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+{"text":"\\section{Introduction} \\label{intro}\n\nNoncommutative Quantum Mechanics is an extensively studied subject \\cite{Gamboa:2001,Nair:2001,Duval:2001,Pei-Ming:2002,Horvathy:2002,Zhang:2004,Acatrinei:2005,Bastos:2006ps,2010CMaPh.299..709B,Bertolami:2005jw,Bernardini} and its interest arises for many reasons, more particularly from the fact that noncommutativity is present string theory and quantum gravity and black hole models (see e.g. \\cite{SW,Connes,BastosQCBH}). NCQM can be viewed as the low-energy and the finite number of particles limit of noncommutative field theories and its main difference from standard quantum mechanics is the inclusion of an additional set of commutation relations for position and momentum operators. The Heisenberg-Weyl algebra for these operators,\n\n\\begin{equation} \\label{heisenberg}\n\\left[\\hat{x}_i,\\hat{x}_j\\right]=0, \\hspace{40pt}\n\\left[\\hat{p}_i,\\hat{p}_j\\right]=0, \\hspace{40pt}\n\\left[\\hat{x}_i,\\hat{p}_j\\right]=\\mathrm{i}\\hbar\\delta_{ij}\n\\end{equation}\nis deformed to the NC algebra:\n\n\\begin{equation} \\label{NC algebra}\n\\left[\\hat{q}_i,\\hat{q}_j\\right]=\\mathrm{i}\\theta_{ij}, \\hspace{40pt}\n\\left[\\hat{\\pi}_i,\\hat{\\pi}_j\\right]=\\mathrm{i}\\eta_{ij}, \\hspace{40pt}\n\\left[\\hat{q}_i,\\hat{\\pi}_j\\right]=\\mathrm{i}\\hbar\\delta_{ij},\n\\end{equation}\nwhere $\\theta_{ij}$ and $\\eta_{ij}$ are anti-symmetric real matrices. The two sets of variables, $\\{\\hat{x}_i,\\hat{p}_i\\}$ and $\\{\\hat{q}_i,\\hat{\\pi}_i\\}$ are related by a non-canonical linear transformation usually refered to as Darboux transformation, also known as Seiberg-Witten (SW) map. It is known that, although this map is not unique, all physical observables are independent of the chosen map \\cite{Bastos:2006ps,2010CMaPh.299..709B}. Moreover, since the NC operators are defined in the same Hilbert space as the commutative ones, one can obtain a representation of them, up to some order of the noncommutative parameters, without the need for the Darboux transformation. However, in most cases, it is simpler to use this transformation in order to recover some known aspects of quantum mechanics. \n\nBesides the well-known operator formulation of quantum mechanics, a phase-space formulation of NCQM has been constructed \\cite{Bastos:2006ps,2010CMaPh.299..709B} which allows for a straightforward implementation of noncommutativity. This formulation is useful for treating general problems such as, for instance, in cases where the potential is not specified. In this case, the position noncommutativity may be treated by a change in the product of functions to the Moyal $\\star$-product, defined as:\n\n\\begin{equation}\nA(x)\\star_{\\theta}B(x):=A(x)\\mathrm{e}^{(\\mathrm{i}\/2)(\\overleftarrow{\\partial_{x_i}})\\theta_{ij}(\\overrightarrow{\\partial_{x_j}})}B(x),\n\\end{equation}\nand the momentum noncommutativity is introduced via a Darboux transformation. In the case of simple potentials, the use of the Darboux transformation ensures on its own, up to some order of the noncommutative parameter, a suitable noncommutative formulation.\n\nThroughout the following sections, whenever need, the Darboux transformation to be used is as follows \\cite{Bastos:2006ps}:\n\n\\begin{equation} \\label{Darboux}\n\\hat{q}_i=\\hat{x}_i-\\frac{\\theta_{ij}}{2\\hbar}\\hat{p}_j, \\qquad \\hat{\\pi}_i=\\hat{p}_i+\\frac{\\eta_{ij}}{2\\hbar}\\hat{x}_j.\n\\end{equation}\n\n\n\n\n\n\n\\section{Gauge Invariance}\n\nIn order to study the effects of NCQM we shall consider some physical systems of interest and investigate the implications of the NC deformation. The first example to consider is that of a particle with mass $m$ and charge $q$ in a magnetic field, with the Hamiltonian given by\n\\begin{equation}\n\\hat{H}= \\frac{1}{2m}\\left[\\boldsymbol{\\hat{\\pi}}-q\\boldsymbol{A}(\\boldsymbol{q})\\right]^2.\n\\end{equation}\n\nIn order to study this system we use the Moyal $\\star$-product for the product of terms and then use the Darboux transformation, Eq. (\\ref{Darboux}), to write the noncommuting Hamiltonian in terms of the commuting variables, $\\hat{x}$ and $\\hat{p}$. Thus, considering,\n\\begin{equation}\n\\hat{H}(\\hat{q},\\hat{\\pi})\\Psi(q)=\\hat{H}(\\hat{x},\\hat{\\pi})\\star_\\theta\\Psi(x)=\\hat{H}(\\hat{x},\\hat{\\pi})\\mathrm{e}^{(\\mathrm{i}\/2)(\\overleftarrow{\\partial_{x_i}})\\theta_{ij}(\\overrightarrow{\\partial_{x_j}})}\\Psi(x),\n\\end{equation}\nat first order in the parameter $\\theta$,\n\\begin{gather}\n\\left[\\hat{H}(\\hat{x},\\hat{\\pi})+\\frac{\\mathrm{i}\\theta_{ab}}{2}\\partial_a\\hat{H}(\\hat{x},\\hat{\\pi})\\partial_b\\right]\\Psi(x)= \\nonumber\\\\\n=\\left[\\frac{1}{2m}\\left(\\boldsymbol{\\hat{\\pi}}^2-2q\\boldsymbol{\\hat{\\pi}}\\cdot\\boldsymbol{A}(\\boldsymbol{q})+q^2A^2(q)\\right)+\\frac{\\mathrm{i}\\theta_{ab}}{2}\\partial_a\\left(q^2A^2(x)-2q\\boldsymbol{A}(x)\\cdot\\hat{\\pi}\\right)\\partial_b\\right]\\Psi(x)\n\\end{gather}\n\n\nIf we now consider that $\\theta_{ab}=\\theta\\epsilon_{ab}$, where $\\epsilon_{ab}$ is the 2-dimentional antisymmetric symbol, the effective noncommutative Hamiltonian, at first order in $\\theta$, becomes:\n\\begin{equation}\n\\hat{H}=\\frac{1}{2m}\\left(\\boldsymbol{\\hat{\\pi}}^2-2q\\boldsymbol{\\hat{\\pi}}\\cdot\\boldsymbol{A}(\\boldsymbol{q})+q^2A^2(q)\\right)+\\frac{\\mathrm{i}}{4m}\\left[\\nabla\\left(q^2A^2(x)-2q\\boldsymbol{A}(x)\\cdot\\hat{\\pi}\\right)\\times\\nabla\\right]\\cdot\\boldsymbol{\\theta}\n\\end{equation}\nwhere $\\boldsymbol{\\theta}=\\theta(1,-1,1)$. We now make use of the Darboux transformation, Eq. (\\ref{Darboux}), in the momentum operator (which is now the only noncommutative operator in the Hamiltonian) to obtain:\n\\begin{multline} \\label{eq ncemg}\n\\hat{H}=\\frac{1}{2m}\\left[\\left(\\hat{\\boldsymbol{p}}-q\\boldsymbol{A}(\\boldsymbol{x})\\right)^2-\\frac{1}{\\hbar}(\\hat{\\boldsymbol{x}}\\times\\hat{\\boldsymbol{p}})\\cdot\\boldsymbol{\\eta}-\\frac{q}{\\hbar}(\\hat{\\boldsymbol{x}}\\times\\boldsymbol{A}(\\boldsymbol{x}))\\cdot\\boldsymbol{\\eta}+\\frac{1}{4\\hbar^2}\\eta^2\\epsilon_{ij}\\epsilon_{ik}\\hat{x}_j\\hat{x}_k\\right] \\\\\n-\\frac{1}{4m\\hbar}\\left[\\nabla\\left(q^2A^2(\\boldsymbol{x})-2q\\boldsymbol{A}(\\boldsymbol{x})\\cdot\\hat{\\boldsymbol{p}}-\\frac{q}{\\hbar}(\\hat{\\boldsymbol{x}}\\times\\boldsymbol{A}(\\boldsymbol{x}))\\cdot\\boldsymbol{\\eta}\\right)\\times\\hat{\\boldsymbol{p}}\\right]\\cdot\\boldsymbol{\\theta},\n\\end{multline}\nwhere, as in the case of $\\theta$, $\\boldsymbol{\\eta}=\\eta(1,-1,1)$. We aim now to see how a gauge transformation modifies the Hamiltonian and study the condition under which the Hamiltonian is gauge invariant. Gauge invariance must be imposed, otherwise a gauge change would lead to a modification of the system energy for the same physical configuration. For this purpose, we consider a gauge transformation to the vector potential $\\boldsymbol{A}\\rightarrow\\boldsymbol{A}'=\\boldsymbol{A}+\\boldsymbol{\\nabla}\\alpha$, where $\\alpha$ is a scalar function of position. Consider now the first set of terms in the Hamiltonian, Eq. (\\ref{eq ncemg}). Under the stated transformation, we get:\n\\begin{equation}\n\\begin{split}\n\\frac{1}{2m}\\left[\\left(\\hat{\\boldsymbol{p}}-q\\boldsymbol{A}(\\boldsymbol{x})-q\\boldsymbol{\\nabla}\\alpha\\right)^2-\\frac{1}{\\hbar}(\\hat{\\boldsymbol{x}}\\times\\hat{\\boldsymbol{p}})\\cdot\\boldsymbol{\\eta}- \\right. \\\\\n\\left. -\\frac{q}{\\hbar}(\\hat{\\boldsymbol{x}}\\times\\boldsymbol{A}(\\boldsymbol{x}))\\cdot\\boldsymbol{\\eta}-\\frac{q}{\\hbar}(\\hat{\\boldsymbol{x}}\\times\\boldsymbol{\\nabla}\\alpha)\\cdot\\boldsymbol{\\eta}+\\frac{1}{4\\hbar^2}\\eta^2\\epsilon_{ij}\\epsilon_{ik}\\hat{x}_j\\hat{x}_k\\right].\n\\end{split}\n\\end{equation}\n\nIf we now change the wave function on which the Hamiltonian acts, to $\\Psi=\\mathrm{e}^{\\mathrm{i}q\\alpha\/\\hbar}\\Psi'$, the first set of extra terms in Eq. (\\ref{eq ncemg}) coming from the gauge transformation will be cancelled and so we may conclude that this set of therms is not problematic. However, this is not true for the second set of terms which is transformed to,\n\n\\begin{equation}\n\\left[\\nabla\\left(q^2(A(\\boldsymbol{x})+\\boldsymbol{\\nabla}\\alpha)^2-2q\\boldsymbol{A}(\\boldsymbol{x})\\cdot\\hat{\\boldsymbol{p}}-2q\\boldsymbol{\\nabla}\\alpha\\cdot\\hat{\\boldsymbol{p}}-\\frac{q}{\\hbar}(\\hat{\\boldsymbol{x}}\\times\\boldsymbol{A}(\\boldsymbol{x}))\\cdot\\boldsymbol{\\eta}-\\frac{q}{\\hbar}(\\hat{\\boldsymbol{x}}\\times\\boldsymbol{\\nabla}\\alpha)\\cdot\\boldsymbol{\\eta}\\right)\\times\\hat{\\boldsymbol{p}}\\right]\\cdot\\boldsymbol{\\theta}.\n\\end{equation}\n\nIf we now consider the wave function transformation, $\\Psi=\\mathrm{e}^{\\mathrm{i}q\\alpha\/\\hbar}\\Psi'$, we verify that the gauge transformation is not cancelled due to the momentum operator outside the divergence acting on the exponential. Thus, the phase transformation that absorbs the gauge transformation terms in the first part of the Hamiltonian, Eq. (\\ref{eq ncemg}), does not do so for the second set of terms. This comes from the fact that, in the first term, the change in $\\boldsymbol{A}$ can be seen as a change in $\\hat{\\boldsymbol{p}}$, and a constant change in momenta can always be absorbed by a phase change. The same does not occur for the change in the second term, making it impossible to accommodate it into a change in phase. Therefore, in order to make the Hamiltonian gauge invariant, this term must vanish. To accomplish this for any $\\boldsymbol{A}$, $\\theta$ must vanish. This result is consistent to an explicit computation in the context of the Hamiltonian of fermionic fields \\cite{Bertolami:2011rv}.\n\n\n\\section{Gravitational Quantum Well and the Equivalence Principle in NCQM}\n\nA very interesting system to directly connect gravity to quantum mechanics is the gravitational quantum well \\cite{Landau,LF,Nesvizhesky}. As we shall see, this connection can be used to constrain quantum measurements of gravity phenomena and to test the equivalence principle (see also Refs. \\cite{Bertolami:2003,Bastos:2010au}). It is easy to show that this principle holds for usual quantum mechanics, in the sense that a gravitational field is equivalent to an accelerated  reference frame. We shall see that this also holds in the context of NCQM for isotropic noncommutativity parameters. In the following we shall study the noncommutative GQW \\cite{Bertolami:2005jw} and its connection to accelerated frames of reference.