delta+4\\xi)s_0<0$ and $l\\geq l_0(M,C_\\star,\\alpha,\\tau,\\xi,\\tau',\\delta, {r_1},s_0,d)>0$}. Applying the {\\bf Perturbation argument} \\eqref{pl2}--\\eqref{pl3} again implies that $\\mathcal{A}'$ has a left inverse $\\mathcal{V}$ satisfying\n\\begin{align*}\n\\|\\mathcal{V}\\|_{s_0}&\\leq 2\\|\\mathcal{W}_0\\|_{s_0}\\leq C(C_\\star,s_0,d)l^{\\alpha\\tau+(2+2\\xi)s_0},\\\\\n\\|\\mathcal{V}\\|_s&\\leq C(s)(\\|\\mathcal{W}_0\\|_s+\\|\\mathcal{W}_0\\|_{s_0}^2\\|\\mathcal{R}\\|_s) \\ ({\\rm by}\\ \\eqref{pl3})\\\\\n&\\leq C(C_\\star,s,s_0,d)l^{(1+\\xi)(s+s_0)+\\alpha\\tau}\\\\%+ C(s,s_0,d)\\|\\mathcal{A}'\\|_s\\ ({\\rm by}\\ \\eqref{w0s}\\ {\\rm and}\\ \\eqref{aps})\\\\\n\\nonumber&\\ \\ +C(M,C_\\star,s,s_0,d)(1+\\|\\mathcal{T}_X\\|_{s})l^{\\tau'+2\\alpha\\tau+(4+4\\xi+\\delta)s_0}(l^{s}\\|\\mathcal{T}_X\\|_{s_0}+\\|\\mathcal{T}_X\\|_{s+s_0})\\\\\n&\\leq C(M,C_\\star,s,s_0,d)(1+\\|\\mathcal{T}_X\\|_{s})l^{\\tau'+2\\alpha\\tau+(4+4\\xi+\\delta)s_0}(l^{(1+\\xi)s}\\|\\mathcal{T}_X\\|_{s_0}+\\|\\mathcal{T}_X\\|_{s+s_0}).\n\\end{align*}\n\n\n\n\\end{proof}\n\n\n{\\bf Step 3: Completion of proof}\n\nCombining \\eqref{auh}, \\eqref{ugub} and \\eqref{apub} implies\n\\begin{align*}\nu_G=\\mathcal{M}h+\\mathcal{N}u_B,u_B=\\mathcal{V}\\mathcal{Z}h.\n\\end{align*}\nThus\n\\begin{align*}\n(\\mathcal{A}^{-1})_B^X=\\mathcal{V}\\mathcal{Z}, \\ (\\mathcal{A}^{-1})_G^X=\\mathcal{M}+\\mathcal{N}(\\mathcal{A}^{-1})^X_B.\n\\end{align*}\nThen for $s\\geq s_0$, we can obtain by using the \\textbf{Interpolation property} \\eqref{ip1} and the \\textbf{Smoothing property} \\eqref{sp2}\n\\begin{align*}\n\\|(\\mathcal{A}^{-1})_B^X\\|_s&\\leq C(s)(\\|\\mathcal{V}\\|_s\\|\\mathcal{Z}\\|_{s_0}+\\|\\mathcal{V}\\|_{s_0}\\|\\mathcal{Z}\\|_s)\\\\\n&\\leq C(M,C_\\star,s,s_0,d)(1+\\|\\mathcal{T}_X\\|_{s})^2l^{2\\tau'+2\\alpha\\tau+(5+4\\xi+2\\delta)s_0}(l^{(1+\\xi)s}\\|\\mathcal{T}_X\\|_{s_0}+\\|\\mathcal{T}_X\\|_{s+s_0}) \\\\\n&\\ \\ +C(M,C_\\star,s,s_0,d)(1+\\|\\mathcal{T}_X\\|_{s})l^{2\\tau'+\\alpha\\tau+(3+2\\delta+2\\xi)s_0}(l^{s}\\|\\mathcal{T}_X\\|_{s_0}+\\|\\mathcal{T}_X\\|_{s+s_0})\\\\\n&\\ \\ ({\\rm by}\\ \\eqref{zs}\\ {\\rm and}\\ \\eqref{vs})\\\\\n&\\leq C(M,C_\\star,s,s_0,d)(1+\\|\\mathcal{T}_X\\|_{s})^2l^{2\\tau'+2\\alpha\\tau+(5+4\\xi+2\\delta)s_0}(l^{(1+\\xi)s}\\|\\mathcal{T}_X\\|_{s_0}+\\|\\mathcal{T}_X\\|_{s+s_0}).\n\\end{align*}\nWe obtain the similar bound for $\\|(\\mathcal{A}^{-1})_G^X\\|_s$. Thus for any $s\\in[s_0,{r_1}]$, we obtain\n\\begin{align*}\n\\|\\mathcal{A}^{-1}\\|_s&\\leq \\|(\\mathcal{A}^{-1})_B^X\\|_s+\\|(\\mathcal{A}^{-1})_G^X\\|_s\\\\\n& C(M,C_\\star,s,s_0,d)(1+\\|\\mathcal{T}_X\\|_{r_1})^2l^{2\\tau'+2\\alpha\\tau+(5+4\\xi+2\\delta)s_0}(l^{(1+\\xi)s}\\|\\mathcal{T}_X\\|_{s_0}+(2L)^{s_0} \\|\\mathcal{T}_X\\|_{{r_1}})\\\\\n&\\leq C(M,C_\\star,r_1,s_0,d)\\|\\mathcal{T}_X\\|_{{r_1}}^2 L^{\\alpha^{-1}(2\\tau'+2\\alpha\\tau+(5+4\\xi+2\\delta)s_0)+s_0+\\alpha^{-1}{(1+\\xi)}s}\\\\\n&\\leq L^{\\tau'+\\frac{1+\\xi}{\\alpha} s},\n\\end{align*}\nwhere in the last inequality we use the third inequality in \\eqref{para} and $$L>l\\geq l_0(\\|\\mathcal{T}\\|_{{r_1}}, M,C_\\star,\\alpha,\\tau,\\xi,\\tau',\\delta, {r_1},s_0,d)>0.$$\n\n\nThis finishes the proof of the \\textbf{Coupling Lemma}.\n\n\n\\end{proof}\n\n\n\nWe are in a position to finish the proof of Theorem \\ref{msa}.\n\n{\\bf Deterministic part}\n\n\\begin{lem}\\label{jlem}\nFix $J\\in 2\\mathbb{N}$. Assume \\eqref{para} holds true and $1+\\xi<\\alpha$. Assume further that any pairwise disjoint $(E,\\delta)$-{\\bf bad} $l$-cubes contained in $\\Lambda_{L}(n)$ has number at most $J-1$ and $\\|G_{\\Lambda_L(n)}(E)\\|\\leq L^{\\tau}$. Then for $$l\\geq l_0(M,J,\\alpha,\\tau,\\xi,\\tau',\\delta, {r_1},s_0,d)>0,$$ we have $\\Lambda_L(n)$ is $(E,\\frac{1+\\xi}{\\alpha})$-{\\bf good}.\n\\end{lem}\n\n\\begin{proof}[{\\bf Proof of Lemma \\ref{jlem}}]\nThe main point here is to obtain the \\textit{separation property} of $(E,\\delta)$-\\textbf{bad} $l$-cubes contained in $\\Lambda_L(n)$.\n\nDenote by $\\Lambda_{l}(k^{(1)}),\\cdots,\\Lambda_l(k^{(t_0)})\\subset\\Lambda_L(n)$ all the $(E,\\delta)$-\\textbf{bad} $l$-cubes. Obviously, $t_0\\leq (2L+1)^d$.\nWe first claim that there exists $\\widetilde{Z}=\\{m^{(1)},\\cdots,m^{(t_1)}\\}\\subset Z=\\{k^{(1)},\\cdots,k^{(t_0)}\\}$ such that $t_1\\leq J-1$, $|m^{(i)}-m^{(j)}|>2l$ (for $ i\\neq j$) and\n\\begin{align}\\label{vitali}\n\\bigcup_{1\\leq j\\leq t_0}\\Lambda_l(k^{(j)})\\subset \\bigcup_{1\\leq j\\leq t_1}\\Lambda_{3l}(m^{(j)}).\n\\end{align}\nThis claim should be compared with the Vitali covering argument. We prove this claim as follows. We start from $m^{(1)}=k^{(1)}$. Define $Z_1$ to be the set of all $k^{(j)}$ satisfying $|k^{(j)}-m^{(1)}|\\leq 2l$. If $Z_1=Z$, then we stop the process, and it is easy to check \\eqref{vitali} with $t_1=1.$ Otherwise, we have $Z\\setminus Z_1\\neq \\emptyset$ and we can choose a $m^{(2)}\\in Z\\setminus Z_1$. Similarly, let $Z_2$ be the set of all $k^{(j)}\\in Z\\setminus Z_1$ satisfying $|k^{(j)}-m^{(2)}|\\leq l$. If $Z_2=Z\\setminus Z_1$, we stop the process and \\eqref{vitali} holds with $t_1=2$. Repeating this process and since $t_0<\\infty$, we can obtain \\eqref{vitali} for some $t_1\\leq t_0.$ From the construction, we must have $|m^{(i)}-m^{(j)}|>2l$ for $i\\neq j$, or equivalently $\\Lambda_{l}(m^{(i)})\\cap\\Lambda_l(m^{(j)})=\\emptyset$ for $i\\neq j.$ Recalling the assumption of Lemma \\ref{jlem}, we get $t_1\\leq J-1.$ The proof of this claim is finished.\n\n\n We now separate further the clusters $\\Lambda_{3l}(m^{(1)})\\cap\\Lambda_L(n) ,\\cdots,\\Lambda_{3l}(m^{(t_1)})\\cap\\Lambda_L(n)$. Define a relation $\\bowtie$ on $\\widetilde Z$ as follows. Letting $k,k'\\in \\widetilde Z$, we say $k\\bowtie k'$ if there is a sequence $k_0,\\cdots, k_q\\in\\widetilde Z$ ($q\\geq 1$) satisfying $k_0=k,k_q=k'$ and\n \\begin{align}\\label{shi1}\n |k_j-k_{j+1}|\\leq 2l^{1+\\xi}\\ {\\rm for}\\ \\forall\\ 0\\leq j\\leq q-1.\n \\end{align}\nIt is easy to see $\\bowtie$ is an equivalence relation on $\\widetilde Z.$ As a result, we can partition $\\widetilde Z$ into disjoint equivalent classes (\\textit{w.r.t} $\\bowtie$), say $\\pi_1,\\cdots,\\pi_{t_2}$ with $t_2\\leq t_1$. We also have by \\eqref{shi1}\n\\begin{align}\n\\label{shi2}|k-k'|&\\leq 2Jl^{1+\\xi}\\ {\\rm for}\\ \\forall\\ k,k'\\in\\pi_j,\\\\\n\\label{shi3}{\\rm dist}(\\pi_{i},\\pi_j)&>2l^{1+\\xi}\\ {\\rm for}\\ i\\neq j.\n \\end{align}\nCorrespondingly, we can define\n$$ \\Omega_j=\\bigcup_{y\\in\\pi_j}(\\Lambda_{3l}(y)\\cap\\Lambda_L(n)).$$\nFrom \\eqref{shi2} and \\eqref{shi3}, we obtain\n\\begin{align*}\n{\\rm diam}(\\Omega_j)&\\leq 10Jl^{1+\\xi}\\ {\\rm for}\\ 1\\leq j\\leq t_2,\\\\\n{\\rm dist}(\\Omega_{i},\\Omega_j)&>2l^{1+\\xi}-10l>l^{1+\\xi}\\ {\\rm for}\\ i\\neq j.\n\\end{align*}\nMoreover, since $B=\\bigcup\\limits_{1\\leq j\\leq t_2}\\Omega_j$ contains all the $(E,\\delta)$-\\textbf{bad} $l$-cubes, it follows that if $\\Lambda_l(k)\\subset\\Lambda_L(n)$ and $\\Lambda_l(k)\\cap (\\Lambda_L(n)\\setminus B)\\neq \\emptyset$, then $\\Lambda_l(k)$ must be $(E,\\delta)$-\\textbf{good}. Let $G=\\Lambda_L(n)\\setminus B$. Then every $k\\in G$ is $(E,\\delta)$-\\textbf{good} \\textit{w.r.t} $\\Lambda_L(n)$. Actually, if $k\\in G\\subset\\Lambda_L(n)$, then there exists $\\Lambda_l(k')\\subset \\Lambda_{L}(n)$ with $k\\in\\Lambda_l(k')$ such that ${\\rm dist}(k,\\Lambda_L(n)\\setminus\\Lambda_l(k'))\\geq l$. This $\\Lambda_l(k')$ must be $(E,\\delta)$-\\textbf{good} since $k\\notin B,$ i.e., $k$ is $(E,\\delta)$-\\textbf{good} \\textit{w.r.t} $\\Lambda_L(n).$\n\n\n\n\n\n\n\n\n\nFinally, it suffices to apply Lemma \\ref{klem} with $B=\\bigcup\\limits_{j=1}^{t_2}\\Omega_j$, $G=\\Lambda_L(n)\\setminus B$ and $C_\\star=10J$.\n\n\n\n\n\n\\end{proof}\n\n\n\n\n{\\bf Probabilistic part}\n\n\n\n\n Fix $m,n$ with $|m-n|>2L$ and write $\\Lambda_1=\\Lambda_{L}(m), \\Lambda_2=\\Lambda_{L}(n)$. We define the\nfollowing events for $i=1,2$:\n\\begin{align*}\n&{\\mathbf{A}}_i:\\ \\Lambda_{i}\\ {\\rm is\\ } (E,\\frac{1+\\xi}{\\alpha})-{\\rm\\bf bad},\\\\\n&{\\mathbf{B}}_i: \\ {\\rm either}\\ G_{\\Lambda_i}(E) \\ {\\rm does\\ not\\ exist\\ or}\\ \\|G_{\\Lambda_i}(E)\\|\\geq L^{\\tau},\\\\\n&{\\mathbf{C}}_i:\\ \\Lambda_i\\ {\\rm contains\\ }J\\ {\\rm\\ pairwise\\ disjoint\\ } (E,\\delta)-{\\rm \\bf bad}\\ l-{\\rm cubes},\\\\\n&{\\mathbf{D}}:\\ {\\exists}\\ E\\in I \\ {\\rm so\\ that\\ both}\\ \\Lambda_{1}\\ {\\rm and}\\ \\Lambda_{2}\\ {\\rm are\\ } (E,\\frac{1+\\xi}{\\alpha})-{\\rm\\bf bad}.\n\\end{align*}\nUsing Lemma \\ref{jlem} yields\n\\begin{align}\n\\nonumber\\mathbb{P}(\\mathbf{D})&\\leq \\mathbb{P}\\left(\\bigcup_{E\\in I}({\\mathbf{A}}_1\\cap{\\mathbf{A}}_2)\\right)\\leq \\mathbb{P}\\left(\\bigcup_{E\\in I}\\left(({\\mathbf{B}}_1 \\cup{\\mathbf{C}}_1)\\cap({\\mathbf{B}}_2 \\cup{\\mathbf{C}}_2\\right))\\right)\\\\\n\\nonumber&\\leq \\mathbb{P}\\left(\\bigcup_{E\\in I}({\\mathbf{B}}_1\\cap{\\mathbf{B}}_2)\\right)+\\mathbb{P}\\left(\\bigcup_{E\\in I}({\\mathbf{B}}_1\\cap{\\mathbf{C}}_2)\\right)\\\\\n\\nonumber&\\ \\ +\\mathbb{P}\\left(\\bigcup_{E\\in I}({\\mathbf{C}}_1\\cap{\\mathbf{B}}_2)\\right)+\\mathbb{P}\\left(\\bigcup_{E\\in I}({\\mathbf{C}}_1\\cap{\\mathbf{C}}_2)\\right)\\\\\n\\label{idp}&\\leq \\mathbb{P}\\left(\\bigcup_{E\\in I}({\\mathbf{B}}_1\\cap{\\mathbf{B}}_2)\\right)+3\\mathbb{P}\\left(\\bigcup_{E\\in I}{\\mathbf{C}}_1\\right).\n\\end{align}\nIt is easy to see since (\\textbf{P1})\n\\begin{align}\\label{mp}\n\\mathbb{P}\\left(\\bigcup_{E\\in I}{\\mathbf{C}}_1\\right)\\leq C(d)L^{Jd}(l^{-2p})^{J\/2}\\leq C(d)L^{-J(\\alpha^{-1}p-d)}\n\\end{align}\nWe then estimate the first term in \\eqref{idp}. By (\\textbf{P2}), we obtain\n\\begin{align}\n \\nonumber\\mathbb{P}\\left(\\bigcup_{E\\in I}({\\mathbf{B}}_1\\cap{\\mathbf{B}}_2)\\right)&\\leq \\mathbb{P}\\left({\\rm dist}(\\sigma(H_{\\Lambda_1}),\\sigma(H_{\\Lambda_2}))\\leq 2L^{-\\tau}\\right)\\\\\n \\label{l2p}&\\leq{L^{-2p}}\/{2}.\n\\end{align}\nCombining \\eqref{idp}, \\eqref{mp} and \\eqref{l2p}, we have $\\mathbb{P}({\\mathbf{D}})\\leq C(d)L^{-J(\\alpha^{-1}p-d)}+L^{-2p}\/2$.\n\nThis concludes the proof.\n\\end{proof}\n\n\\section{Validity of (\\textbf{P1}) and (\\textbf{P2})}\nIn this section we will verify the validity of (\\textbf{P1}) and (\\textbf{P2}) in Theorem \\ref{msa}. As a consequence, we prove a complete MSA argument on Green's functions estimate. The regularity of $\\mu$ plays an essential role here.\n\\begin{thm}\\label{ine}\nLet $\\mu$ be H\\\"older continuous of order $\\rho>0$ (i.e., $\\mathcal{K}_\\rho(\\mu)>0$). Fix $0<\\kappa<\\mathcal{K}_\\rho(\\mu)$, $E_0\\in\\R$ and $\\tau'>(p+d)\/\\rho$. Then there exists $$\\underline{L}_0=\\underline{L}_0(\\kappa,\\mu,\\rho,\\tau',p,{r_1},s_0,d)>0$$ such that the following holds: if $L_0\\geq \\underline{L}_0$, then there is some $\\lambda_0=\\lambda_0(L_0,\\kappa,\\rho,p,s_0,d)>0$ and $\\eta=\\eta(L_0,\\kappa,\\rho,p,d)>0$ so that for $\\lambda\\geq \\lambda_0$, we have\n\\begin{align*}\n \\mathbb{P}({\\ \\exists}\\ E\\in [E_0-\\eta,E_0+\\eta] \\ {\\rm s.t.\\ }\\ {\\rm both} \\ \\Lambda_{L_0}(m)\\ {\\rm and}\\ \\Lambda_{L_0}(n)\\ {\\rm are\\ }\\ (E,\\delta)-{\\rm\\bf bad})\n \\leq L_0^{-2p}\n \\end{align*}\nfor all $|m-n|>2L_0$.\n\\end{thm}\n\\begin{rem}\nWe will see in the proof $\\lambda_0\\sim{L_0^{(p+d)\/\\rho}}\\kappa^{-1\/\\rho}$ and $\\eta\\sim {L_0^{-(p+d)\/\\rho}}\\kappa^{1\/\\rho}$. In addition, $\\lambda_0$ and $\\eta$ are independent of $E_0$.\n\\end{rem}\n\n\\begin{proof}\nDefine the event\n\\begin{align*}\n{\\mathbf{R}}_n(\\varepsilon):\\ |V_\\omega(k)-E_0|\\leq \\varepsilon\\ {\\rm \\ for\\ some }\\ k\\in\\Lambda_{L_0}(n),\n\\end{align*}\nwhere $\\varepsilon\\in (0,1)$ will be specified below. Then by \\eqref{mu}, we obtain for $2\\varepsilon\\leq \\kappa_0=\\kappa_0(\\kappa,\\mu)>0$,\n\\begin{align}\n\\nonumber\\mathbb{P}({\\mathbf{R}}_n(\\varepsilon))&\\leq (2L_0+1)^d\\mu\\left([E_0-\\varepsilon,E_0+\\varepsilon]\\right)\\\\\n\\nonumber&\\leq 2^\\rho(2L_0+1)^d\\kappa^{-1} \\varepsilon^\\rho\\\\\n\\label{Rn}&\\leq L_0^{-p},\n\\end{align}\nwhich permits us to set\n\\begin{align*}\n\\varepsilon=2^{-1}3^{-d\/\\rho}\\kappa^{1\/\\rho}L_0^{-(p+d)\/\\rho}.\n\\end{align*}\nIn particular, \\eqref{Rn} holds for $L_0\\geq \\underline{L}_0(\\kappa,\\mu,\\rho,p,d)>0$.