\n\n\\subsection{Fock space formulation of NC Gravitational Quantum Well}\n\nLet us consider the GQW in the context of NCQM.\nTo start with we review some aspects of the usual GQW in standard quantum mechanics. The Hamiltonian is given by:\n\n\\begin{equation} \\label{QGWH}\n\\hat{H}=\\frac{1}{2m}\\hat{\\boldsymbol{p}}^2+mg\\hat{x}_i.\n\\end{equation}\nfor a particle with mass, $m$, in a gravitational field with acceleration, $g$, in the $x_i$ direction.\n\nWith the Fock space treatment in mind we define creation and annihilation operators for this Hamiltonian:\n\\begin{gather}\n\\hat{b}=\\left(\\frac{m^2}{\\hbar^2g}\\right)^{\\frac{1}{3}}\\left[\\hat{x}+\\frac{i}{2}\\left(\\frac{g^2\\hbar}{m^4}\\right)^{\\frac{1}{3}}\\hat{p}_x\\right], \\label{annihilation}\\\\\n\\hat{b}^\\dagger=\\left(\\frac{m^2}{\\hbar^2g}\\right)^{\\frac{1}{3}}\\left[\\hat{x}-\\frac{i}{2}\\left(\\frac{g^2\\hbar}{m^4}\\right)^{\\frac{1}{3}}\\hat{p}_x\\right], \\label{creation}\n\\end{gather}\nwhere the definition concerns only for the $x$ direction, as the $y$ component of the Hamiltonian is just that of a free particle. The normalization factors is chosen so that the operators $\\hat{b}$ and $\\hat{b}^\\dagger$ are dimensionless. The Hamiltonian can then be rewritten as \n\\begin{equation} \\label{comm hamiltonian}\n\\hat{H}=K_1\\left(\\hat{\\Gamma}_x+\\hat{\\Gamma}_y\\right)+K_2\\left(\\hat{b}^\\dagger_x+\\hat{b}_x\\right),\n\\end{equation}\nwhere\n\\begin{gather}\n\\hat{\\Gamma}_i=\\hat{b}^\\dagger_i\\hat{b}_i+\\hat{b}_i\\hat{b}^\\dagger_i-\\hat{b}^\\dagger_i\\hat{b}^\\dagger_i-\\hat{b}_i\\hat{b}_i, \\\\\nK_1=\\frac{1}{16}\\left(\\frac{\\hbar^3m^2}{g}\\right)^{2\/3}, \\\\\nK_2=\\frac{mg}{2}\\left(\\frac{\\hbar^2g}{m^2}\\right)^{1\/3}.\n\\end{gather}\n\nGiven the form of the Hamiltonian, it is evident that it is not diagonal in this representation, so it is not particularly useful for calculations of eigenstates and eigenvalues. This is expected from the usual solution to this problem, in which the energies involve the zeros of the Airy function, $Ai(x)$. We now examine the noncommutative Hamiltonian \\cite{Bertolami:2005jw},\n\n\\begin{equation} \\label{NCGQWH}\n\\hat{H}^{NC}=\\frac{1}{2m}\\left[\\hat{p}_x^2+\\hat{p}_y^2\\right]+mg\\hat{x}+\\frac{\\eta}{2m\\hbar}(\\hat{x}\\hat{p}_y-\\hat{y}\\hat{p}_x)+\\frac{\\eta^2}{8m\\hbar^2}\\left(\\hat{x}^2+\\hat{y}^2\\right);\n\\end{equation}\nwhich is the equation of a particle under the influence of a gravitational field plus a fictitious ``magnetic field\", $\\overrightarrow{B_{NC}}=-(\\eta\/q\\hbar)\\overrightarrow{\\mathrm{e}_z}$, plus an harmonic restoring force. Through the definitions, Eqs. (\\ref{annihilation}) and (\\ref{creation}), it can be rewriten it, up to first order in $\\theta$ and $\\eta$, as:\n\\begin{equation} \\label{noncomm hamiltonian}\n\\hat{H}^{NC}=K_1\\left(\\hat{\\Gamma}_x+\\hat{\\Gamma}_y\\right)+K_2\\left(\\hat{b}^\\dagger_x+\\hat{b}_x\\right)+\\frac{i\\eta}{4m\\hbar^{\\frac{2}{3}}}\\left(\\hat{b}^\\dagger_y\\hat{b}_x-\\hat{b}^\\dagger_x\\hat{b}_y\\right).\n\\end{equation}\n\nIt should be pointed out that this treatment considers only first order terms in either $\\eta$ or $\\theta$, although the latter does not show up in the Hamiltonian as its effect can be absorbed by a phase factor of the wave function. Noting the similarities between both commutative and noncommutative Hamiltonians, we might ask wether there is a transformation that can turn one into the other. That might be an interesting finding as, then, noncommutativity, at least for this system, could be regarded as a modification to the commutative case, and noncommutative eigenfunctions could be constructed using commutative ones, which are well known. Furthermore, it would make noncommutativity the result of a transformation of variables, and not a fundamental property of the system under study.  In order to pursue this analysis, we must introduce an operator transformation in which the new operators, $\\hat{a}_i$ and $\\hat{a}^\\dagger_i$ for $i=x,y$, obey the same commutation relations as the original operators. Thus we define,\n\n\\begin{gather} \\label{a operators}\n\\hat{b}_i:=\\sum_{j=1}^2u_{ij}\\hat{a}_j+s_{ij}\\hat{a}^\\dagger_j, \\\\\n\\hat{b}^\\dagger_i:=\\sum_{j=1}^2u^*_{ij}\\hat{a}^\\dagger_j+s^*_{ij}\\hat{a}_j,\n\\end{gather}\nwhere we impose the commutation relations\n\n\\begin{equation}\n\\left[\\hat{a}_i,\\hat{a}^\\dagger_j\\right]=\\delta_{ij},\n\\end{equation}\nand all the other commutation relations vanish. These conditions introduce a set of constraints on the parameters $u_{ij}$ and $s_{ij}$, namely:\n\n\\begin{gather}\n\\lvert u_{11}\\lvert^2-\\lvert s_{11} \\lvert^2+\\lvert u_{12}\\lvert^2-\\lvert s_{12}\\lvert^2=1, \\nonumber\\\\\n\\lvert u_{21}\\lvert^2-\\lvert s_{21} \\lvert^2+\\lvert u_{22}\\lvert^2-\\lvert s_{22}\\lvert^2=1.