\n\nSuppose now $\\omega\\notin {\\mathbf{R}}_n(\\varepsilon)$. Then for all $|E-E_0|\\leq \\varepsilon\/2$ and $k\\in\\Lambda_{L_0}(n)$, we have\n\\begin{align*}\n|V_\\omega(k)-E|&\\geq |V_\\omega(k)-E_0|-|E-E_0|\\geq \\varepsilon\/2,\n\\end{align*}\nwhich permits us to set $\\eta=\\varepsilon\/2$. Moreover, for $\\mathcal{D}=R_{\\Lambda_{L_0}(n)}(V_\\omega(\\cdot)-E)R_{\\Lambda_{L_0}(n)}$, we have by Definition \\ref{snorm} that $\\|\\mathcal{D}^{-1}\\|_s\\leq C(d)\/\\varepsilon$ for $s\\geq s_0$. Notice that\n\\begin{align*}\n\\|\\lambda^{-1}\\mathcal{T} \\mathcal{D}^{-1}\\|_{s_0}\\leq C(s_0,d)\\lambda^{-1}\\varepsilon^{-1}\\leq 1\/2\n\\end{align*}\nif $$\\lambda\\geq \\lambda_0=2C(s_0,d)\\varepsilon^{-1}.$$\n We assume $\\lambda\\geq\\lambda_0$. Then by the \\textbf{Perturbation argument} (i.e., \\eqref{pl2}--\\eqref{pl3}) and $$H_{\\Lambda_{L_0}(n)}-E=R_{\\Lambda_{L_0}(n)}\\lambda^{-1} \\mathcal{T} R_{\\Lambda_{L_0}(n)}+\\mathcal{D},$$ we have\n\\begin{align*}\n\\|G_{\\Lambda_{L_0}(n)}(E)\\|_{s_0}&\\leq 2\\|\\mathcal{D}^{-1}\\|_{s_0}\\leq C(d)\\varepsilon^{-1},\n\\end{align*}\nand for $s\\geq s_0$,\n\\begin{align*}\n\\|G_{\\Lambda_{L_0}(n)}(E)\\|_{s}&\\leq C(s,d)(\\varepsilon^{-1}+\\lambda^{-1}\\varepsilon^{-2})\\\\\n&\\leq C(s,s_0,d)\\varepsilon^{-1}\\ ({\\rm since\\ }\\lambda\\geq \\lambda_0\\sim\\varepsilon^{-1}).\n\\end{align*}\nWe restrict $s_0\\leq s\\leq {r_1}$ in the following. In order to show $\\Lambda_{L_0}(n)$ is $(E,\\delta)$-\\textbf{good}, it suffices to let\n\\begin{align}\\label{ll0}\nC({r_1},s_0,d)\\varepsilon^{-1}=C(\\rho,{r_1},s_0,d)\\kappa^{-1\/\\rho}L_0^{(p+d)\/\\rho}\\leq L_0^{\\tau'},\n\\end{align}\nwhich indicates we can allow $L_0\\geq \\underline{L}_0(\\kappa,\\mu,\\rho,\\tau',p,{r_1},s_0,d)>0.$ We should remark here \\eqref{ll0} makes sense since $\\tau'>(p+d)\/\\rho$.\n\nFinally, for $|m-n|>2L_0$ and $\\lambda\\geq \\lambda_0$, we have by the \\textit{i.i.d} assumption of the potentials that\n\\begin{align*}\n &\\mathbb{P}({\\ \\exists}\\ E\\in [E_0-\\eta,E_0+\\eta] \\ {\\rm s.t.\\ }\\ {\\rm both}\\ \\Lambda_{L_0}(m)\\ {\\rm and}\\ \\Lambda_{L_0}(n)\\ {\\rm are\\ }\\ (E,\\delta)-{\\rm\\bf bad})\\\\\n &\\leq \\mathbb{P}(\\mathbf{R}_m(\\varepsilon))\\mathbb{P}(\\mathbf{R}_n(\\varepsilon))\\\\\n &\\leq L_0^{-2p}\\ ({\\rm by\\ }\\eqref{Rn}).\n \\end{align*}\n\\end{proof}\n\n\n\n\n\n\n\nWe then turn to the verification of (\\textbf{P2}). This will follow from an argument of Carmona-Klein-Martinelli \\cite{CKM87}.\n\\begin{lem}\\label{we}\n Let $\\mu$ be H\\\"older continuous of order $\\rho>0$ (i.e., $\\mathcal{K}_\\rho(\\mu)>0$). Then for any $0<\\kappa<\\mathcal{K}_\\rho(\\mu)$, we can find $\\kappa_0=\\kappa_0(\\kappa,\\mu)>0$ so that\n \\begin{align*}\n\\mathbb{P}({\\rm dist}(E,\\sigma(H_{\\Lambda_L(n)}))\\leq \\varepsilon) \\leq \\kappa^{-1} 2^\\rho(2L+1)^{d(1+\\rho)}\\varepsilon^\\rho\n\\end{align*}\n for all $E\\in\\R$, $n\\in\\Z^d$ and for all $\\varepsilon>0$, $L>0$ with $\\varepsilon(2L+1)^d\\leq \\kappa_0.$\n\n\\end{lem}\n\\begin{proof}\nNotice that the long-range term $\\lambda^{-1} \\mathcal{T}$ in our operator is \\textit{non-random}. Then the proof becomes similar to that in the Schr\\\"odinger operator case by Carmona-Klein-Martinelli \\cite{CKM87}. We omit the details here.\n\\end{proof}\n\n\n\n\n\n\nWe can then verify (\\textbf{P2}) in Theorem \\ref{msa}.\n\n\n\n\n \n\n\n\n\n\n\\begin{thm}[{\\bf Verification of} (\\textbf{P2})]\\label{vp2}\nLet $\\mu$ be H\\\"older continuous of order $\\rho>0$ (i.e., $\\mathcal{K}_\\rho(\\mu)>0$). Fix $0<\\kappa<\\mathcal{K}_\\rho(\\mu)$. Then For $L\\geq \\underline{L}_0(\\kappa,\\mu,\\rho,\\tau,p,d)>0$ and\n\\begin{align}\\label{tr}\n\\tau>(2p+(2+\\rho)d)\/\\rho,\n\\end{align} we have\n \\begin{align*}\\mathbb{P}({\\rm dist}(\\sigma(H_{\\Lambda_{L}(m)}),\\sigma(H_{\\Lambda_{L}(n)}))\\leq 2L^{-\\tau})\\leq L^{-2p}\/2\\end{align*} for all $|m-n|>2L$.\n\\end{thm}\n\\begin{proof}\nApply Lemma \\ref{we} with $\\varepsilon=2L^{-\\tau}$. Then we have by the \\textit{i.i.d} assumption of potentials, \\eqref{tr} and $L\\geq \\underline{L}_0(\\kappa,\\mu,\\rho,\\tau,p,d)>0$ that\n\\begin{align*}\n&\\mathbb{P}({\\rm dist}(\\sigma(H_{\\Lambda_{L}(m)}),\\sigma(H_{\\Lambda_{L}(n)})))\\\\\n&\\leq \\sum_{E\\in \\sigma(H_{\\Lambda_{L}(m)})}\\mathbb{P}({\\rm dist}(E,\\sigma(H_{\\Lambda_L(n)}))\\leq 2L^{-\\tau})\\\\\n&\\leq \\kappa^{-1} 4^\\rho(2L+1)^{d(2+\\rho)}L^{-\\rho\\tau}\\\\\n&\\leq L^{-2p}\/2.\n\\end{align*}\n\\end{proof}\n\nFinally, we provide a complete MSA argument on Green's functions estimate.\n\n\n\\begin{thm}\\label{kthm}\nLet $\\mu$ be H\\\"older continuous of order $\\rho>0$ (i.e., $\\mathcal{K}_\\rho(\\mu)>0$). Fix $E_0\\in\\R$ with $|E_0|\\leq 2(\\|\\mathcal{T}\\|+M)$, and assume \\eqref{para}, \\eqref{tr} hold true. Assume further that $(1+\\xi)\/\\alpha\\leq \\delta$, and $p>\\alpha d+2\\alpha p\/J$ with $J\\in 2\\N$. Then for $0<\\kappa<\\mathcal{K}_\\rho(\\mu)$, there exists $$\\underline{L}_0=\\underline{L}_0(\\kappa,\\mu,\\rho,\\|\\mathcal{T}\\|_{r_1},M,J,\\alpha,\\tau,\\xi,\\tau',\\delta, p,{r_1},s_0,d)>0$$ such that the following holds: For $L_0\\geq \\underline{L}_0$, there are some $\\lambda_0=\\lambda_0(L_0,\\kappa,\\rho,p,s_0,d)>0$ and $\\eta=\\eta(L_0,\\kappa,\\rho,p,d)>0$ so that for $\\lambda\\geq \\lambda_0$ and $k\\geq 0$, we have\n\\begin{align*}\n \\mathbb{P}({\\ \\exists}\\ E\\in [E_0-\\eta,E_0+\\eta] \\ {\\rm s.t.\\ }\\ {\\rm both}\\ \\Lambda_{L_k}(m)\\ {\\rm and}\\ \\Lambda_{L_k}(n)\\ {\\rm are\\ }\\ (E,\\delta)-{\\rm\\bf bad})\n \\leq L_k^{-2p}\n \\end{align*}\nfor all $|m-n|>2L_k$, where $L_{k+1}=[L_k^{\\alpha}]$ and $L_0\\geq \\underline{L}_0$.\n\\end{thm}\n\\begin{rem}\n\\begin{itemize}\n\\item[]\n\\item In this theorem we also have $\\lambda_0\\sim{L_0^{(p+d)\/\\rho}}\\kappa^{-1\/\\rho}$ and $\\eta\\sim {L_0^{-(p+d)\/\\rho}}\\kappa^{1\/\\rho}$. Usually, to prove the localization we can choose $L_0\\sim \\underline{L}_0$. The key point of the MSA scheme is that the largeness of disorder (i.e., $\\lambda_0$) depends only on the initial scales. The later iteration steps do not increase $\\lambda$ further. We also observe that $\\lambda_0$ and $\\eta$ are free from $E_0$.\n\n \\item In order to apply Theorem \\ref{msa}, we restrict $|E_0|\\leq 2(\\|\\mathcal{T}\\|+M)$ in this theorem. Actually, we have $\\sigma(H_\\omega)\\subset[-\\|\\mathcal{T}\\|-M, \\|\\mathcal{T}\\|+M]$.\n\\end{itemize}\n\\end{rem}\n\n\\begin{proof}\nLet $\\underline{L}_{00}=\\underline{L}_{00}(\\kappa,\\mu,\\rho,\\tau',p,{r_1},s_0,d)>0$ be given by Theorem \\ref{ine}.\nWe choose\n\\begin{align}\\label{ul0}\n\\underline{L}_0=\\max\\{\\underline{L}_{00},\\underline{l}_0\\},\n\\end{align}\nwhere $\\underline{l}_0=\\underline{l}_0(\\|\\mathcal{T}\\|_{r_1}, M, J, \\alpha, \\tau, \\xi, \\tau', \\delta, p, {r_1}, s_0, d)$ is given by Theorem \\ref{msa}.\n\nThen applying Theorem \\ref{ine} with $L_0\\geq\\underline{L}_0$, ${\\lambda}_0={\\lambda}_0(L_0,\\kappa,\\rho,p,s_0,d)$ and $\\eta=\\eta(L_0,\\kappa,\\rho,p,d)$ yields\n\\begin{align*}\n \\mathbb{P}({\\ \\exists}\\ E\\in [E_0-\\eta,E_0+\\eta] \\ {\\rm s.t.\\ }\\ {\\rm both}\\ \\Lambda_{L_0}(m)\\ {\\rm and}\\ \\Lambda_{L_0}(n)\\ {\\rm are\\ }\\ (E,\\delta)-{\\rm\\bf bad})\n \\leq L_0^{-2p}\n \\end{align*}\nfor all $|m-n|>2L_0$ and $\\lambda\\geq \\lambda_0$.\n\nLet $L_{k+1}=[L_k^\\alpha]$ and $L_0\\geq\\underline{L}_0$.\n\nAssume for some $k\\geq 0$ the following holds:\n\\begin{align*}\n \\mathbb{P}({\\ \\exists}\\ E\\in [E_0-\\eta,E_0+\\eta] \\ {\\rm s.t.\\ }\\ {\\rm both}\\ \\Lambda_{L_k}(m)\\ {\\rm and}\\ \\Lambda_{L_k}(n)\\ {\\rm are\\ }\\ (E,\\delta)-{\\rm\\bf bad})\n \\leq L_k^{-2p}\n \\end{align*}\nfor all $|m-n|>2L_k$.\nObviously, we have by \\eqref{ul0} that ${L}_k\\geq {L}_0\\geq \\underline{L}_0\\geq \\underline{l}_0>0$. Then applying Theorem \\ref{msa} (with $l=L_k,L=L_{k+1}, I=[E_0-\\eta,E_0+\\eta]$) and Theorem \\ref{vp2} yields\n\\begin{align*}\n \\mathbb{P}({\\ \\exists}\\ E\\in [E_0-\\eta,E_0+\\eta] \\ {\\rm s.t.\\ }\\ {\\rm both}\\ \\Lambda_{L_{k+1}}(m)\\ {\\rm and}\\ \\Lambda_{L_{k+1}}(n)\\ {\\rm are\\ }\\ (E,\\delta)-{\\rm\\bf bad})\n \\leq L_{k+1}^{-2p}\n \\end{align*}\nfor all $|m-n|>2L_{k+1}$.\n\nThis finishes the proof of the whole MSA argument.\n\n\\end{proof}\n\n\\section{Proof of Theorem \\ref{mthm}}\nRecall the Poisson's identity: Let $\\psi=\\{\\psi(n)\\}\\in {\\C}^{\\Z^d}$ satisfy $H_\\omega \\psi=E\\psi$. Assume further $G_{\\Lambda}(E)$ exists for some $\\Lambda\\subset{\\Z}^d$. Then for any $n\\in\\Lambda$, we have\n\\begin{align}\\label{pi}\n\\psi(n)=-\\sum_{n'\\in\\Lambda,n''\\notin\\Lambda}\\lambda^{-1} G_\\Lambda(E)(n,n')\\mathcal{T}(n',n'')\\psi(n'').\n\\end{align}\n\nWe then introduce the Shnol's Theorem of \\cite{H19} in long-range operator case, which is useful to prove our localization. We begin with the following definition.\n\\begin{defn\nLet $\\varepsilon>0$. An energy $E$ is called an $\\varepsilon$-generalized eigenvalue if there exists some $\\psi\\in{\\C}^{\\Z^d}$ satisfying $\\psi(0)=1, |\\psi(n)|\\leq C(1+|n|)^{d\/2+\\varepsilon}$ and $H_\\omega \\psi=E\\psi$. We call such $\\psi$ the $\\varepsilon$-generalized eigenfunction.\n\\end{defn}\nThe Shnol's Theorem for $H_\\omega$ reads\n\n\\begin{lem}[\\cite{H19}]\\label{shn} Let $r-2d>\\varepsilon>0$ and let $\\mathcal{E}_{\\omega,\\varepsilon}$ be the set of all $\\varepsilon$-generalized eigenvalues of $H_\\omega$. Then we have $\\mathcal{E}_{\\omega,\\varepsilon}\\subset\\sigma(H_\\omega),\\ \\nu_\\omega(\\sigma(H_\\omega)\\setminus \\mathcal{E}_{\\omega,\\varepsilon})=0$, where $\\nu_\\omega$ denotes some complete spectral measure of $H_\\omega$\n\\end{lem}\n\\begin{rem}\nTo prove pure point spectrum of $H_\\omega$, it suffices to show each $E\\in\\mathcal{E}_{\\omega,\\varepsilon}$ is indeed an eigenvalue of $H_\\omega$. Actually, since all eigenvalues of $H_\\omega$ are at most countable, it follows from Lemma \\ref{shn} that all spectral measures of $H_\\omega$ support on a countable set, and thus are of pure point. In the following we even obtain polynomially decaying of each generalized eigenfunction of $H_\\omega$ for $\\mathbb{P}$ a.e. $\\omega$. This yields the \\textit{power-law} localization.\n\\end{rem}\n\nIn what follows we fix $L_0=\\underline{L}_0,\\lambda_0,\\eta$ and $I=[E_0-\\eta,E_0+\\eta]$ in Theorem \\ref{kthm}.\n\nRecalling Theorem \\ref{kthm}, we have for $\\lambda\\geq\\lambda_0$ and $k\\geq 0$,\n\\begin{align}\\label{pbe}\n\\mathbb{P}({\\exists}\\ E\\in I \\ {\\rm s.t.\\ }\\ {\\rm both} \\ \\Lambda_{L_k}(m)\\ {\\rm and}\\ \\Lambda_{L_k}(n)\\ {\\rm are\\ \\ } (E,\\delta)-{\\rm{\\bf bad}})\n \\leq L_k^{-2p}\n \\end{align}\nfor all $|m-n|>2L_k$, where $L_{k+1}=[L_{k}^{\\alpha}]$ and $L_0\\gg1$.\n\n\n\nWe then prove of our main result on the \\textit{power-law} localization.\n\n\\begin{proof}[\\bf Proof of Theorem \\ref{mthm}]\nWe choose appropriate parameters satisfying \\eqref{para}, \\eqref{tr}, $(1+\\xi)\/\\alpha\\leq \\delta$, and $p>\\alpha d+2\\alpha p\/J$ with $J\\in 2\\N$. For this purpose, we can set by direct calculation the following\n\\begin{align*}\n\\alpha=6, \\delta=1\/2, \\xi=2.\n\\end{align*}\nLet $0<\\varepsilon\\ll1$ (will be specified later). We define $J_\\star=J_\\star(d,\\varepsilon)$ to be the smallest even integer satisfying $p=6d+\\varepsilon$ and $p>6d+\\frac{12}{J_\\star}p. $\nAs a consequence, we can set\n\\begin{align*}\n\\tau=(14\/\\rho+1)d+O(\\varepsilon+\\varepsilon\/\\rho), s_0=d\/2+\\varepsilon, \\tau'=(42\/\\rho+11\/2)d+O(\\varepsilon+\\varepsilon\/\\rho).\n\\end{align*}\nRecalling Remark \\ref{rgdec}, we set $\\zeta=19\/20$. Then for\n${r_1}\\geq (94\/\\rho+13)d$, we obtain $\\tau'0$, then we have by \\eqref{gdec} that\n\\begin{align}\n\\label{z1}\\|G_{\\Lambda_L(n)}(E)\\|&\\leq L^{(42\/\\rho+23\/4)d+O(\\varepsilon+\\varepsilon\/\\rho)},\\\\\n\\label{z2}|G_{\\Lambda_L(n)}(E)(n',n'')|&\\leq | n'-n''|^{-{r_1}\/20}\\ {\\rm for}\\ |n'-n''|\\geq L\/2.\n\\end{align}\n\n\n\nFor any $k\\geq 0$, we define the set $A_{k+1}=\\Lambda_{L_{k+1}}\\setminus\\Lambda_{2L_k}$ and the event\n\\begin{align*}{\\mathbf{E}}_k: \\ \\exists\\ E\\in I \\ {\\rm s.t.\\ both }\\ \\Lambda_{L_k}\\ {\\rm and}\\ {\\Lambda_{L_k}(n)}\\ \\ {\\rm (for\\ }{\\forall\\ n\\in A_{k+1})}\\ {\\rm are\\ }\\ (E,1\/2)-{\\rm {\\bf bad}}.\\end{align*}\nThus from $p=6d+\\varepsilon, \\alpha=6$ and \\eqref{pbe},\n\\begin{align*}\n\\mathbb{P}({\\mathbf{E}}_k)&\\leq (2L_{k}^6+1)^dL_k^{-2(6d+\\varepsilon)}\\leq C(d)L_k^{-(6d+2\\varepsilon)},\\\\\n\\sum_{k\\geq 0}\\mathbb{P}({\\mathbf{E}}_k)&\\leq \\sum_{k\\geq 0} C(d)L_k^{-(6d+2\\varepsilon)}<\\infty.\n\\end{align*}\nBy the Borel-Cantelli Lemma, we have $\\mathbb{P}({\\mathbf{E}}_k\\ {\\rm occurs \\ infinitely \\ often})=0.