\n\\end{gather}\n\nConsidering Eq. (\\ref{comm hamiltonian}) in terms of operators $\\hat{b}_i$ and $\\hat{b}^\\dagger_i$ and using the definitions, Eq. (\\ref{a operators}), we get the Hamiltonian in terms of the operators $\\hat{a}_i$ and $\\hat{a}^\\dagger_i$ as \n\n\\begin{multline} \\label{new hamiltonian}\n\\hat{H}=K_1\\left[\\gamma_1\\hat{a}^\\dagger_x\\hat{a}_x+\\gamma_1\\hat{a}_x\\hat{a}^\\dagger_x+\\gamma_2\\hat{a}^\\dagger_x\\hat{a}^\\dagger_x+\\gamma^*_2\\hat{a}_x\\hat{a}_x+\\gamma_3\\hat{a}^\\dagger_y\\hat{a}_y+\\gamma_3\\hat{a}_y\\hat{a}^\\dagger_y+\\gamma_4\\hat{a}^\\dagger_y\\hat{a}^\\dagger_y+ \\right. \\\\\n\\left. +\\gamma^*_4\\hat{a}_y\\hat{a}_y+2\\gamma_5\\hat{a}^\\dagger_x\\hat{a}^\\dagger_y+2\\gamma^*_5\\hat{a}_x\\hat{a}_y+2\\gamma_6\\hat{a}^\\dagger_x\\hat{a}_y+2\\gamma^*_6\\hat{a}^\\dagger_y\\hat{a}_x\\right]+ \\\\\n+K_2\\left[\\hat{a}^\\dagger_x\\left(u^*_{11}+s_{11}\\right)+\\hat{a}_x\\left(s^*_{11}+u_{11}\\right)+\\hat{a}^\\dagger_y\\left(u^*_{12}+s{12}\\right)+\\hat{a}_y\\left(s^*_{12}+u_{12}\\right)\\right],\n\\end{multline}\nwhere, for simplicity, we have defined,\n\n\\begin{subequations}\n\\begin{equation}\n\\gamma_1:=\\lvert u_{11}\\lvert^2+\\lvert s_{11}\\lvert^2-u^*_{11}s^*_{11}-u_{11}s_{11}+\\lvert u_{21}\\lvert^2+\\lvert s_{21}\\lvert^2-u^*_{21}s^*_{21}-u_{21}s_{21},\n\\end{equation}\n\\begin{equation}\n\\gamma_2:=2u^*_{11}s_{11}-\\left(u^*_{11}\\right)^2-s_{11}^2+2u^*_{21}s_{21}-\\left(u^*_{21}\\right)^2-s_{21}^2,\n\\end{equation}\n\\begin{equation}\n\\gamma_3:=\\lvert u_{12}\\lvert^2+\\lvert s_{12}\\lvert^2-u^*_{12}s^*_{12}-u_{12}s_{12}+\\lvert u_{22}\\lvert^2+\\lvert s_{22}\\lvert^2-u^*_{22}s^*_{22}-u_{22}s_{22},\n\\end{equation}\n\\begin{equation}\n\\gamma_4:=2u^*_{12}s_{12}-\\left(u^*_{12}\\right)^2-s_{12}^2+2u^*_{22}s_{22}-\\left(u^*_{22}\\right)^2-s_{22}^2,\n\\end{equation}\n\\begin{equation}\n\\gamma_5:=u^*_{11}s_{12}+s_{11}u^*_{12}-u^*_{11}u^*_{12}-s^*_{11}s^*_{12}+u^*_{21}s_{22}+s_{21}u^*_{22}-u^*_{21}u^*_{22}-s^*_{21}s^*_{22},\n\\end{equation}\n\\begin{equation}\n\\gamma_6:=u^*_{11}u_{12}+s_{11}s^*_{12}-u^*_{11}s^*_{12}-s^*_{11}u^*_{12}+u^*_{21}u_{22}+s_{21}s^*_{22}-u^*_{21}s^*_{22}-s^*_{21}u^*_{22}.\n\\end{equation}\n\\end{subequations}\n\nComparing the Hamiltonian in Eq. (\\ref{new hamiltonian}) to the one in Eq. (\\ref{noncomm hamiltonian}), we can immediately set the conditions for the $\\gamma_i$'s\n\n\\begin{subequations}\n\\begin{equation}\n\\gamma_1=1,\n\\end{equation}\n\\begin{equation}\n\\gamma_2=-1,\n\\end{equation}\n\\begin{equation}\n\\gamma_3=1,\n\\end{equation}\n\\begin{equation}\n\\gamma_4=-1,\n\\end{equation}\n\\begin{equation}\n\\gamma_5=0,\n\\end{equation}\n\\begin{equation}\n\\gamma_6=\\mathrm{i}\\frac{\\eta}{4m\\hbar^{\\frac{2}{3}} K_1}:=i\\eta\\mathrm{c} , \\mathrm{c}\\in \\mathbb{R}.\n\\end{equation}\n\\end{subequations}\n\n\nFurthermore, comparing the terms that are linear in the $\\hat{a}$ operators, we get two additional equations for the $u$ and $s$ parameters,\n\n\\begin{subequations}\n\\begin{equation}\nu^*_{11}+s_{11}=1,\n\\end{equation}\n\\begin{equation}\nu^*_{12}+s_{12}=0.\n\\end{equation}\n\\end{subequations}\n\nIn total we now have 16 variables and a total of 16 distinct equations constraining the values of this variables. Hence, this system of equations has either a single solution or none. It is found that this system has no solution for $\\eta\\neq 0$, which can be verified using well known Mathematica or MatLab procedures. Therefore, it is not possible to describe, as expected, the noncommutative Hamiltonian as a mixture of eigenstates of the commutative Hamiltonian, and so it is a completely different problem. Once again we stress that this result is only valid at first order in both noncommutative parameters. However, it is reassuring to confirm that, at least at this level, noncommutativity is indeed a completely different problem than the commutative one.\n\n\\subsection{Equivalence Principle}\n\nHaving verified that the noncommutative Hamiltonian of the GQW is in fact a different problem than the commutative one, we can try to examine the issue of the noncommutative Equivalence Principle. We have seen that the only parameter having an effect on the eigenstates and eigenvalues is $\\eta$, as the $\\theta$ factor can be absorbed into a phase factor in the wave function of the system. The WEP states that, locally, any gravitational field is equivalent to an accelerated reference frame. This is one of the basic tenets of General Relativity and holds with great accuracy (see e.g. Ref. \\cite{Bertolami:2012}, chapter 22, for a review of the experimental status of relativity). In standard QM, for the GQW, this can be verified to hold in a quite simple way. In the context of NCQM we will show how it can be verified in what follows next. For this purpose we consider the noncommutative GQW Schr$\\ddot{\\mathrm{o}}$dinger equation,\n\n\\begin{equation}\n\\hat{H}^{NC}_g\\Psi=\\left[\\frac{1}{2m}\\left(\\hat{\\pi}_x^2+\\hat{\\pi}_y^2\\right)+mg\\hat{Q}_x\\right]\\Psi=E\\Psi\n\\end{equation}\nand applying the Darboux transformation to write it in terms of the commutative variables, that is, Eq. (\\ref{NCGQWH}):\n\n\\begin{equation} \\label{schrodinger of NC GQW}\n\\left[\\frac{1}{2m}\\left(\\hat{p}_x^2+\\hat{p}_y^2\\right)+mg\\hat{x}+\\frac{\\eta}{2m\\hbar}(\\hat{x}\\hat{p}_y-\\hat{y}\\hat{p}_x)+\\frac{\\eta^2}{8m\\hbar^2}\\left(\\hat{x}^2+\\hat{y}^2\\right)\\right]\\Psi=\\mathrm{i}\\hbar\\frac{\\partial\\Psi}{\\partial t},\n\\end{equation}\nwhere we have considered the time dependent problem as we have to use a change of coordinates evolving in time. We now consider the noncommutative free particle equation:\n\n\\begin{equation} \\label{free hamiltonian}\n\\left[-\\frac{\\hbar^2}{2m}\\left(\\frac{\\partial^2}{\\partial x^2}+\\frac{\\partial^2}{\\partial y^2}\\right)-\\frac{\\mathrm{i}\\eta}{2m}(x\\frac{\\partial}{\\partial y}-y\\frac{\\partial}{\\partial x})+\\frac{\\eta^2}{8m\\hbar^2}\\left(x^2+y^2\\right)\\right]\\Psi=\\mathrm{i}\\hbar\\frac{\\partial\\Psi}{\\partial t},\n\\end{equation}\nand introduce a change of coordinates defined as\n\n\\begin{subequations} \\label{acc coordinates}\n\\begin{equation}\nx'=x+\\sigma(t)\n\\end{equation}\n\\begin{equation}\ny'=y\n\\end{equation}\n\\end{subequations}\n\nIn order for the WEP to be preserved we require that\n\n\\begin{equation} \\label{equality}\n\\hat{H}^{NC}_g(\\hat{\\boldsymbol{x}},\\hat{\\boldsymbol{p}})\\Psi(x,y)=\\hat{H}^{NC}_{free}(\\hat{\\boldsymbol{x'}},\\hat{\\boldsymbol{p'}})\\Psi'(x',y'),\n\\end{equation}\nwhere $\\hat{H}^{NC}_g$ is the noncommutative GQW Hamiltonian and $\\hat{H}^{NC}_{free}$ is the noncommutative Hamiltonian of a free particle and $\\Psi'(x',y')=\\mathrm{e}^{\\mathrm{i}\\phi(x',y')}\\Psi(x',y')$, so that the eigenfunctions are the same, but by a phase. Starting from the free particle Hamiltonian we write it in terms of an accelerated reference frame coordinates, and thus,\n\\begin{subequations} \\label{acc differentials}\n\\begin{equation}\n\\frac{\\partial}{\\partial x'}\\Psi(x',y')=\\frac{\\partial}{\\partial x}\\Psi(x,y),\n\\end{equation}\n\\begin{equation}\n\\frac{\\partial}{\\partial y'}\\Psi(x',y')=\\frac{\\partial}{\\partial y}\\Psi(x,y),\n\\end{equation}\n\\begin{equation}\n\\frac{\\partial}{\\partial t'}\\Psi(x',y')=\\left(\\frac{\\partial}{\\partial t}-\\frac{\\mathrm{d}\\sigma(t)}{\\mathrm{d}t}\\frac{\\partial}{\\partial x}\\right)\\Psi(x,y).\n\\end{equation}\n\\end{subequations}\n\nHence, combining Eqs. (\\ref{acc coordinates}) and (\\ref{acc differentials}), the right-hand side of Eq. (\\ref{free hamiltonian}) becomes:\n\\begin{multline}\n\\left[-\\frac{\\hbar^2}{2m}\\left(\\frac{\\partial^2}{\\partial x^2}+\\frac{\\partial^2}{\\partial y^2}\\right)-\\frac{\\mathrm{i}\\eta}{2m}\\left(x\\frac{\\partial}{\\partial y}-y\\frac{\\partial}{\\partial x}\\right)-\\frac{\\mathrm{i}\\eta}{2m}\\sigma(t)\\frac{\\partial}{\\partial y}+\\frac{\\eta^2}{8m\\hbar^2}\\left(x^2+y^2\\right)+ \\right. \\\\\n\\left. \\frac{\\eta^2}{8m\\hbar^2}\\left(-2x\\sigma(t)+\\sigma^2(t)\\right)\\right]\\Psi'(x,y)=\\mathrm{i}\\hbar\\left(\\frac{\\partial}{\\partial t}-\\frac{\\mathrm{d}\\sigma(t)}{\\mathrm{d}t}\\frac{\\partial}{\\partial x}\\right)\\Psi'(x,y).\n\\end{multline}\n\nIn order to check if Eq. (\\ref{equality}) is consistent we must either compute the phase $\\phi$ or prove there is no wave function which holds for the mentioned relation. For this we consider the relation between $\\Psi$ and $\\Psi'$ and compute the action of the operators on the wave function $\\Psi'(x',y')=\\mathrm{e}^{\\mathrm{i}\\phi(x',y')}\\Psi(x',y')$. The obtained result is as follows:\n\\begin{multline} \\label{full equation}\n\\left[-\\frac{\\hbar^2}{2m}\\left(\\frac{\\partial^2}{\\partial x^2}+\\frac{\\partial^2}{\\partial y^2}\\right)-\\frac{\\mathrm{i}\\eta}{2m}\\left(x\\frac{\\partial}{\\partial y}-y\\frac{\\partial}{\\partial x}\\right)+\\frac{\\eta^2}{8m\\hbar^2}\\left(x^2+y^2\\right)\\right]\\Psi'+\\left[-\\frac{\\mathrm{i}\\hbar^2}{2m}\\frac{\\partial^2\\phi}{\\partial x^2}+\\frac{\\hbar^2}{2m}\\frac{\\partial\\phi}{\\partial x}^2 \\right. \\\\\n\\left. -\\frac{\\mathrm{i}\\hbar^2}{2m}\\frac{\\partial^2\\phi}{\\partial y^2}+\\frac{\\hbar^2}{2m}\\frac{\\partial\\phi}{\\partial y}^2+\\frac{\\eta}{2m}y\\frac{\\partial\\phi}{\\partial x}-\\frac{\\eta}{2m}x\\frac{\\partial\\phi}{\\partial y}+\\frac{\\eta}{2m}\\sigma(t)\\frac{\\partial\\phi}{\\partial y}-\\frac{\\eta^2}{4m\\hbar^2}x\\sigma(t)+\\frac{\\eta^2}{4\\hbar^2}\\sigma^2(t)+ \\right. \\\\\n\\left. \\hbar\\frac{\\partial\\phi}{\\partial t}+\\hbar\\frac{\\mathrm{d}\\sigma}{\\mathrm{d}t}\\frac{\\partial\\sigma}{\\partial x}\\right]\\Psi'+\\left[-\\frac{\\mathrm{i}\\hbar^2}{2m}\\frac{\\partial\\phi}{\\partial x}+\\mathrm{i}\\hbar\\frac{\\mathrm{d}\\sigma}{\\mathrm{d}t}\\right]\\frac{\\partial\\Psi'}{\\partial x}+\\left[-\\frac{\\mathrm{i}\\hbar^2}{m}\\frac{\\partial\\phi}{\\partial y}-\\frac{\\mathrm{i}\\eta}{2m}\\sigma(t)\\right]\\frac{\\partial\\Psi'}{\\partial t}=\\mathrm{i}\\hbar\\frac{\\partial\\Psi'}{\\partial t}.\n\\end{multline}\n\nNow, for the purpose of retrieving the noncommutative GQW we must compare both Schr$\\ddot{\\mathrm{o}}$dinger equations to set constraints on the form of the phase $\\phi$. Imposing that the term multiplying the derivative of $\\Psi'$ vanishes, we get:\n\\begin{equation}\n\\frac{\\partial\\phi}{\\partial x}=\\frac{m}{\\hbar}\\frac{\\mathrm{d}\\sigma}{\\mathrm{d}t},\n\\end{equation}\nwhich implies, taking into account the fact that $\\sigma$ only depends on time, that:\n\\begin{equation} \\label{first form}\n\\phi=\\frac{m}{\\hbar}\\frac{\\mathrm{d}\\sigma}{\\mathrm{d}t}x+f(y,t).