$\nIf we set $\\Omega_0$ to be the event s.t. ${\\mathbf{E}}_k\\ {\\rm occurs \\ only \\ finitely \\ often}$, then $\\mathbb{P}(\\Omega_0)=1.$\n\n\n\nLet $E\\in I$ be an $\\varepsilon_1$-generalized eigenvalue and $\\psi$ be its generalized eigenfunction, where $0<\\varepsilon_1\\ll1$ will be specified later. In particular, $\\psi(0)=1$. Suppose now there exist infinitely many $L_k$ so that $\\Lambda_{L_k}$ are $(E,1\/2)$-\\textbf{good}. Then from the Poisson's identity \\eqref{pi} and \\eqref{z1}--\\eqref{z2}, we obtain since ${r_1}\\geq (100\/\\rho+15)d$\n\\begin{align}\n\\nonumber1=|\\psi(0)|&\\leq \\sum_{n'\\in \\Lambda_{L_k}, n''\\notin \\Lambda_{L_k}}C(d) |G_{\\Lambda_{L_k}}(E)(0,n')|\\cdot|n'-n''|^{-r}(1+|n''|)^{d\/2+\\varepsilon_1}\\\\\n\\nonumber&\\leq {\\rm(I)}+{\\rm (II)},\n\\end{align}\nwhere\n\\begin{align*}\n&{\\rm(I)}=\\sum_{|n'|\\leq L_k\/2, |n''|>{L_k}}C(\\varepsilon_1,d)L_k^{(42\/\\rho+23\/4)d+O(\\varepsilon+\\varepsilon\/\\rho)}(|n''|\/2)^{-(100\/\\rho+15)d}|n''|^{d\/2+\\varepsilon_1},\\\\\n&{\\rm(II)}=\\sum_{L_k\/2\\leq |n'|\\leq L_k , |n''|>L_k}C(d) |n'|^{-(5\/\\rho+3)d}| n'-n''|^{-(100\/\\rho+15)d}|n''|^{d\/2+\\varepsilon_1}.\n\\end{align*}\nFor ${\\rm(I)}$, we have by \\eqref{ldec},\n\\begin{align*}\n{\\rm(I)}&\\leq C(\\varepsilon_1,d)L_k^{(42\/\\rho+27\/4)d+O(\\varepsilon+\\varepsilon\/\\rho)}L_k^{-(100\/\\rho+15-3\/2)d\/2+O(\\varepsilon+\\varepsilon\/\\rho+\\varepsilon_1)}\\\\\n&\\leq C(\\varepsilon_1,d)L_k^{-8d\/\\rho+O(\\varepsilon+\\varepsilon\/\\rho+\\varepsilon_1)}\\to 0\\ ({\\rm as \\ } L_k\\to\\infty).\n\\end{align*}\nFor ${\\rm(II)}$, we have also by \\eqref{ldec},\n\\begin{align*}\n{\\rm(II)}&\\leq C(\\varepsilon_1,d)L_k^d\\sum_{|n''|>L_k} |n''|^{-(5\/\\rho+3-1\/2)d+O(\\varepsilon_1)}\\\\\n&\\leq C(\\varepsilon_1,d)L_k^{-(5\\rho\/2+1\/4)d+O(\\varepsilon_1)}\\to 0\\ ({\\rm as \\ } L_k\\to\\infty).\n\\end{align*}\nThis implies that for any $\\varepsilon_1$-generalized eigenvalue $E$, there exist only finitely many $L_k$ so that $\\Lambda_{L_k}$ are $(E,1\/2)$-\\textbf{good}.\n\nIn the following we fix $\\omega\\in\\Omega_0$.\n\n\nFrom the above analysis, for ${r_1}\\geq (100\/\\rho+15)d$, we have shown there exists $k_0(\\omega)> 0$ such that for $k\\geq k_0$ all $\\Lambda_{L_k}(n)$ with $n\\in A_{k+1}$ are $(E,1\/2)$-\\textbf{good}. We define another set $\\widetilde A_{k+1}=\\Lambda_{L_{k+1}\/10}\\setminus \\Lambda_{10 L_k}$. Obviously, $\\widetilde A_{k+1}\\subset A_{k+1}$. We will show for ${r}\\geq\\max\\{(100\/\\rho+23)d, 331d\\}$, $\\varepsilon,\\varepsilon_1\\ll1$ and $k\\geq k_1=k_1(\\kappa,\\mu,\\rho,{r},d,\\omega)>0$ the following holds true:\n \\begin{align}\\label{dec}\n|\\psi(n)|\\leq |n|^{-{r}\/600}\\ {\\rm for\\ } n\\in\\widetilde A_{k+1}.\n\\end{align}\n Once \\eqref{dec} was established for all $k\\geq k_1$, it follows from $\\bigcup_{k\\geq k_1}\\widetilde A_{k+1}=\\{n\\in{\\Z}^d:\\ |n|\\geq 10L_{k_1}\\}$ that $|\\psi(n)|\\leq {|n|^{-{r}\/600}}$ for all $|n|\\geq 10L_{k_1}$. This implies that $H_\\omega$ exhibits the \\textit{power-law} localization on $I$. In order to finish the proof of Theorem \\ref{mthm}, it suffices to cover $[-\\|\\mathcal{T}\\|-M,\\|\\mathcal{T}\\|+M]$ by intervals of length $\\eta$.\n\nWe let $r=r_1+8d$.\n\nWe try to prove \\eqref{dec}. Notice that $\\omega\\in\\Omega_0$ and $n\\in \\widetilde A_{k+1}\\subset A_{k+1}$. We know $\\Lambda_{L_k}(n)\\subset A_{k+1}$ is $(E,1\/2)$-\\textbf{good}. Then recalling \\eqref{pi} again, we have\n\\begin{align*}\n|\\psi(n)|&\\leq \\sum_{n'\\in \\Lambda_{L_k}(n), n''\\notin \\Lambda_{L_k}(n)}C(d) |G_{\\Lambda_{L_k}(n)}(E)(n,n')|\\cdot|n'-n''|^{-r}(1+|n''|)^{d\/2+\\varepsilon_1}\\\\\n&\\leq {\\rm (III)}+{\\rm (IV)},\n\\end{align*}\nwhere\n\\begin{align*}\n&{\\rm (III)}\\\\\n&=\\sum_{|n'-n|\\leq L_k\/2, |n''-n|>{L_k}}C(d)L_k^{(42\/\\rho+23\/4)d+O(\\varepsilon+\\varepsilon\/\\rho)}(|n''-n|\/2)^{-r}(1+L_{k+1}+|n''-n|)^{d\/2+\\varepsilon_1},\\\\\n&{\\rm (IV)}=\\sum_{L_k\/2\\leq |n'-n|\\leq L_k , |n''-n|>{L_k}}C(d) |n'-n|^{-{r_1}\/20}|n'-n''|^{-r}(1+L_{k+1}+|n''-n|)^{d\/2+\\varepsilon_1}.\n\\end{align*}\nFor {\\rm (III)}, we have by \\eqref{ldec},\n\\begin{align*}\n{\\rm (III)}&\\leq C(\\varepsilon_1,r,d)L_{k+1}^{d\/2+\\varepsilon_1}L_k^{(42\/\\rho+27\/4)d+O(\\varepsilon+\\varepsilon\/\\rho)}\\sum_{|n''-n|>L_k} |n''-n|^{-r+d\/2+\\varepsilon_1}\\\\\n&\\leq C(\\varepsilon_1,r,d)L_k^{(42\/\\rho+39\/4)d+O(\\varepsilon+\\varepsilon\/\\rho+\\varepsilon_1)} L_k^{(-r+3d\/2+\\varepsilon_1)\/2}\\\\\n&\\leq C(\\varepsilon_1,r,d)L_k^{-r\/2+(42\/\\rho+21\/2)d+O(\\varepsilon+\\varepsilon\/\\rho+\\varepsilon_1)}.\n\\end{align*}\nFor {\\rm (IV)}, we also have by \\eqref{ldec},\n\\begin{align*}\n{\\rm (IV)}&\\leq C(\\varepsilon_1,{r_1},d)L_{k+1}^{d\/2+\\varepsilon_1}L_k^d\\sum_{|n''-n|>L_k} |n''-n|^{-{r_1}\/20+d\/2+\\varepsilon_1}\\\\\n&\\leq C(\\varepsilon_1,{r_1},d)L_k^{4d+6\\varepsilon_1} L_k^{(-{r_1}\/20+3d\/2+\\varepsilon_1)\/2}\\\\\n&\\leq C(\\varepsilon_1,{r_1},d)L_k^{-{r_1}\/40+19d\/4+7\\varepsilon_1}.\n\\end{align*}\nCombining the above estimates and since ${r}\\geq\\max\\{(100\/\\rho+23)d, 331d\\}$, we have\n\\begin{align*}\n|\\psi(n)|\\leq C(\\varepsilon_1,{r_1},d)L_k^{(-{r_1}\/40+19d\/4+16\\varepsilon+7\\varepsilon_1)\/6} \\leq |n|^{-{r}\/600},\n\\end{align*}\nwhere we use $|L_k|\\geq |n|^{1\/6}\\gg1$ for $n\\in \\widetilde{A}_{k+1}$, and $\\varepsilon,\\varepsilon_1\\ll1$.\n\n\n\\end{proof}\n\n\\section*{Acknowledgements}\nI would like to thank Svetlana Jitomirskaya for reading earlier versions of the paper\nand her constructive suggestions. The author is grateful to Xiaoping Yuan for his encouragement.\n\nThis work was supported by NNSF of China grant 11901010\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction} \\label{sec:intro}\nX-ray and TeV gamma-ray observations clarify that supernova remnants (SNRs) are the acceleration sites of the cosmic rays up to the TeV range \n\\citep[e.g.,][]{1995Natur.378..255K,2013Sci...339..807A}.\nThe accelerated electrons with such energies emit non-thermal X-ray emission via synchrotron radiation which is characterized by power-law energy distribution in an X-ray band. \nThus, understanding of the non-thermal properties in X-ray is of great importance to study the nature of the accelerated electrons.\n\nThe diffusive shock acceleration mechanism \\citep[e.g.,][]{2008ApJ...678..939Z} is believed to be a relevant mechanism, \nwhich can explain the power-law spectrum of the non-thermal X-ray emission. \nRecently, it is reported that several SNRs exhibit the spatial shape variations of the non-thermal X-ray spectra with a physical scale of $\\sim$1-5 pc \\citep[e.g.,][]{2015ApJ...799..175S,2017ApJ...835...34T}. \nSeveral authors \\citep[e.g.,][]{2015ApJ...799..175S,2017ApJ...835...34T} suggest that the origin of the variations is due to the spatial difference of the cosmic-ray acceleration efficiency related to the surrounding interstellar gas distribution. \n\nSuperbubbles (SBs) are formed by combined phenomena of stellar winds from massive stars in OB associations and eventual supernovae (SNe) of those stars \\citep[e.g.,][]{1980ApJ...238L..27B}. \nThe morphology of hot gas in SBs is expected to be similar to that of a bubble blown by stellar winds of an isolated massive star \\citep{1977ApJ...218..377W}. \nThe kinetic energy in some of SBs exceed that in a supernova ($E_{\\rm K} \\sim10^{51}$ {\\rm erg}). \nSBs are filled with hot gas ($\\sim$10$^6$ K) heated by stellar wind and SN ejecta.\nNon-thermal X-ray emission has been detected from a number of Galactic and extragalactic SBs, \nsuch as RCW38 \\citep{2002ApJ...580L.161W}, Westerlund 1 \\citep{2006ApJ...650..203M}, IC 131 \\citep{2009ApJ...707.1361T}, N11 \\citep{2009ApJ...699..911M}, N51D \\citep{2004ApJ...605..751C}, \nand 30 Dor C \\citep{2004ApJ...602..257B,2009PASJ...61S.175Y,2015A&A...573A..73K}, \nwhich suggests a potential accelerating power exceeding an SNR, even though the detection of the non-therrmal emission in N11 and N51D is now doubtful due to the fluctuation of background point sources \\citep{2010ApJ...715..412Y}.\nHowever, very few studies have been performed to investigate the spatial variation of the non-thermal X-ray emission in the SBs and thus the cosmic-ray acceleration mechanism in the SBs has yet to be elucidated fully. \n\nThe SB, 30 Dor C, in Large Magellanic Cloud (LMC) was discovered by \\citet{1968MNRAS.139..461L}. \nIt is believed that 30 Dor C has formed via stellar winds of the LH90 OB association \\citep{1993A&A...280..426T} and several SNe. \nThe SB has the strongest non-thermal X-ray emission among SBs and a large diameter of $\\sim$80 pc.\nThus, it is one of ideal laboratories for studying the non-thermal emission mechanisms associated with a SB. \nThe SB includes not only the non-thermal emission but also the thermal emission from the shock-heated interstellar medium \\citep[e.g.,][]{2004ApJ...602..257B,2009PASJ...61S.175Y,2015A&A...573A..73K}. \nTherefore, comparing the spatial distribution of the non-thermal and thermal emissions can provide us with hints to reveal association between the acceleration efficiency and the environment potentially.\nIn this paper, we aim at investigating spatial variation of the physical properties in 30 Dor C with an unprecedented high resolution of $\\sim$10 pc. \n\nThe paper is organized as follows: section 2 presents the observations of 30 Dor C with $XMM-newton$ and the data reduction, and sections 3 and 4 describe our analysis method and the results, and the discussions on the spatial variation of the physical properties, respectively. \nIn section 5, we summarize our results and discussions. \nAt 30 Dor C assuming a distance of $\\sim$50~kpc to the LMC, $1\\arcmin$ corresponds to 15~pc. \nIn this paper, we used HEAsoft v6.21, XSPEC version 12.9, the $XMM-Newton$ Source Analysis Software (XMM-SAS) packaged in SAS 15.0.0 for our spectral analysis and a metal-abundance table tabulated in \\citet{1989GeCoA..53..197A}. \nUnless otherwise stated, the error ranges show the 90\\% confidence level from the center value.\n\\section{OBSERVATION AND DATA REDUCTION}\nWe, first, retrieved all of the data available for 30 Dor C in the $XMM-Newton$ Science Archive, and then selected the data taken from the pn instrument in European Photon Image Camera \\citep[EPIC,][]{2001A&A...365L...1J} with rich ($>$50 ks) net exposure time after the removal of the background flare periods to take advantage of the larger effective area than those of the EPIC-MOS instrument and avoid the systematic error between the detectors. \n\\setcounter{table}{0}\n\\begin{deluxetable*}{cccrc}[b!]\n\\tablecaption{Observation Log for 30 Dor C \\label{tab:obs_list}}\n\\tablecolumns{5}\n\\tablenum{1}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Obs. ID} &\n\\colhead{R.A.} &\n\\colhead{Dec.} & \\colhead{Date} & \\colhead{Exposure (ks)\\tablenotemark{a}} \\\\\n\\colhead{} & \\colhead{(J2000.0)} &\n\\colhead{(J2000.0)} & \\colhead{} & \\colhead{pn}\n}\n\\startdata\n 0104660101 & 05h35m27.99s & -69d16m11.0s & 2000 Nov. 17 & 3 \\\\\n\n\n\n 0406840301 & 05h35m27.99s & -69d16m11.1s & 2007 Jan. 1 & 63 \\\\\n 0506220101 & 05h35m28.30s & -69d16m13.0s & 2008 Jan. 11 & 68 \\\\\n 0556350101 & 05h35m28.30s & -69d16m13.0s & 2009 Jan. 30 & 63 \\\\\n\n 0601200101 & 05h35m28.30s & -69d16m13.0s & 2009 Dec. 11 & 71 \\\\\n 0650420101 & 05h35m28.30s & -69d16m13.0s & 2010 Dec. 12 & 51 \\\\\n 0671080101 & 05h35m28.30s & -69d16m13.0s & 2011 Dec. 02 & 61 \\\\\n 0690510101 & 05h35m28.30s & -69d16m13.0s & 2012 Dec. 11 & 60 \\\\\n\\enddata\n\\tablenotetext{a}{All exposure times show flare-filtered exposure times.\\unskip}\n\\end{deluxetable*}\n\nThe basic information of the observations is shown in Table \\ref{tab:obs_list}.\nWe generated calibrated event files with the SAS tools {\\tt epchain}. \nTime intervals with high background rates ($>$ 0.4 cts) seen in light curves of an off-source region in 10--12 keV were discarded in each observation.\nThe event lists were then filtered further, keeping only 0\u2013-4 patterns in an energy range of 0.4--12 keV.\n\n\\begin{figure}[!t]\n\\vspace{0.5cm}\n \\begin{center}\n\\hspace*{-1.0cm}\n \\includegraphics[width=100mm,angle=0]{fig1.eps}\n \\end{center}\n \\caption{\n$XMM-Newton$ EPIC-pn image of 30 Dor C in 0.3--1 keV (red), 1--2 keV (green),\n and 2--7 keV (blue), respectively. Spectra were extracted from the square regions ($\\sim$0.$^\\prime$7 $\\times$ 0.