\n\\end{equation}\n\nConsidering that the last term on the left-hand side of Eq. (\\ref{full equation}) must vanish, and Eq. (\\ref{first form}), it follows that\n\\begin{equation}\n\\frac{\\hbar^2}{m}\\frac{\\partial f}{\\partial y}=-\\frac{\\eta}{2m}\\sigma(t)\\space\\Rightarrow\\space f(y,t)=-\\frac{\\eta}{2\\hbar^2}\\sigma(t)y+\\mu(t);\n\\end{equation}\nreplacing this result into the second term of Eq. (\\ref{full equation}) and comparing with the Hamiltonian, Eq. (\\ref{schrodinger of NC GQW}), yields\n\\begin{equation}\nm\\frac{\\mathrm{d}^2\\sigma}{\\mathrm{d}t^2}x+\\nu(t)=mgx\n\\end{equation}\nwhere $\\nu(t)$ is the sum of all time dependent terms and can be made to vanish through a suitable choice of the function $\\mu(t)$. There is only one non-vanishing remaining term and in order to Eq. (\\ref{equality}) to hold we must impose that\n\\begin{equation}\n\\frac{\\mathrm{d}^2\\sigma}{\\mathrm{d}t^2}=g\\space\\Rightarrow\\space\\sigma(t)=\\sigma_0+vt+\\frac{1}{2}gt^2\n\\end{equation}\n\nThus, we can see that Eq. (\\ref{equality}) holds as far as\n\\begin{equation}\nx'=x+\\sigma_0+vt+\\frac{1}{2}gt^2\n\\end{equation}\nwhich corresponds to an accelerated reference frame. The WEP is then verified to hold for NCQM at least as long as we consider that the noncommutative parameters are isotropic. Hence, bounds on the WEP turn out to be limits on the isotropy of the NC parameters.\n\nFinally, the phase difference between the wavefunctions $\\Phi$ and $\\Phi'$ is given by:\n\\begin{equation}\n\\Psi=\\mathrm{e}^{\\mathrm{i}\\left(\\frac{m}{\\hbar}\\frac{\\mathrm{d}\\sigma}{\\mathrm{d}t}x-\\frac{\\eta}{2\\hbar^2}\\sigma(t)y+\\mu(t)\\right)}\\Psi'\n\\end{equation}\nand, as it has been analysed in Ref. \\cite{Bastos:2008b}, this does not give rise to any physically meaningful effect.\n\n\\subsection{Anisotropic noncommutativity}\n\nAs we have seen in the last subsection, the WEP holds in NCQM, unless NC parameters are anisotropic, i.e. $\\eta_{xy}\\neq\\eta_{xz}$. In what follows we use the bounds on the WEP to constrain the difference between components of the $\\eta$ matrix. The ensued discussion is similar to the one carried out in Ref. \\cite{Bastos:2010au} in the context of the entropic gravity proposal \\cite{Verlinde:2010hp}.\nThe noncommutative Hamiltonian for the GQW is given by Eq. (\\ref{NCGQWH}).\nIn order to find the eigenstates for this problem we use perturbation theory up to first order in $\\eta$, which is sufficient to obtain differences in the energy spectrum for different directions of the gravitational field. For this purpose we define\n\\begin{equation}\n\\hat{H}^{NC}=\\hat{H}_0^{NC}+\\hat{V},\n\\end{equation}\nwhere we consider $\\hat{V}$ a perturbation to the exactly soluble Hamiltonian $\\hat{H}_0^{NC}$, defined by\n\n\\begin{subequations}\n\\begin{equation}\n\\hat{H}_0^{NC}:=\\frac{\\hat{p}_x}{2m}+\\frac{\\hat{p}_y}{2m}+mg\\hat{x},\n\\end{equation}\n\\begin{equation} \\label{perturbation}\n\\hat{V}:=\\frac{\\eta}{2m\\hbar}\\left(\\hat{y}\\hat{p}_x-\\hat{x}\\hat{p}_y\\right)+\\frac{\\eta^2}{8m\\hbar^2}\\left(\\hat{x}^2+\\hat{y}^2\\right).\n\\end{equation}\n\\end{subequations}\n\nSince we are only interested in the corrections of order $\\eta$, we can disregard the second term in $\\hat{V}$. The soluble Hamiltonian is that of a free particle in the $y$ direction and that of the GQW in the $x$ direction. Solutions to these problems are well-known and are given by (e.g. Ref. \\cite{Landau})\n\n\\begin{equation} \\label{solution}\n\\Psi_{nk}(x,y)=A_nAi\\left(\\left(\\frac{2m^2g}{\\hbar^2}\\right)^{1\/3}\\left(x-\\frac{E_{n}}{mg}\\right)\\right)\\chi(y),\n\\end{equation}\nwhere $Ai(z)$ is the Airy function, $\\chi(y)$ is the solution for the free particle, and $E_{n}$ and $A_n$ are the energy eigenvalues in the $x$ direction and the normalization factor for the Airy function, given, respectively, by,\n\n\\begin{equation}\nE_{n}=-\\left(\\frac{mg^2\\hbar^2}{2}\\right)^{1\/3}\\alpha_n,\n\\end{equation}\n\n\\begin{equation}\nA_n=\\left[\\left(\\frac{\\hbar^2}{2m^2g}\\right)^{1\/3}\\int_{\\alpha_n}^{+\\infty}\\mathrm{d}zAi^2(z)\\right]^{-1\/2},\n\\end{equation}\nwhere $\\alpha_n$ are the zeros of the Airy function. The energy eigenvalues in the $y$ direction are given by,\n\\begin{equation}\nE_{y}=\\frac{\\hbar^2k^2}{2m},\n\\end{equation}\nwhere $k$ is the momentum of the particle. The change in energy is given by the expectation value of the operator $\\hat{V}$ in a general state given by Eq. (\\ref{solution}) and, the leading order perturbation to the energy of the system in any state, is given by,\n\n\\begin{equation}\n\\Delta E_n=\\bra{\\Psi_{nk}}\\hat{V}\\ket{\\Psi_{nk}}=\\frac{\\eta k}{2m}\\left[\\left(\\frac{2m^2g}{\\hbar^2}\\right)^{-2\/3}\\mathrm{I}_1^{(n)}+\\frac{E_n}{mg}\\right].\n\\end{equation}\nIt must be noted that we computed the energy eigenvalues for the case of a two dimensional Hamiltonian in the $xy$ plane, so we can write,\n\\begin{equation}\nE_{nk}^{xy}=-\\left(\\frac{mg^2\\hbar^2}{2}\\right)^{1\/3}\\alpha_n+\\frac{\\hbar^2k^2}{2m}+\\frac{\\eta_{xy} k}{2m}\\left[\\left(\\frac{2m^2g}{\\hbar^2}\\right)^{-2\/3}\\mathrm{I}_1^{(n)}+\\frac{E_n}{mg}\\right].