$^\\prime$7) with a region number. \n One- (non-thermal), two- (one-temperature and non-thermal), and three-component (two-temperature and non-thermal) models are finally adopted in magenta-, green-, and red-color box regions. The background spectrum was extracted from the yellow rectangle. \n}\n \\label{fig:30_Dor_C_img}\n\\end{figure}\n\n\\section{Analysis and Results}\nIn order to conduct spatially detailed X-ray spectral analysis for 30 Dor C, spectrum for each region is extracted from all the seven data and the extracted spectra are fitted simultaneously to reduce the statistical error.\nEach spectrum is rebinned to have at least 25 counts per an energy bin to allow the use of the $\\chi^2$-statistic.\nThe energy range in 0.5--7 keV was used in our analysis to avoid detector noise and the EPIC-pn fluorescence line forest just above 7 keV. \nThe SAS tasks \\texttt{rmfgen} and {\\tt arfgen} were utilized to create redistributed matrix files (RMF) \nand ancillary response files (ARF) respectively. \n\n\\subsection{X-ray Background Evaluation\\label{analysis_and_results}}\n\\begin{deluxetable*}{lcc}[b!]\n\\tablecaption{Best fit parameters obtained by using all seven observations for the background region \\label{tab:bgd_para}}\n\\tablecolumns{3}\n\\tablenum{2}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Component} &\n\\colhead{Parameter} &\n\\colhead{Best-fit Value} \n}\n\\startdata\n\\multicolumn{3}{c}{Absorption} \\\\\n\\hline \nGalactic (phabs) & $N_{\\rm H, Gal.}$ & 0.06 (fixed)\\tablenotemark{a}\\\\\nLMC (vphabs)\\tablenotemark{b} & $N_{\\rm H, LMC}$ & 0.20${\\pm0.16}$ \\\\\n\\hline\n\\multicolumn{3}{c}{Astrophyical X-ray Forground and Background} \\\\ [2pt]\n\\hline\nLHB (apec)\\tablenotemark{c} & $kT$ (keV) & 0.1 (fixed)\\tablenotemark{d}\\\\\n & $Norm$\\tablenotemark{e} & $<$12 \\\\\nGH (apec)\\tablenotemark{c} & $kT$ (keV) & 0.22${\\pm 0.01}$\\\\\n & $Norm$\\tablenotemark{e} & 42.5$^{+4.0}_{-4.8}$ \\\\\nCXB (powerlaw) & $\\Gamma$ & 1.4 (fixed)\\tablenotemark{f}\\\\\n & 2-10 keV intensity\\tablenotemark{g} & 6.0$\\pm1.3$ \\\\\n\\hline\n\\multicolumn{3}{c}{Thermal emission in LMC} \\\\ [2pt]\n\\hline\nISM (vapec)\\tablenotemark{b} & $kT$ (keV) & 0.89$^{+0.04}_{-0.03}$ \\\\\n & $Norm$\\tablenotemark{e} & 83$^{+17}_{-15}$ \\\\ \n\\hline\n$\\chi^2\/d.o.f$ & & 509\/375 \\\\\n\\enddata\n\\tablenotetext{a}{Fixed to the Galactic column density from the HI maps \\citep[][]{1990ARAA..28..215D}. The unit is 10$^{22}$ cm$^{-2}$.\\unskip}\n\\tablenotetext{b}{Fixed to the representative LMC values \\citep[][]{1992ApJ...384..508R,2014PASJ...66...26S}.}\n\\tablenotetext{c}{Fixed to a solar abundance table tabulated in \\citet{1989GeCoA..53..197A}.}\n\\tablenotetext{d}{Fixed to the value derived from \\cite{2009PASJ...61..805Y}. }\n\\tablenotetext{e}{Normalization of the apec model divided by a solid angle $\\Omega$. \n$Norm = (1\/\\Omega)$ $n_{\\rm e}n_{\\rm H}$ d$V\/[4((1 + z) D_A)^2 ]$ in unit of $10^{14}\\ {\\rm cm}^5~{\\rm str}^{-1}$, \nwhere, $z$, $n_{\\rm e}$, $n_{\\rm H}$, $D_A$, and $V$ are the redshift, the electron and hydrogen number densities (cm$^{-3}$), \nthe angular diameter distance (cm) and the emission volume (cm$^3$), respectively.}\n\\tablenotetext{f}{Fixed to an averaged value derived in \\cite{2002PASJ...54..327K}.}\n\\tablenotetext{g}{The unit is 10$^{-8}$ erg s$^{-1}$ cm$^{-2}$ str$^{-1}$.}\n\\end{deluxetable*}\n\n\n\\begin{figure}\n \\begin{center}\n\\hspace*{-1.0cm}\n \\includegraphics[width=100mm,angle=0]{fig2.eps}\n \\end{center}\n \\caption{\nA representative spectrum of a background region (Obs. ID: 0601200101) with the best-fit model. The dashed (cyan), dashed-dotted (blue), bold (magenta), solid (red) and solid (black) lines show the LHB, GH, CXB, ISM in the LMC and an artificial Al line, respectively.\n }\n \\label{fig:bgd_spec}\n\\end{figure}\nFirstly, we selected a source-free area in the vicinity of 30 Dor C to reduce spatial variation of the detector noise in the field of view shown in figure \\ref{fig:30_Dor_C_img} as a background region.\nThen, we conducted spectral analysis to confirm whether the region is suitable or not as a background region.\nIn order to create quiescent particle background (QPB) spectra, we used $XMM-Newton$ Extended Source Analysis Software (XMM-ESAS), packaged in SAS 15.0.0. \nThe QPB spectra were subtracted from the spectrum of each region in each observation. \\par \nFor our spectral analysis, we used the following phyically motivated model:\n\\begin{eqnarray*}\n(apec_{\\rm LHB} + phabs_{\\rm Galaxy}*(apec_{\\rm GH} + \\\\\nvphabs_{\\rm LMC} * (powerlaw_{\\rm CXB}))\n\\end{eqnarray*}\nThe X-ray background emission is comprised of three components \\citep[e.g.,][]{2009PASJ...61..805Y}, \nsuch as an unabsorbed thermal (k$T$ $\\sim$0.1 keV) emission \nfrom the Local Hot Bubble (LHB), an absorbed thermal (k$T$ $\\sim$0.2--0.3 keV) emission from the Galactic halo (GH), and an absorbed powerlaw \n\\citep[$\\Gamma = 1.4$, see ][]{2002PASJ...54..327K} \nwhich is known as cosmic X-ray background (CXB). \nWe used collisionally-ionized optically-thin thermal plasma model APEC \\citep{2001ApJ...556L..91S} for the LHB and GH in XSPEC. The metal abundance of these models is fixed to a solar abundance. \nBecause the temperature of the LHB component was not constrained well, the temperature is fixed to be a typical value of 0.1 keV \\citep{2009PASJ...61..805Y}.\nThe absorption by our Galaxy and the LMC was also taken into account.\nWe used a photo-electric absorption model in XSPEC, namely phabs, as the Galactic absorption model. \nThe column density $N_{\\rm H}$ was fixed at 6 $\\times$ 10$^{20}$ cm$^{-2}$ \\citep{1990ARAA..28..215D} \nin the direction of 30 Dor C, assuming the solar abundance. \nThe absorption by the LMC is modeled with vphabs, in which we can set each metal abundance separately.\nThe metal abundance was fixed to the representative LMC values \\citep[C=0.30 $Z_\\odot$, O=0.26 $Z_\\odot$, Ne=0.33 $Z_\\odot$,][]{1992ApJ...384..508R,2014PASJ...66...26S}, while the absorption column density is set to be free.\nThe background spectrum, however, can not be described with the model above and there is a significant residual feature around $\\sim$1 keV corresponding to emission lines from complex Fe L.\nWe hence added another thermal component, $apec_{\\rm LMC}$, with a different temperature as follows:\n\\begin{eqnarray*}\n(apec_{\\rm LHB} + phabs_{\\rm Galaxy}*(apec_{\\rm GH} + \\\\\nvphabs_{\\rm LMC} * (apec_{\\rm LMC} + powerlaw_{\\rm CXB})).\n\\end{eqnarray*}\nThe fitting results improved significantly and the spectrum with the best-fit model is shown in figure \\ref{fig:bgd_spec}. \nThe best-fit parameters are summarized in Table \\ref{tab:bgd_para}. \nThe plasma temperature of the added thermal component is consistent with that of the ISM in the LMC \\citep[e.g.,][]{2002A&A...392..103S}. \nThe 2--10 keV surface brightness of the power-law component \nwas $(6.0\\pm1.3) \\times 10^{-8}$ erg s$^{-1}$ cm$^{-2}$ str$^{-1}$ and the value is in good agreement with the expected CXB intensity \\citep{2002PASJ...54..327K}. \nWe confirmed that the best fit parameters are consistent with those obtained in each observation and thus all the spectra were fitted simultaneously to reduce the statistical error.\nAny further components such as a soft proton model are not required.\nThus, we concluded that the region and model are appropriate to evaluate the X-ray background components including the ISM of the LMC in our analysis.\n\\begin{figure*}[ht]\n \\begin{center}\n\\hspace{-0.5cm}\n \\includegraphics[width=175mm,angle=0]{fig3.eps}\n \\end{center}\n \\caption{\nExamples of the spectra with the best-fit non-thermal-, two-, and three-, component models. (1) non-thermal-component model for the spectrum in region 52. (2) two-component model for the spectrum in region 13. (3) three-component model for the spectrum in region 1. The magenta solid, blue dashed-dotted and orange dotted lines show non-thermal, low-temperature and high-temperature components, respectively. \nThe spectra are extracted from obsid 0601200101.\n}\n \\label{fig:typical_spectra}\n\\end{figure*} \n\n\\subsection{Spectral Analysis for the 30 Dor C Field}\nIn order to investigate spatial variation of the non-thermal X-ray emission in 30 Dor C, \nwe divided the 30 Dor C region into 70 regions of 10 pc $\\times$ 10 pc (0.$^\\prime$7 $\\times$ 0.$^\\prime$7) grids in unprecedented detail.\nThe region number is shown in figure \\ref{fig:30_Dor_C_img}.\nIn our spectral analysis, the background spectrum defined in $\\S\\ref{analysis_and_results}$ was subtracted from each region in each observation.\n\nAs indicated in previous studies \\cite[e.g.,][]{2004ApJ...602..257B,2009PASJ...61S.175Y,2015A&A...573A..73K}, not only non-thermal emission but also thermal emission is sometimes required at the same time to explain the observed spectra.\nActually, some spectra show significant enhancement around 0.6 and\/or 1 keV corresponding to emission lines of highly-ionized oxygen \/ Fe L-shell complex, respectively.\nWe attempted to apply three models in the following order, (1) non-thermal model, (2) two-component (non-thermal and one-temperature thermal) model, and (3) three-component (non-thermal and two-temperature thermal) model.\nFor regions where the fit significantly ($\\geq$ 99\\% in an F test) improved by adding an additional thermal component, we adopted the two- or three-component model.\nA collisionally-ionized optically-thin thermal plasma model, APEC, was used also for the thermal plasma in the regions except a young SNR, MCSNR~J0536-6913, associated with 30 Dor (see the region number 25 in figure \\ref{fig:30_Dor_C_img}).\nThe metal abundance of the low- \/ high-temperature plasmas mainly emitting oxygen \/ Fe L-shell lines is fixed to those reported in \\citet{2009PASJ...61S.175Y} \/ \\citet{1992ApJ...384..508R}.\nThe intrinsic absorption column density in the LMC is applied for the both plasmas and linked to that of the non-thermal model.\nThe only spectrum around MCSNR~J0536-6913 was well expressed with a combination of the non-thermal and non-equilibrium ionization collisional plasma models due to a heavy contamination from the SNR as shown in \\cite{2015A&A...573A..73K} and thus we removed the results in our discussion.\n\nMost of the spectra in the east region can be well described with the two- or three-component model, \nwhereas most of the spectra in the west region can be well fitted with the non-thermal-component model as shown in Figure \\ref{fig:30_Dor_C_img}. \nFigure \\ref{fig:typical_spectra} shows examples of the spectra with the best-fit non-thermal-, two-, and three-component models.\nWe investigated the photon index and intensity of the non-thermal component, the temperature and intensity of the thermal component, and the intrinsic absorption column density of the LMC.\nThe best-fit parameters in the best fit model are summarized in Table \\ref{tab:fitting_results}. \n\nFigures \\ref{fig:map} (a) and (b) show the distributions of the photon index and absorption corrected 2--10 keV intensity of the non-thermal component, respectively. \nIt is found for the first time that the non-thermal component is detected significantly in all the 70 regions covering the entire region of 30 Dor C.\nTheir typical relative error is $\\sim$8\\%. \nThe photon index shows spatial variation of $\\sim$2.0--3.7. \nThe areas with the relatively steep \/ flat photon indices are distributed in the east \/ west regions, respectively. \nEven though this sort of high spatial resolution spectral analysis had not been performed so far, the trend is consistent with the previous studies \\cite[e.g.,][]{2015A&A...573A..73K}.\nThe intensity in 2--10 keV significantly varies by more than an order of magnitude ($\\sim$4\u2013-130 $\\times$ 10$^{-8}$ erg s$^{-1}$ cm$^{-2}$ str$^{-1}$) in the field and is relatively large in the west region of the shell structure.\nTheir typical relative error is $\\sim$15\\%. \n\n\\begin{figure*}[!t]\n \\begin{center}\n \\includegraphics[width=175mm,angle=0]{fig4.eps}\n \\end{center}\n \\caption{Maps of the best-fit parameters: (a) the photon index $\\Gamma$ and (b) absorption-corrected intensity in 2--10 keV [10$^{-8}$ erg s$^{-1}$ cm$^{-2}$ str$^{-1}$], of the non-thermal component. \nSmoothed white contours of the $XMM-newton$ EPIC-pn images (0.5--7 keV) are overlaid in the maps. \n}\n \\label{fig:map}\n\\end{figure*}\n\nThe temperature and normalization of the thermal component mainly emitting oxygen lines vary from $\\sim$0.2 to $\\sim$0.3 keV and from $\\sim$0.2 to $\\sim$7 [$10^{17}$ cm$^{-5}$ str$^{-1}$], respectively.\nBecause the low-temperature thermal plasma is not detected in the source-free region of the vicinity of 30 Dor C and the morphology apparently forms a shell-like structure as shown in figure \\ref{fig:30_Dor_C_img}, the plasma may be associated with 30 Dor C.\nSuch low-temperature plasma with a temperature of $\\sim$0.1--0.3 keV is found also in other SBs \\cite[e.g.,][]{2001ApJS..136..119D,2010ApJ...715..412Y} and detected mainly in the east region as previously reported in \n\\citet{2004ApJ...602..257B,2009PASJ...61S.175Y,2015A&A...573A..73K}. \nThe temperature and normalization of the thermal component mainly emitting Fe L-shell lines vary from $\\sim$0.9 to $\\sim$1.2 keV and from $\\sim$0.1 to $\\sim$0.2 [$10^{17}$ cm$^{-5}$ str$^{-1}$], respectively. \nThe temperature is consistent with that of the observed in the source-free region within the statistical error. \nAccording to the results of \\citet{2002A&A...392..103S}, \nthe flux of ISM in LMC varies by more than twice depending on the regions.\nThe normalization of the high-temperature components is consistent with that of the observed in background spectra within the variation.\nWhile the results suggest that the high-temperature plasma may be due to the spatial variation of the ISM in the LMC,\nthe origin of the component is beyond our scope.\nWe confirmed that the uncertainty, e.g., in the metal abundance, does not affect our results for the non-thermal component significantly.\n\nThe intrinsic absorbing column density $N_{\\rm H}$ of the LMC ranges from $\\sim$0.3 to $\\sim$2 $\\times$ 10$^{22}$ cm$^{-2}$ and a typical relative error is $\\sim$15 \\%.\nThe intrinsic absorption in the east area of 30 Dor C seems to be relatively small ($\\sim$0.6 $\\times$ 10$^{22}$ cm$^{-2}$) while large ($\\sim$1 $\\times$ 10$^{22}$ cm$^{-2}$) in the west of the shell-like structure.\n\nWe confirmed that our representative results for our spectral analysis on the temperature of the thermal components, photon index of the non-thermal components, and intrinsic absorbing column density in the LMC are consistent with those of the previous studies \\cite[e.g.,][]{2004ApJ...602..257B,2004ApJ...611..881S,2009PASJ...61S.175Y,2015A&A...573A..73K}. \n\n\\input{bestfit_180709.tex}\n\n\\section{DISCUSSION}\nWe conducted the spatially resolved spectral analysis of 30 Dor C with a physical scale of $\\sim$10 pc in X-ray for the first time. We revealed that the non-thermal emission exists in all the regions covering the whole area of 30 Dor C and extracted the distribution of the physical properties such as the photon index and absorption-corrected intensity of the non-thermal component. \nWe found that the spectral shape changes and therefore the physical properties vary in this field.\nIn this section, we discussed, in particular, the origin of the spatial variation of the non-thermal X-ray properties to study the mechanism of cosmic-ray acceleration in SBs.\n\nSome SNRs also show spatial variation of the photon index and intensity of the non-thermal component \\citep[e.g.,][]{2015ApJ...799..175S,2017ApJ...835...34T}.\nIn particular, pc-scale spatially resolved spectral analysis reveals that the photon index closely correlates with the synchrotron X-ray intensity \\citep[e.g.,][]{2015ApJ...799..175S}.\nThus, we also extracted the relation between the photon index and the synchrotron X-ray intensity in the same manner as shown in figure \\ref{fig:properties}(a). \nThere is a clear negative correlation with a correlation coefficient of $\\sim$-0.5.\nOne of the interpretations is due to a shock-cloud interaction.\nMagnetohydrodynamic numerical simulations in \\cite{2012ApJ...744...71I} predict that such spatial variation can be produced by the shock-cloud interaction because the synchrotron X-ray intensity is positively correlated with the strength of the enhanced magnetic field due to the turbulence occurred in the interaction. \nThe photon index also can be changed by the enhanced magnetic field since the particle acceleration occurs efficiently.\nTherefore, it is naturally expected that the photon index gets flatter with increasing the synchrotron X-ray intensity.\n\nAccording to the scenario, the larger the photon index is, the smaller the cut-off energy in the energy distribution of\nelectrons should be. When we applied a broken power-law model instead of the power-law model for the non-thermal\nX-ray emission, however, no constraint was given to the breaking energy with the X-ray spectra alone. The correlation\nbetween the photon index and the cut-off energy has been observed in some SNRs when X-ray synchrotron spectra\nare analyzed using the SRCUT model \\citep{1998ApJ...493..375R,1999ApJ...525..368R} combined with radio synchrotron spectra\n\\citep{2004A&A...425..121R,2005ApJ...632..294B,2005ApJ...621..793B}. The spatially-resolved flux and spectral index\nof the radio synchrotron emission of 30 Dor C have been obtained in \\citet{2015A&A...573A..73K}, but as the authors say \nit is difficult to obtain their reliable values for the entire region of 30 Dor C due to the contaminations of \na foreground molecular cloud and of thermal radio emission. This situation prevents us from analyzing the\nmulti-wavelength spectra from radio to X-ray. The analysis of the spatially-resolved spectral energy distribution is\nleft to future works.\n\n\\cite{2017ApJ...843...61S} presents the molecular cloud distribution around 30 Dor C and it seems that there is a positive correlation between the synchrotron X-ray intensity and the amount of the molecular cloud.\nThe detailed comparison between X-ray and radio observations will be discussed (Yamane et al. in prep.).\n\n\\cite{2017ApJ...835...34T} argues that efficient acceleration occurs in the low density environment implying that the photon index steepens with increasing the normalization of the thermal component observationally based on the pc-scale spectral analysis results.\nThus, we also extracted the relation as shown in figure \\ref{fig:properties}(b).\nOne can see a positive correlation with a correlation coefficient of $\\sim$0.4 and thus similarities for SNRs are found in terms of the correlations between the non-thermal properties themselves and the non-thermal and thermal properties, which suggests the possibility that the same acceleration mechanism works also in the supperbubble.\n\n\\begin{figure*}\n \\begin{center}\n \\includegraphics[width=170mm,angle=0]{fig5.eps}\n \\end{center}\n\\vspace{-0cm}\n\\caption{Correlation plots for (a) the photon index vs. the 2--10 keV intensity [10$^{-8}$ erg s$^{-1}$ cm$^{-2}$], (b) Normalization of the low temperature thermal components [$10^{17}$ cm$^{-5}$ str$^{-1}$] vs. photon index, respectively.\n }\n\n\\label{fig:properties}\n\\end{figure*}\n\nThe bright non-thermal X-ray emission in 30 Dor C was detected. \nHowever, no other SBs exhibit such a bright non-thermal emission. \nThe SBs, where non-thermal X-ray emission has been significantly detected, \nare only RCW38 \\citep{2002ApJ...580L.161W}, Westerlund 1 \\citep{2006ApJ...650..203M} and IC 131 \\citep{2009ApJ...707.1361T}. \nThis sort of variation is observed also in SNRs and \\cite{2012ApJ...746..134N} discussed the time evolution of the non-thermal component as a function of the radius which can be an indicator of the dynamical age of the SNR as described in \\cite{1977ApJ...218..377W}.\nAs an analogy of the SNR case, we also investigated the relation between the non-thermal luminosity and the radius of the SBs as shown in figure \\ref{fig:SB_nonthermal}. \nThe non-thermal luminosity goes up with increasing the radius up to $\\sim$40 pc, whereas it then appears to be decreases. \n30 Dor C is located around the peak, which suggests that the system is currently on a phase of high energy particle acceleration.\n\n\\begin{figure}[t]\n\\vspace*{0.7cm}\n \\begin{center}\n \\includegraphics[width=90mm,angle=0]{fig6.eps}\n \\end{center}\n \\caption{\nNon-thermal X-ray luminosity in 2--10 keV as a function of the radius for superbubbles. \n }\n \\label{fig:SB_nonthermal}\n\\end{figure}\n\n\n\n\\section{SUMMARY}\nWe conducted a detailed spatial analysis using the large amount of $XMM-Newton$ archival data for 30 Dor C to study spatial variation of mainly the non-thermal component.\nThe 30 Dor C field was divided into 70 regions with a physical scale of $\\sim$10 pc and we found for the first time that the non-thermal emission exists in all the regions covering the whole field of 30 Dor C.\nThe extracted spectra in the east region can be described well with\na combination of the thermal and non-thermal models,\nwhereas the spectra in the west region can be well fitted with\nthe non-thermal model alone. \nThe photon index and intensity in 2--10 keV indicate the spatial variation of $\\sim$2.0--3.7 and $\\sim$(4--130) $\\times$ 10$^{-8}$ erg s$^{-1}$ cm$^{-2}$ str$^{-1}$) in the field and the negative correlation between the non-thermal physical properties is observed.\nThe temperature and normalization of the thermal component also vary within a range of $\\sim$0.2--0.3 keV \nand $\\sim$0.2--7 $\\times$ 10$^{17}$ cm$^{-5}$ str$^{-1}$, respectively. \nThe positive correlation between the photon index and the normalization of the thermal component is also observed as is the case in SNRs, suggesting that the same acceleration mechanism dominates also in the supperbubble.\\\\\n\n\nThis research was supported by a grant from the Hayakawa Satio Fund awarded by the Astronomical Society of Japan.\nHM is supported by JSPS KAKENHI Grand Number JP 15640356.\nIM acknowledge supports from the by JSPS KAKENHI Grand Number JP 26220703.\nHS is supported by JSPS KAKENHI Grand Number JP 16K17664.\nThe authors are grateful to the anonymous referee for his\/her comprehensive comments and useful suggestions, which improved the paper very much.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{INTRODUCTION}\n\nProduction, detection, and study of quark-gluon plasma (QGP) constitutes one of\nthe most important challenges of present day nuclear physics. \nThere are plausible reasons to believe that this deconfined state of strongly\ninteracting matter may be produced in collisions involving heavy nuclei. This\nexpectation has led to a great deal of excitement, and a number of international\ncollaborative efforts are underway\nto identify the signatures of QGP. Of these, \nlepton pairs ($e^+e^-$ or $\\mu^+ \\mu^-$) and photons are considered \nas one of the more \nreliable probes of this hot and dense phase since their\nmean free path is quite large compared to typical nuclear size enabling\nthem to escape without any final state interaction.\nTheir abundance and spectral distributions are also a rapidly varying\nfunction of the temperature and thus they furnish most valuable information\nabout the nascent plasma. \n\nSpurred by these expectations a considerable theoretical\neffort has been devoted to the\nstudy of large mass dileptons and high $p_T$ photons, which may have their\norigin mostly in the early stages of the QGP.\nA number of experiments\n\\footnote{ \n\\noindent Was this the face that launched a thousand ships?\\\\\nAnd burnt the topless towers of Ilium?\\\\\nSweet Helen, make me immortal with a kiss..\\\\\n -- Faustus, Christopher Marlowe (1564--1593)\n}, viz.,\nWA80, WA93, WA98, HELIOS, CERES, and NA38 experiments at\nthe CERN SPS, the PHENIX experiment at the BNL RHIC,\nand the ALICE experiment at the CERN LHC, are designed towards\nmeasuring the electromagnetic radiations from relativistic heavy ion collisions.\n It should be remembered, though,\nthat photons and dileptons are emitted at every stage of the evolution of the\nsystem, and they carry rather more precise\nimprints of the circumstances of their\n`birth'. We shall see that the intense glow of soft photons and low mass\ndileptons can provide reliable and useful information about the later\nstages of the interacting system.\n\nWe concentrate in particular on soft photons and low-mass dileptons \nproduced by bremsstrahlung processes whose\nenergies are low enough to enable us to use the so-called soft-photon\napproximation \\cite{ruckl}. This brings in a unique advantage as, \nbut for a `known' phase\nspace factor (see later) arising due to the finite mass of the dileptons, the\nbasic cross-sections for the two processes become identical. The dileptons\nof different invariant masses, however, are affected differently by \nthe transverse flow,\nwhich should be substantial towards the last moments of the interacting system.