\n\\end{equation}\nThus an anisotropy in the momentum space breaks the equivalence principle. \n\nConsider now the NC GQW for a particle moving along the $y$ direction with a gravitational field in the $x$ direction and the same equation for a particle traveling along the $x$ direction with a gravitational field in the $z$ direction. Assuming that the test particles have the same momentum in the direction in which they are free, hence:\n\n\\begin{equation} \\label{deltag}\nmx(g_x-g_z)=\\frac{k}{2m}\\left[\\left(\\frac{2m^2g}{\\hbar^2}\\right)^{2\/3}\\mathrm{I}_1^{(n)}+\\frac{E_n}{mg}\\right]\\left(\\eta_{xy}-\\eta_{yz}\\right),\n\\end{equation}\nwhere $x$ is the position of the test particle. Thus, using the bound on the WEP for two different directions (see e.g. Ref. \\cite{PhysRevLett.100.041101}):\n\n\\begin{equation} \\label{torsion balance}\n\\frac{\\Delta a}{a}:=\\frac{|a_1-a_2|}{a}\\lesssim 10^{-13},\n\\end{equation}\nplus data from Ref. \\cite{Nesvizhesky} , namely that $k=1.03\\times 10^8\\>\\>m^{-1}$ and $x=12.2\\>\\>\\mu m$ for the eigenstate of lower energy and $g=9.80665\\>\\>m\/s^2$, Eq. (\\ref{deltag}) yields:\n\n\\begin{equation} \\label{relation}\n\\frac{\\Delta g}{g}=1.4\\times 10^{60}\\Delta\\eta.\n\\end{equation}\n\nApplying the bound from Eq. (\\ref{torsion balance}) to Eq. (\\ref{relation}), the bound for $\\Delta\\eta$ is computed to be:\n\n\\begin{equation}\n\\Delta\\eta\\lesssim 10^{-73} \\>\\mathrm{kg}^2\\mathrm{m}^2\\mathrm{s}^{-2},\n\\end{equation}\nwhich bounds the noncommutative momentum anisotropy in a quite stringent way. In natural units:\n\n\\begin{equation}\n\\sqrt{\\Delta\\eta}\\lesssim10^{-10}\\>\\>\\mathrm{eV}.\n\\end{equation}\n\n\n\\section{Lorentz invariance}\nLorentz symmetry is a fundamental cornerstone of all known physical theories. Thus, it is natural to consider experimental bounds on this invariance to constrain noncommutativity which explicitly violates Lorentz symmetry . A major tool for these tests is the relativistic dispersion relation,\n\\begin{equation} \\label{dispersion relation}\nE^2=p^2c^2+m^2c^4.\n\\end{equation} \n\nThis relation is tested with great accuracy at very high energies. Indeed, ultra-high energy cosmic rays allow for constraining this relationship for an extra quadratic term on the energy to the $1.7\\times 10^{-25}$ level \\cite{Bertolami:1999da}. This estimate is confirmed through direct measurements by the Auger collaboration \\cite{Auger}. \n\nThus, assuming a correction of the dispersion relation proportional to $E^2$ at the $1.7\\times 10^{-25}$ level \\cite{Bertolami:1999da}, then it is possible to constrain the $\\eta$ parameter, that is:  \n\\begin{equation}\n\\eta \\leqslant (1.7\\times 10^{-25}) E^2,\n\\end{equation}\nhence for ultra-high energy cosmic rays, with $E\\sim 10^{20} \\> \\mathrm{eV}$, we can establish that $\\sqrt{\\eta}\\leqslant 4.1\\times 10^{7}\\>\\mathrm{eV}$, which is not at all a very stringent upper bound. A much more constraining bound can be set through low-energy tests of Lorentz symmetry. Indeed, assuming limits arising from the nuclear Zeeman levels, one can establish that $\\eta \\leqslant  10^{-22}E^2$, which for $E \\sim \\mathrm{MeV}$ \\cite{PhysRevLett.57.3125}, implies that $\\sqrt{\\eta}\\leqslant 10^{-11}\\>\\mathrm{MeV}\\simeq 10^{-5}\\>\\>\\mathrm{eV}$. This result is competitive with the most stringent bound on $\\eta$, namely $\\sqrt{\\eta}\\leqslant 2 \\times 10^{-5}\\>\\>\\mathrm{eV}$ \\cite{Bertolami:2011rv}, obtained from the hydrogen hyperfine transition, the most accurate experimental result ever obtained.  \n\n\n\\section{Discussion and Conclusions}\n\nIn this work we have addressed several issues on NCQM. Gauge invariance of the electromagnetic field is verified to hold only if the parameter $\\theta$ vanishes, which is consistent with previous work for fermionic fields \\cite{Bertolami:2011rv}. This result implies that, for abelian gauge theories, spatial directions do commute and noncommutative effects are expected only for the momenta.\n\n\nAlso, we have compared the GQW Hamiltonian in the context of NCQM with the Hamiltonian for the same problem in QM. Using the Fock space formalism with creation and annihilation operators, we found no evidence for a connection between this two problems at first order in the parameter $\\eta$. This shows that NCQM poses a different problem from QM at least in the context of GQW.\nFollowing this result, we studied the WEP in the noncommutative scenario. It is concluded that this principle holds for NCQM in the sense that an accelerated frame of reference is locally equivalent to a gravitational field, as long as noncommutativity is isotropic. If an anisotropy is introduced in the noncommutative parameters, using data from Refs. \\cite{PhysRevLett.100.041101,Nesvizhesky}, we set a bound on the anisotropy of the $\\eta$ parameter, $\\sqrt{\\Delta\\eta}\\lesssim10^{-10}\\>\\>\\mathrm{eV}$. It is then clear that the anisotropy of the noncommutative momentum parameter is many orders of magnitude smaller than the NC parameter itself. This result also states that the existence of a preferential observer to whom the spatial $x$,$y$ and $z$ directions are well defined is limited to the same degree as the anisotropy factor.