\n A comparison of the yield of dileptons of different masses and photons could \nthen provide us with a valuable information about the flow. This\nidentity of the cross-section for \nthe basic process is not available, for example, between \nsingle photons originating from Compton and annihilation processes \n(in the QGP), and nuclear reactions of the type, say, \n$\\pi\\rho\\,\\rightarrow\\,\\pi \\gamma$ in the hadronic matter or\nquark or pion annihilation processes for the dileptons.\n\nIt is also important to understand this contribution in quantitative detail\nas it is quite likely that the lowering of the mass of $\\rho$ mesons,\ndue to high baryonic densities reached in such collisions, for\nexample, will populate the mass region well below the $m_\\rho$\nin case of dileptons. Thus for example, Li et al\n\\cite{li} postulate that the mass of the `primordial' $\\rho$ mesons may have\n dropped to 370 MeV in S+Au collisions at the CERN SPS studied by the \nCERES group \\cite{ceres}. The drop is likely to be even larger for the Pb+Pb\ncollsion at the SPS energies, in their treatment \\cite{ko}.\nThis increase in the net baryonic density is unlikely to be achieved \nat the RHIC or the LHC energies because of the considerable increase in the\ntransparency.\n\nOn the other hand, there are reasons to believe that the pion-form factor\n $F_\\pi(M)$, as\nwell as the decay width ($\\Gamma_\\rho$) of the $\\rho$ meson, e.g., \nmay depend on the\ntemperature, either because of chiral symmetry restoration, or collision\nbroadening, or both. Thus, in the simplest approximation for the chiral\nperturbation theory, modifications of $F_\\pi$ and $\\Gamma(\\rho)$ are given\nby \\cite{gaber}\n\\begin{eqnarray}\nF_\\pi(M,T)&=&F_\\pi(M,0)\\left( 1-\\frac{T^2}{8F_\\pi^2(M,0)}\\right)~,\\nonumber\\\\\n\\Gamma_\\rho(T)&=&\\Gamma_\\rho(0)\/ \\left(1-\\frac{T^2}{4F_\\pi^2(M,0)}\\right),\n\\end{eqnarray}\nwith the mass of the $\\rho$ meson remaining essentially unchanged (see e.g.,\nRef. \\cite{rob} for more recent developments). \nThis corresponds to a\ndecrease in $F_{\\pi}$ by about 30\\% and an increase in $\\Gamma_{\\rho}$ by\na factor of 3 at T=160 MeV, thus affecting the production of dileptons\nfrom annihilation of pions beyond $M\\approx$ 400 MeV or so.\nThe broadening of the $\\Gamma(\\rho)$ \ndue to collisions has been estimated by Haglin\n\\cite{kevin2} as\n\\begin{equation}\n\\Gamma_\\rho(T)=\\Gamma_\\rho(0)+\\left(a+b T +c T^2\\right),\n\\end{equation}\nwhere $a= 0.50$ GeV, $b=-7.16$, and $c=30.16$ GeV$^{-1}$, and corresponds to an \nincrease of about 80\\% at $T=$ 160 MeV, whose effect will be limited to\ndileptons from pion annihilation beyond $M\\approx$ 500 MeV. \n\nRecall that in the early days of the investigations of QGP it was \noften suggested\nthat thermal dileptons having an invariant mass less than $2 m_\\pi$ could\noriginate only from the annihilation of quarks and anti-quarks which were\nassumed to be essentially massless.\n Very soon it was realized (see Ref.\\cite{helmut,kevin1} and references\ntherein) that there\ncould be substantial production of dileptons having lower invariant\nmasses from the bremsstrahlung of pions as well as quarks.\nA better understanding \\cite{braaten,weldon} of the dynamics of hot QCD has \nendowed quarks with a thermal mass of a few hundred MeV. \nIf we believe \\cite{kamp} that\nthe invariant mass of the dileptons will have $M \\geq 2 m_{\\mathrm {th}}$,\nif they originate from the quark annihilation, then the mass window below a\nfew hundred MeV is populated primarily by dileptons from \nbremsstrahlung processes at the colliders. Of course, there would be a\nbackground from Dalitz decays of $\\eta$ and $\\pi^0$ mesons, which will have to\nbe eliminated before the glow of the soft dileptons becomes visible.\nSimilar considerations will apply to soft photons originating from the\nbremsstrahlung of pions.\n\n The QGP likely to be produced in relativistic heavy ion collisions will\nhave enormous internal pressure, and will expand rapidly \\cite{vesa}.\n If the life time of the interacting system is large compared to\n $\\sim R_T\/c_s$, where\n$R_T$ is the transverse size of the system and $c_s$ is the speed \nof sound, the consequences of the transverse expansion will become very evident.\nThe last stages of the interacting system are also likely to be repository\nof the details of the flow, and thus the soft photons and low mass\ndileptons which derive their maximum contributions from this stage will\nalso carry unique information about the flow.\n\n We organise our paper as follows. In Section II we briefly recall\nthe formulation for the bremsstrahlung production of soft photons and low\nmass dileptons.\nSection III describes the results and discussions related to various\napproximations used in the work are given in Section IV. \nFinally we give a brief\nSummary. With this we also conclude our study of soft electromagnetic\nradiations \\cite{dipali,pradip}\ninitiated earlier.\n\n\\section{FORMULATION}\n\n\\subsection{Soft Photon Approximation}\n\nThe production of low mass dileptons and soft photons is most conveniently\nevaluated within a soft-photon approximation. In so far as this approximation\nremains valid, it provides for an easy manipulation of the strong interaction\npart of the scattering and enables us to test the sensitivity of the\nresults to such details. \n\nThus the first question which comes to mind is, how\nreliable is the soft-photon approximation? The existing treatments for the\nbremsstrahlung production of low mass dileptons were critically examined by\nLichard, recently \\cite{peter}. We shall, as in our earlier works\n \\cite{dipali,pradip}, use the correct numerical factor of \n($\\alpha\/3\\pi M^2$) in Eq.(8) (see later)\nand also use the virtual photon current, as suggested by Lichard.\n\nA reasonably accurate check on the soft photon approximation for\nthe quark driven processes can be obtained from a comparison with the\nrate for a zero-momentum soft dilepton production \\cite{eric} in a QGP\nevaluated by using the resummation technique of \nBraaten and Pisarski \\cite{braaten}. This was done recently \\cite{kevin1},\nwith interesting results. To quote, it was found that for dileptons having\nmasses $M\\leq$ 0.1 GeV, the soft photon approximation gives results\nwhich are very close to the findings of QCD perturbation theory.\nThe soft photon approximation was found to lead to results which were\nsmaller by a factor of about 1--4 for $0.1\\leq M \\leq 0.3$ GeV.\nThis has its origin in the fact that the QCD perturbation\ntheory includes the annihilation process, which\ncontributes substantially at larger masses (see figs.1a--c, later).\n This comparison provides two important insights;\nviz., scattering with virtual bremsstrahlung\n(rather than annihilation or Compton-like processes) accounts for most of the \nlow-mass QGP-driven pairs and the soft-photon approximation as applied to\nquark processes is fairly reasonable. We must add that we\nshall depict lowest order annihilation process $q\\bar{q}\\rightarrow e^+e^-$\nclearly and separately. \n\nIn a recent study, Eggers et al. \\cite{hans} have evaluated the bremsstrahlung\nproduction of dileptons from pion driven processes, without using the\nsoft-photon approximation in a One Boson Exchange model for the \ninteraction of pions. One of the many interesting observations in that work\nis that the use of the soft-photon approximation in terms of the invariant\nmass of the dileptons can overestimate the contribution of the bremsstrahlung\nprocesses. We use the soft-photon approximation in terms of the four-\nmomenta of the real or the virtual photons, and thus we feel that we are\nrelatively safe from this criticism. Thus we insist that both $M$ and \n$q$ remain reasonbly small. We have also limited ourselves to\ninvariant masses upto 300 MeV. \nStill it is worthwhile to recall that even though the individual contributions\nfrom different reactons involving pions to the basic cross-section\n ($d\\sigma\/dM$) are off by differing amounts as compared to the predictions\nof the soft-photon approximation, the rates are overestimated by atmost\n a factor of 1.5--2 for $M\\leq$ 0.3 GeV, provided we follow the suggestions\nof Lichard \\cite{peter}, as we have.\n\nIn view of the above discussion, we believe that the soft-photon approximation\nas employed by us is reliable to within a factor of 2.\nStill it should be certainly worthwhile to have the \nresults of this treatment \\cite{hans} for the transverse mass distribution to\nsettle this issue, clearly.\n\n\\subsection{Low Mass Dileptons}\n\nThe mechanism for the production of soft virtual photons from\nbremsstrahlung within a soft photon approximation \nhas been discussed by a number \nof authors ( see Ref.\\cite{dipali} and references therein) \nin great detail and thus we shall only briefly recall the\nformulation in order to fix the notation.\n\n The invariant cross-section for the scattering and at the same time\nproduction of a soft photon of four momentum $q^\\mu=(q^0,\\vec{q})=(E,\\vec{q})$ \nis given by\n\\begin{equation}\nq_0 \\frac{d^4 \\sigma^{\\gamma}}{d^3q dx} = \\frac{\\alpha}{4 \\pi^2}\n\\left \\{ \\sum_{{\\mathrm {pol}} \\lambda} J \\cdot \\epsilon_{\\lambda}~~ \nJ \\cdot \\epsilon_{\\lambda} \\right \\}\\frac{d\\sigma}{dx}\n\\end{equation}\nwhere $d\\sigma\/dx$ is the strong interaction cross-section for the\nreaction $ab\\,\\rightarrow\\, cd$, $\\epsilon_\\lambda$ is the polarization of the\nemitted photon, and\n $J^\\mu$ is the {\\em virtual photon current}\\cite{peter} given by\n\\begin{eqnarray}\nJ^\\mu& =& -Q_a \\frac{2p_a^{\\mu}-q^{\\mu}}{2p_a \\cdot q-M^2}-\nQ_b \\frac{2p_b^{\\mu}-q^{\\mu}}\n{2p_b \\cdot q-M^2}\\nonumber\\\\\n& &+Q_c \\frac{2p_c^{\\mu}+q^{\\mu}}{2p_c \\cdot q+M^2}+\nQ_d \\frac{2p_d^{\\mu}+q^{\\mu}}{2p_d \\cdot q+M^2}.\n\\end{eqnarray}\nIn the above equation the $Q$'s and $p$'s represent the charges (in units of\nproton charge) and the particle four momenta, respectively. \nThe cross-section for the production of dilepton is then obtained as\n\\begin{equation}\nE_+E_-\\, \\frac{d^6 \\sigma^{e^+e^-}}{d^3p_+d^3p_-} = \\frac{\\alpha}{3\\pi^2}\n\\frac{1}{q^2}\\, q_0\\frac{d^3\\sigma^{\\gamma}}{d^3q}\n\\end{equation}\n\nNow the invariant cross-\nsection for dilepton pair production can be written as \\cite{dipali}\n\\begin{eqnarray}\nE_+E_-\\, \\frac{d\\sigma_{ab \\rightarrow cd}^{e^+e^-}}{d^3p_+d^3p_-}=\n\\,& & \\frac{\\alpha^2}{12 \\pi^4 M^2} \\int \n |{\\epsilon \\cdot J}|_{ab \\rightarrow cd}^2\n\\,\\frac{d\\sigma_{ab \\rightarrow cd}}{dt}\\nonumber\\\\\n&\\times & \\delta(q^2-M^2)\\, dM^2\\,\\nonumber\\\\\n& \\times& \\delta^4 \\left(q-(p_++p_-)\\right) \\,d^4q \\,dt.\n\\end{eqnarray}\nThe rate of production of dileptons at temperature $T$\ncan then be written as\n\\begin{eqnarray}\nE\\frac{dN}{d^4xdM^2d^3q}=\\frac{T^6g_{ab}}{16\\pi^4} \n\\int_{z_{\\mathrm {min}}}^{\\infty} \\,dz & &\n\\frac{\\lambda(z^2T^2,m_a^2,m_b^2)}{T^4}\\nonumber\\\\\n&\\times &\n\\Phi(s,s_2,m_a^2,m_b^2)\\,\\nonumber\\\\\n&\\times & K_1(z)\\,\nE\\frac{d\\sigma_{ab}^{e^+e^-}}{dM^2d^3q},\n\\end{eqnarray}\nwhere\nthe cross-section for the process $ab \\rightarrow cd \\,e^+e^-$\nis given by\n\\begin{equation}\nE\\frac{d\\sigma_{ab \\rightarrow cd}^{e^+e^-}}{dM^2d^3q} =\n \\frac{\\alpha^2}{12 \\pi^3 M^2} \\,\\frac{\\widehat{\\sigma}(s)}{E^2},\n\\end{equation}\nwith\n\\begin{equation}\n\\widehat{\\sigma}(s)=\\int_{- \\lambda(s,m_a^2,m_b^2)\/s}^0 \\,dt\\,\n\\frac{d\\sigma_{ab \\rightarrow cd}}{dt}\\,\n\\left(q_0^2\\left|\\epsilon\\cdot J\\right|^2_{ab \\rightarrow cd}\\right),\n\\end{equation}\nand\n\\begin{equation}\n\\Phi(s,s_2,m_a^2,m_b^2)=\\frac{\\lambda^{1\/2}(s_2,m_a^2,m_b^2)}\n {\\lambda^{1\/2}(s,m_a^2,m_b^2)}\\,\\frac{s}{s_2},\n\\end{equation}\n$s_2=s+M^2-2\\sqrt{s}q_0$, and $\\lambda(x,y,z)=x^2-2(y+z)x+(y-z)^2$.\nThe expression for the average of the electromagnetic factor \nover the solid angle, can be found in Ref.\\cite{dipali}. \nThe value of $z_{\\mathrm {min}}$ is obtained from \n$\\lambda(s_2,m_a^2,m_b^2) = 0$. Note that the right hand side of Eq.(8)\nvaries as $1\/M^4$ for ${\\mathbf q}=0$.\n\n The strong interaction differential cross-section\n$d\\sigma_{qq}\/dt$ and $d\\sigma_{qg}\/dt$ for scattering of\nquarks and gluons are obtained from semi-phenomenological expressions\nused earlier by several authors for this purpose\n \\cite{kevin1,dipali,pradip,daniel}. For hot hadronic matter, we\nhave included the leading reactions: $\\pi^+ \\pi^- \\rightarrow \\pi^0\n\\pi^0$, $\\pi^+ \\pi^- \\rightarrow \\pi^+ \\pi^-$, $\\pi^+ \\pi^0 \\rightarrow\n\\pi^+ \\pi^0$, and $\\pi^- \\pi^0 \\rightarrow \\pi^- \\pi^0$ and evaluated\nthe strong scattering cross-section from an effective Lagrangian\nincorporating $\\sigma,$ $\\rho,$ and $f$ meson exchange \n\\cite{kevin1,dipali,pradip}.\n\n The corresponding expressions for the contribution of annihilation\nprocesses $q \\bar q \\rightarrow e^+e^-$ and $\\pi^+ \\pi^- \\rightarrow\ne^+e^-$ are given \\cite{kkmm} by,\n\n\\begin{eqnarray}\nE\\frac{dN}{d^4x dM^2 d^3q}& =& \\frac{\\sigma_a(M)}{4 (2\\pi)^5} M^2 e^{-E\/T}\n\\left[1-\\frac{4m_a^2}{M^2}\\right],\\nonumber\\\\\n\\sigma_{a}(M) &= &F_{a} {\\bar \\sigma(M)}, \\nonumber\\\\\n{\\bar \\sigma(M)}& = &\\frac{4 \\pi \\alpha^2}{3 M^2}\n\\left[1+\\frac{2m_e^2}{M^2}\\right] \\left[1-\\frac{4m_e^2}{M^2}\\right]^{1\/2},\n\\end{eqnarray}\nwhere $F_q=20\/3$ for a QGP consisting of $u$ and $d$ quarks, and\ngluons, and $F_\\pi$ is the pion form factor.\n\n\\subsection{ Soft Photons}\n\nNow we consider soft photon emission through the \nbremsstrahlung process, $ab \\rightarrow\ncd \\gamma$.\nThe invariant cross-section for the above process is obtained from\neq. (4) with $J^{\\mu}$ replaced by \n\\begin{equation}\nJ^\\mu = -Q_a \\frac{p_a^{\\mu}}{p_a \\cdot q}-Q_b \\frac{p_b^{\\mu}}\n{p_b \\cdot q}+Q_c \\frac{p_c^{\\mu}}{p_c \\cdot q}+\nQ_d \\frac{p_d^{\\mu}}{p_d \\cdot q},\n\\end{equation}\nwhich is appropriate for the emission of real photons.\nThe rate of production of photons at temperature $T$\ncan then be written as\n\\begin{eqnarray}\nE\\frac{dN}{d^4xd^3q}=\\frac{T^6g_{ab}}{16\\pi^4}\n & &\\int_{z_{\\mathrm {min}}}^{\\infty} \\,dz\n\\frac{\\lambda(z^2T^2,m_a^2,m_b^2)}{T^4}\\nonumber\\\\\n&\\times &\n\\Phi(s,s_2,m_a^2,m_b^2)\\, K_1(z)\\,\nE\\frac{d\\sigma_{ab}^{\\gamma}}{d^3q},\n\\end{eqnarray}\nwhere\n\\begin{equation}\nE\\frac{d\\sigma_{ab}^{\\gamma}}{d^3q} = \\frac{\\alpha}{4\\pi^2}\\, \n\\frac{{\\widehat\n\\sigma}(s)}{E^2},\n\\end{equation}\nwith ${\\widehat\\sigma}(s)$ defined as before (Eq.(9)) with $J^\\mu$ \nreplaced by real photon\ncurrent Eq.(12).\nEven at the risk of repetition, we would like to add that {\\it if we \nput $M = $ 0 in the phase-space factor $\\Phi_2$ and use the real photon\ncurrent in Eq.(7) and Eq.(9)}, we shall have\n\\begin{equation}\nE \\frac{dN_{e^+e^-}}{d^4xdM^2d^3q} \\equiv \\frac{\\alpha}{3 {\\pi} M^2}\\,\\,\nE \\frac{dN_{\\gamma}}{d^4xd^3q},\n\\end{equation}\nwhich also remains true in limit $M \\rightarrow $0.\nThus a comparison of the expression Eq.(13) with Eq.(7) \nimmediately shows that one\nmay use the results for photons and dileptons (with different masses) \nwith advantage to get information about, say, the evolution of the system.\n\nThe annihilation and the Compton processes $q \\bar q \\rightarrow \\gamma g$\nand $q(\\bar q)g \\rightarrow q(\\bar q)g \\gamma$ have already been studied\nin great detail by a number of authors \\cite{Kapusta}. We only mention \nthe result for a comparison:\n\\begin{equation}\nE\\frac{dN_{\\gamma}^{C+ann}}{d^4xd^3q} = \\frac{5}{9}\\, \\frac{\\alpha \\alpha_s}\n{2\\pi^2}\\, T^2\\, e^{-E\/T}\\, \\ln \\left( \\frac{2.912ET}{6m_q^2}+1 \\right)\n\\end{equation}\nwhere $m_q = \\sqrt{ 2\\pi \\alpha_s\/3}\\,T$ is the thermal mass of the\nquarks.\nIn the hadronic sector we consider the processes\n$\\pi \\rho \\rightarrow a_1 \\rightarrow \\pi \\gamma$, $\\pi \\rho\n\\rightarrow \\pi \\gamma$ \n for which rates\nhave been evaluated and parametrized in a convenient form \n\\cite{Kapusta,Xiong,Nadeu}. \n\\begin{equation}\nE\\frac{dN_{\\pi \\rho \\rightarrow a_1 \\rightarrow \\pi \\gamma}}{d^4xd^3q} =\n 2.4T^{2.15}\\,\\exp\\left[-1\/(1.35TE)^{0.77}-E\/T\\right]\\\\,\n\\end{equation}\n\\begin{equation}\nE\\frac{dN_{\\pi \\rho \\rightarrow \\pi \\gamma}}{d^4xd^3q} =\n T^{2.4}\\,\\exp\\left[-1\/(2TE)^{3\/4}-E\/T\\right]\\\\.\n\\end{equation}\nThe decay $\\omega \\rightarrow \\pi \\gamma$ during the life-time of the\ninteracting system is obtained from,\n\\begin{eqnarray}\nE\\frac{dN_{\\omega \\rightarrow \\pi \\gamma}}{d^4xd^3q} =\n\\frac{3m_{\\omega}\\Gamma_{\\omega \\rightarrow \\pi \\gamma}}{16\\pi^3E_0E}\n& &\\int_{E_{min}}^{\\infty}\\, \n dE_{\\omega}\\,E_{\\omega}f_{B E}(E_{\\omega})\n\\nonumber\\\\\n&\\times &\\left[1+f_{B E}(E_{\\omega}-E)\\right]\n\\end{eqnarray}\n\nHere $E_{min} = m_{\\omega}(E^2 + E^2_0)\/2EE_0$ and $E_0$ is the photon\nenergy in the rest frame of the $\\omega$ meson. Recall that for low energy\nphotons the reactions $\\pi\\pi\\,\\rightarrow\\,\\rho\\gamma$ and\nthe bremsstrahlung process $\\pi\\pi\\,\\rightarrow\\, \\pi\\pi\\gamma$ are equivalent,\nand including both of them would amount to a double\ncounting \\cite{pradip,redlich}.\n\n\\subsection{ Initial Conditions}\n\n We have considered central collisions of lead nuclei at CERN SPS, BNL RHIC,\nand CERN LHC energies. We assume that the collision leads to a thermalized\nand chemically equilibrated quark gluon plasma at an initial time\n $\\tau_i=$ 1 fm\/$c$ and initial temperature $T_i$.\n Further assuming an isentropic\nexpansion, one may relate \\cite{hk} the initial conditions to the \nmultiplicity density ($dN\/dy$);\n\\begin{equation}\nT_i^3 \\tau_i = \\frac{2 \\pi^4}{45 \\zeta(3) \\pi R_T^2 4a_k}\\, \\frac{dN}{dy},\n\\end{equation}\nwhere $R_T$ is the transverse radius of the lead nucleus and \n$a_k = 37\\pi^2\/90$ for a system consisting\nof massless u and d quarks, and gluons. The evolution\nof the system is obtained from a boost-invariant longitudinal\nexpansion and cylindrically symmetric transverse expansion \\cite{vesa}.\nWe further assume a first order phase transition to a hadronic matter \nconsisting of $\\pi$, $\\rho$, $\\omega$, and $\\eta$ mesons,\n($a_k \\approx 4.6\\pi^2\/90$), at $T=$ 160 MeV \\cite{our}.\nAfter all the quark matter has adiabatically converted to hadronic matter,\nthe system enters a hadronic phase and undergoes a freeze-out at $T=$ 140 MeV.\n\nThe particle rapidity density is taken as \\cite{kms} 624, 1735, and 5624\nrespectively. One may obtain much larger initial temperatures for the\nsame multiplicity densities by assuming more rapid thermalization of\nthe plasma. An upper limit for this is obtained by taking \n$\\tau_{i} \\simeq 1\/3 T_{i}$. We shall argue later, that the multiple\nscattering effects in the early dense QGP will, however, suppress the\nsoft radiations considerably, and thus the choice of\n$\\tau_i$ =1 fm\/$c$ should provide an interesting trade-off between\nthese competing effects.\n\n\\subsection{ Space-time Integration}\n\nThe dilepton transverse mass yield is then obtained by convoluting\nthe rates for their emission from QGP and hadronic matter with the space- \ntime history of the system:\n\\begin{eqnarray}\n\\frac{dN}{dM^2d^2M_Tdy}=\\int & &\\,\\tau \\,d\\tau\\, r\\, dr\\, d\\phi\\, d\\eta\n\\left[f_Q\\,E\\frac{dN^q}{d^4x dM^2 d^3q}\\right.\\nonumber\\\\\n&+&\\left.(1-f_Q)\\,E\\frac{dN^{\\pi}}\n{d^4x dM^2 d^3q}\\right]~,\n\\end{eqnarray}\nwhere $f_Q(r, \\tau)$ gives the fraction of the quark matter\nin the system.\n\n Similarly the photon spectrum is obtained by convoluting the rates for the\nemission of photons from QGP and the hadronic matter with the space time\nhistory of the system; \n\\begin{eqnarray}\n\\frac{dN}{d^2q_Tdy}=\\int & &\\,\\tau \\,d\\tau \\,r \\,dr\\, d\\phi \\,d\\eta\n\\left[f_Q\\, E\\frac{dN^q}{d^4x d^3q}\\right.\\nonumber\\\\\n&+&\\left.(1-f_Q)\\,\n E\\frac{dN^{\\pi}}{d^4x d^3q}\\right].\n\\end{eqnarray}\n\n\n\\section{RESULTS}\n\n\\subsection{Low mass Dileptons }\n\nIn order to ascertain the relative importance of the contributions of the\n quark bremsstrahlung, \npionic bremsstrahlung, quark annihilation, and pionic annihilation processes\nto low mass dileptons we show the rates for different values of $M$\nat $T=$ 160 MeV. All the results for the quark annihilation processes are\nobtained by taking $m_q=$ 5 MeV. If we adopt the view that \n$m_q=m_{\\mathrm {th}}$,\nas indeed, we have taken while evaluating the bremsstrahlung contributions,\n then the quark annihilation contribution will be absent\n\\cite{kamp} in this mass range.\nIn any case, we see that the quark driven bremsstrahlung processes\noutshine the pion driven bremsstrahlung contribution (fig.1a--c).\n\n\\begin{figure}\n\\psfig{figure=fig1a.ps,height=2.25in,width=3.25in}\n\\vskip 0.2cm\n\\psfig{figure=fig1b.ps,height=2.25in,width=3.25in}\n\\vskip 0.2cm\n\\psfig{figure=fig1c.ps,height=2.25in,width=3.25in}\n\\vskip 0.4cm\n\\caption{\n The production rate of low mass dielectrons from quark\nand pion bremsstrahlung at $T = $ 160 MeV.\nIn addition, the contribution of quark annihilation process is given\nfor a comparison. These results are shown for (a) $M =$ 0.1 GeV, (b)\n $M =$ 0.2 GeV, and (c) $M =$ 0.3 GeV respectively.}\n\\end{figure}\n\nThe results for the transverse mass distribution for the low mass dileptons\nat SPS energies are given in fig.2a--c.\nWe now see that the pion driven processes dominate the yield at all masses\nas the 4- volume occupied by the hadronic matter is much larger.\nThis is also evident from the invariant mass distribution (fig.2d).\nConsidering that the pion annihilation threshold limits the mass to $M>2m_\\pi$\nand even the quark annihilation may contribute only to masses larger than this,\nwe do find an intense glow of low mass dileptons, once the background from\nthe Dalitz decays of $\\pi^0$ and $\\eta$ mesons is removed. The recent\nexperience with the CERES experiment \\cite{ceres} has shown \nthat this could be possible to some extent. Recall also that the CERES\ndata for the S+Au system \\cite{ceres} shows a contribution from the\nbremsstrahlung processes \\cite{ssg}. It will be interesting to find a\nconfirmation of these early observations from the results for the Pb+Pb\nsystem as well.\n\n\\begin{figure}\n\\psfig{figure=fig2a.ps,height=2.25in,width=3.25in}\n\\vskip 0.2cm\n\\psfig{figure=fig2b.ps,height=2.25in,width=3.25in}\n\\vskip 0.2cm\n\\psfig{figure=fig2c.ps,height=2.25in,width=3.25in}\n\\vskip 0.2cm\n\\psfig{figure=fig2d.ps,height=2.25in,width=3.25in}\n\\vskip 0.4cm\n\\caption{(a--d): The transverse mass distribution of low mass dielectrons\nat SPS energies including bremsstrahlung process and annihilation \nprocess in the quark matter and the hadronic matter. We give the results\nfor invariant mass M equal to\n 0.1 GeV (a), 0.2 GeV (b), and 0.3 GeV (c) \nrespectively. The invariant mass distribution of low mass\ndielectrons are also shown (d).}\n\\end{figure}\n\nThe transverse mass distribution at RHIC energies (fig.3a--c) reveals another\ninteresting aspect. The transverse mass distribution at lower $M_T$ is\ndominated by the pion contribution. However at larger $M_T$, the contributions\nof the quark driven and pion driven processes are similar. This is a \nreflection of the larger temperature in the quark phase, and a larger effect\nof the transverse flow during the hadronic phase. If we look only at the \ninvariant mass distribution (fig.3d), this interesting aspect does not show up.\n\n\\begin{figure}\n\\psfig{figure=fig3a.ps,height=2.25in,width=3.25in}\n\\vskip 0.2cm\n\\psfig{figure=fig3b.ps,height=2.25in,width=3.25in}\n\\vskip 0.2cm\n\\psfig{figure=fig3c.ps,height=2.25in,width=3.25in}\n\\vskip 0.2cm\n\\psfig{figure=fig3d.ps,height=2.25in,width=3.25in}\n\\vskip 0.4cm\n\\caption{(a--d): Same as fig.~2, for RHIC energies.}\n\\end{figure}\n\nWe have further found (not shown here for reasons of space) that, at \nLHC energies the quark\ndriven bremstrahlung processes start dominating over the pion driven \nbremstrahlung processes even at relatively smaller $M_T$, as the slopes of\nthe quark-driven processes are much smaller.\n This aspect remains true even\nin the invariant mass spectrum, and the contributions become similar at $M=$\n0.3 GeV. \n\nThese results also clearly reveal the rapidly changing\nimportance of the different processes considered here leading to low mass\ndileptons, as the available energy (initial conditions) changes and\nas the invariant mass $M$ assumes varying values. When detailed results\nare available these considerations may help resolve different contributions.\n\nWe envisage an increase by a factor of 2--4 in the dilepton yield as we go from\nSPS to RHIC energies, and by a factor of 15-20 as we go from SPS to \nLHC energies. Thus the existence of a longlived interacting system would be\ncharacterized by an intense glow of low mass dileptons.\nThis means a large increase in the electromagnetic signals, as compared to\nthe estimates done by using only pion and quark annihilations.\n \nIt is well known that ratios of particle spectra can sensitively reveal the\ndetails of the variations of the underlying processes. We have seen in\nEq.(8) that\nthe transverse mass-spectra for low mass dileptons are proportional to\n$1\/M^2$. In figs.4--6 we have \nplotted the ratio of $M^{2} \\,dN\/d^{2}M_{T}dM^{2}dy$ at $M =$ 0.1 GeV to \nthat for $M =$0.2 GeV and $M =$0.3 GeV both with (solid line) and without \n(dashed line) the transverse flow at SPS, RHIC, and LHC respectively.\nWe have verified that the (oscillatory) structure seen in the results without \nthe transverse flow has its origin in the structure in $\\pi \\pi$ scattering\ncross-section,\nwhich is sampled in the process. One can also show that if there is no\ntransverse expansion of the system then the ratios as depicted here would\nbe independent of the initial temperature, that is they would be identical\nfor SPS, RHIC, and LHC energies, which is also seen from these figures. \nA deviation from this universal behaviour is indicative of the increasing\nimportance of the transverse flow as one increases the initial temperature\nof the system, which in turn decides the overall life-time of the system.\n\n\n\\begin{figure}\n\\psfig{figure=fig4.ps,height=2.25in,width=3.25in}\n\\vskip 0.4cm\n\\caption{ The ratio of $M^2$ weighted\n differential dielectron yield $M^2 dN\/ dM^2d^2M_{T}dy$ at $M =$ 0.1 GeV \nto that at $M =$ 0.2 GeV and $M =$ 0.3\nGeV as a function of transverse mass $M_T$ for SPS energies.\n The solid curve gives the total\ncontribution (quark matter + hadronic matter) with the transverse \nflow. Similarly the dashed curve gives the total contribution without\nthe transverse flow.}\n\\end{figure}\n\\begin{figure}\n\\psfig{figure=fig5.ps,height=2.25in,width=3.25in}\n\\vskip 0.4cm\n\\caption{Same as fig.~4 for RHIC energies. The definition of the\nsolid and the dashed curves are same as in fig.~4.} \n\\end{figure}\n\\begin{figure}\n\\psfig{figure=fig6.ps,height=2.25in,width=3.25in}\n\\vskip 0.4cm\n\\caption{Same as fig.~4 for LHC energies. The definition of the\nsolid and the dashed curves are same as in fig.~4.} \n\\end{figure}\n\n\nFinally, while investigating the dependence of our results on the\nfreeze-out temperature \na successively increasing dependence on this last stage of the\ninteracting system was seen as we go from SPS to RHIC to LHC energies. \nThe largest\nsensitivity is thus seen for the LHC energies (see fig.~7)\n, where the life-time of the\ninteracting system is longest, giving the transverse flow effects ample\nscope to come into full play.\n\n\\begin{figure}\n\\psfig{figure=fig7.ps,height=2.25in,width=3.25in}\n\\vskip 0.4cm\n\\caption{Sensitivity of the low mass dielectron spectra to the\nfreeze- out temperature at LHC energies for $M =$ 0.1 GeV.}\n\\end{figure}\n\n\\subsection{ Soft Photons}\n\nIn a manner similar to the above, we have plotted the rates for \ndifferent photon producing processes at $T=160$ MeV (fig.8).\nWe see that the quark and pion driven bremsstrahlung processes dominate\nupto energies of a few hundred MeV, after which they fall rapidly.\n\n\\begin{figure}\n\\psfig{figure=fig8.ps,height=2.25in,width=3.25in}\n\\vskip 0.4cm\n\\caption{Soft photon production rate at $T =$ 160 MeV \nfrom quark and pion bremsstrahlung, Compton + annihilation\nprocesses and the sum of the main hadronic reactions as shown in the figure.}\n\\end{figure}\n\nSpace-time integrated results for RHIC energies are\nshown in fig.9. We find that soft photons having transverse\nmomenta of upto a few hundred MeV mostly originate from pion driven\nbremsstrahlung processes, and once again the existence of a longlived\ninteracting system is revealed by an intense glow of soft photons,\nonce the background of decay photons is removed.\n\n\\begin{figure}\n\\psfig{figure=fig9.ps,height=2.25in,width=3.25in}\n\\vskip 0.4cm\n\\caption{The transverse momentum distribution of soft photons from \ndifferent mechanisms at RHIC energies. The sum of the contribution \n$\\pi \\rho \\rightarrow \\pi \\gamma$, $\\pi \\rho \\rightarrow a_1 \\rightarrow\n\\pi \\gamma$ and the decay $\\omega \\rightarrow \\pi \\gamma$ is referred as\n{\\it 'reactions'}.}\n\\end{figure}\nEven though the relative importance of the various contributions was\nfound to be similar at SPS and LHC energies, we envisage an increase \nby a factor of 2--4 in the yield\nof photons having $p_T=$ 200 MeV, as we go from SPS to RHIC and an increase\nby a factor of almost 10 as we go from SPS to LHC energies. The\ncomplete dominance of soft photons in determining the multiplicity\nof photons produced is seen from fig.10. Note that we have included only\nphotons having $p_T>$ 100 MeV, for this discussion, as we know that the\nyield for lower $p_T$ is subject to Landau Pomeranchuk effect.\nIt may be noted, however, that unlike the case of dileptons, the contribution\nof the reaction $\\pi \\pi \\rightarrow \\rho \\gamma$ which is equivalent\nto $\\pi \\pi \\rightarrow \\pi \\pi \\gamma$ was included in the esimates of\nsingle photons \\cite{Kapusta}, and thus the increase above does not \nnecessarily mean a new source. It merely points to a rapid rise in the\nyield of single photons having $p_T < $ 300--400 MeV.\n\\begin{figure}\n\\psfig{figure=fig10.ps,height=2.25in,width=3.25in}\n\\vskip 0.4cm\n\\caption{Soft photons vs. photons from Compton plus annihilation\nprocesses from the QGP and hadronic reactions at SPS, RHIC, and LHC energies\nfrom central collision of two lead nuclei.} \n\\end{figure}\n\n\\section{DISCUSSION}\n\nThere are a number of aspects which should be discussed before we draw our\nconclusions. We have already discussed the validity of the\nsoft-photon approximation in sect. II.\n\nIt is well-known that bremsstrahlung radiation could be very large for\nlight particles and one may worry about this aspect for the radiations\nfrom the quark matter. We have, however, used the thermal mass of quarks\nwhile evaluating the $d\\sigma\/dt$ as well as the kinematics of the collision,\nwhich is appropriate for fermions moving in a hot medium. We have already\nstated that if we extend this argument to quark \nannihilation as well, then the bremsstrahlung processes become the leading\ncontributors to this mass range for dileptons.\n\nAny study of soft electromagnetic radiations must address the question \nof Landau- Pomeranchuk \\cite{Landau} suppression of \nsuch processes in a dense medium.\nThe Landau- Pomeranchuk effect provides that if the formation time of a\nparticle is more than the time between two collisions, the emission of\nthe particle could be considerably suppressed due to destructive\ninterference of multiple scatterings. We would like to draw the\nattention of the readers to arguments developed in Ref. \\cite{pradip}\nearlier, about the extent of the modifications to our predictions due to this.\nIt can be argued that our results\nfor the sum of the radiations from the interacting system will remain \nfairly free from the effects of Landau- Pomeranchuk suppression\ntill we restrict ourselves to photons and dileptons having energies \nlarger than a few hundred MeV. The Landau- Pomeranchuk suppression\ncould be severe for lower energies as, indeed, \ndemonstrated by Cleymans et al. \\cite{jean}.\n\nThese considerations have an interesting connotation. Recall that\nwe have taken the initial time as 1 fm\/c. It is quite likely that the\nQGP may be thermalized much more quickly \\cite{kms}, and then we can have\na larger initial temperature for the same multiplicity of the\nparticles. We have seen earlier that the partonic density can then be\nmuch higher, and thus there would be a suppression of soft radiations.\nThus our choice of $\\tau_{i} = $1 fm\/c ensures that we start our \nevaluations {\\it after} the Landau- Pomeranchuk effect has lost its \ndominating effect, and that our estimates remain reasonable.\n A more complete treatment will include the Landau\nPomeranchuk effect and thus these suppressions would be automatically,\nand more properly accounted for.\n\nWe have approximated the hadronic phase as a non-interacting gas of $\\pi$,\n$\\rho$, $\\omega$, and $\\eta$ mesons. How will the results differ for\na richer hadronic matter, which would result in a reduced life-time for the\nmixed-phase? A richer equation of state for hadornic matter will also imply\na smaller speed of sound, and the attendant slower cooling of the system,\nand a longer life time for the hadronic phase. Thus, it was found recently\nthat the results for single photons \\cite{cape} with a\ntruncated equation of state\nas used here and a resonance gas containing all hadrons, for the \nhadronic matter left the final results essentially unaltered. Similar results\nshould be expected here, due to the similarity of the rates for the\nquark and pion driven processes (see. fig.1).\n\n\nAll our evaluations are made with the assumption that the QGP, as produced\ninitially, is in kinetic and chemical equilibrium, and that its evolution is\nisentropic. It is quite likely that the plasma as produced in relativistic\nheavy ion collisions is neither in kinetic nor in chemical equilibrium.\nHow will this affect our findings? Even though the kinetic equilibrium\ncould be achieved quickly enough, the chemical equilibration itself may \nnot be achieved at all \\cite{klaus,biro,mustafa}. The contributions of the\nQGP part is then easily obtained by introducing the products $\\lambda_i\n\\lambda_j$, where $\\lambda_i$ is the fugacity of the parton species $i$,\nin our expressions \\cite{strik}. \nNeedless to add that the overall contribution could\ncome down by a factor of upto 10 or more depending upon the initial\nconditions. So far there is no treatment which could model the\nhadronization of QGP which is far from chemical equilibrium. It is not\neven clear that such a matter will go through a mixed phase. The description\nof the hadronic phase (if any) also gets uncertain. However, a very\ninteresting outcome of this scenario could be a complete absence of radiations\nfrom the hadronic processes, if the QGP phase is not followed by an interacting\nhadronic matter living for some finite time! This could be of great interest.\n\nWhat could be other sources of low mass dileptons? It was suggested some \ntime ago \\cite{ssg} that $\\pi \\rho \\rightarrow \\pi\\, e^+e^-$ could contribute\nto low mass dileptons. This has now been evaluated \\cite{kevin2}, \nand it is found\nto contribute less than the bremsstrahlung processes at lower masses. However\nthe bremsstrahlung contribution decreases rapidly and for $M>$ 300 MeV,\nand the above reaction contributes at a level of 10--50\\% of the pionic annihilation.\nIt will be of interest to study the transverse mass distribution of this \nreaction, as it is likely to be different. \n\n\\section{SUMMARY}\n\nWe have calculated the transverse mass distribution of low mass\ndileptons and transverse momentum distribution of soft photons\nfrom central collision of two lead nuclei at CERN SPS, BNL RHIC,\nand CERN LHC energies. We assume that the collision leads to a\nthermalized and chemically equilibrated quark gluon plasma at the\nproper time $\\tau_{i} = $1 fm\/c. The plasma then expands, cools,\nand gets into a mixed phase at $T = $160 MeV. After all the quark\nmatter is adiabatically converted to hadronic matter, it cools again, \nand undergoes a freeze-out at $T = $140 MeV. We have considered a \nboost invariant longitudinal and cylindrically symmetric transverse\nexpansion. This is, to our knowledge, the first treatment of the \ndynamics of soft electromagnetic radiations in such collisions, \nwith transverse expansion, whose effect is seen to be large when\nthe life-time of the interacting system is large.\n\nWe find that the formation of such a system may be characterized \nby an intense glow of soft electromagnetic radiations, whose\nfeatures depend sensitively on the last stage of evolution, once we remove the\nbackground of decay photons or dileptons.\n\nWe are grateful to Hans Eggers and Kevin Haglin for very many useful\n discussions during the course of this work.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}