\n\nAdditionally, the breaking of Lorentz symmetry is examined in the context of NCQM. Assuming a violation of the relativistic dispersion relation proportional to $E^2$, bounds from ultra-high energy cosmic rays (see Refs. \\cite{Bertolami:1999da,Auger}) imply that $\\sqrt{\\eta}\\leq 4.1\\times 10^7\\>\\>\\mathrm{eV}$. Considering instead bounds arising from nuclear Zeeman levels, one can obtain that $\\sqrt{\\eta}\\leq 10^{-5}\\>\\>\\mathrm{eV}$, which is competitive with bounds arising from the hydrogen hyperfine transition $\\sqrt{\\eta}\\leq 2\\times 10^{-6}\\>\\>\\mathrm{eV}$ \\cite{Bertolami:2011rv}, the most stringent bound ever obtained.  \n\n\\vspace{0.5cm} \n\n\\noindent\n{ \\bf Acknowledgements}\n\n\\noindent\nThe authors would like to thank Catarina Bastos for relevant discussions on the matter of this work.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe theory of optimal transport has drawn much attention in recent years.\nIts applications to geometry and PDEs have in particular been largely \ndisseminated. In this paper, we would like to show its effectiveness in a \ndynamical context. We are interested in arguably the simplest dynamical\nsystem where the action on measures is significantly different from the action \non points, namely expanding circle maps.\n\nAnother goal of the paper is to examplify the rigorous differential structure\ndefined by N. Gigli \\cite{Gigli}, for the simplest possible compact manifold.\nNote that one can use absolutely continuous curves to define the almost \neverywhere differentiability of maps, see in particular\n\\cite{Gigli2} where this method is applied to the exponential map.\nOther previous uses of variants of this manifold structure\ninclude the definition of gradient flows, as in the pioneering \\cite{Otto} and in \n\\cite{Ambrosio-Gigli-Savare}, and of curvature, as in \\cite{Lott}.\nBut up to our knowledge, no example of explicit derivative of a measure-defined\nmap at a given point had been computed.\n\n\\subsection{An important model example}\n\nLet us first consider the usual degree $d$\nself-covering map of the circle $\\mathbb{S}^1 = \\mathbb{R}\/\\mathbb{Z}$ defined by\n\\[\\Phi_d(x) = dx \\mod 1.\\]\nIt acts on the set $\\mathscr{P}(\\mathbb{S}^1)$ of Borel probability measures,\nendowed with the topology of weak convergence, by the push-forward map\n$\\Phi_{d\\#}$. \n\nA map like $\\Phi_d$ can act by composition on the right on a\nfunction space (e.g. Sobolev spaces). \nThe adjoint of this map is usually called\na Perron-Frobenius operator or a transfer operator, and a great \ndeal of effort has been made to understand these\noperators, especially their spectral properties (see for example \\cite{Baladi}). \nOne can consider\n$\\Phi_{d\\#}$ as an analogue for possibly singular measures of the Perron-Frobenius\noperator of $\\Phi_d$.\n\nAs pointed out by the referee of a previous version of this paper, using the\nfinite-to-one maps \n\\[(x_1,\\ldots,x_n)\\mapsto \\frac1n\\delta_{x_1}+\\dots+\\frac1n\\delta_{x_n}\\]\nit is easy to prove that $\\Phi_{d\\#}$ is topologically transitive and has infinite\ntopological entropy. To refine this last remark, we shall prove that $\\Phi_{d\\#}$ \nhas positive metric mean dimension (a metric dynamical invariant of infinite-entropy\nmaps).\n\\begin{theo}\\label{theo:mdim}\nFor all integer $d\\geqslant 2$ and all exponent $p\\in[1,+\\infty)$ we have \n\\[\\mathop \\mathrm{mdim}_M\\nolimits(\\Phi_{d\\#},\\mathop \\mathrm{W}\\nolimits_p)\\geqslant p(d-1)\\]\nwhere $\\mathop \\mathrm{W}\\nolimits_p$ is the Wasserstein metric with cost $|\\cdot|^p$.\n\\end{theo}\nThe definition of Wasserstein metrics is given below; for the definiton\nof metric mean dimension and the proof of the above result, see \nSection \\ref{sec:entropy}.\nExcept in this result, we shall only use the quadratic Wasserstein metric.\n\nOur main goal is to study the first-order dynamics of $\\Phi_{d\\#}$ near the uniform\nmeasure $\\lambda$. The precise setting will be exposed latter; let us just give\na few elements. The tangent space $T_\\mu$ to $\\mathscr{P}(\\mathbb{S}^1)$ at a measure\n$\\mu$ that is absolutely continuous with continuous density identifies\nwith the Hilbert space $L^2_0(\\mu)$ of all vector fields \n$v:\\mathbb{S}^1\\to\\mathbb{R}$ that are $L^2$ with respect to $\\mu$,\nand such that $\\int v \\,\\lambda =0$. More generally,\nif $\\mu$ is atomless $T_\\mu$ identifies with a Hilbert subspace\n$L^2_0(\\mu)$ of $L^2(\\mu)$.\n\nWe have a kind of exponential map: $\\exp_\\mu(v)=\\mu+v:=(\\mathrm{Id}+v)_\\#\\mu$. \nThen we say that a map $f$ acting on $\\mathscr{P}(\\mathbb{S}^1)$\nhas G\\^ateau derivative $L$ at $\\mu$ if $f(\\mu)$ has no atom\nand $L:L^2_0(\\mu)\\to L^2_0(f(\\mu))$ is a continous linear operator\nsuch that for all $v$ we have\n\\[\\mathop \\mathrm{W}\\nolimits(f(\\mu+tv),f(\\mu)+tLv)=o(t).\\]\n\nOur first differentiability result is the following.\n\\begin{theo}\\label{theo:diff}\nThe map $\\Phi_{d\\#}$ has a G\\^ateaux derivative at $\\lambda$,\nequal to $d$ times\nthe Perron-Frobenius operator of $\\Phi_d$ acting on $L^2_0(\\lambda)$.\nIn particular its spectrum is the disc of radius $d$ and all numbers of modulus $<d$ are\neigenvalues with infinite multiplicity.\n\\end{theo}\nThis result is detailled as Theorem \\ref{theo:differential} and Proposition \\ref{prop:spec}\nbelow. We shall also see that $\\Phi_{d\\#}$ is not Fr\\'echet differentiable.\n\n\\subsection{General expanding maps}\n\nThe next step is to consider the action on measures of expanding\ncircle maps. In Section \\ref{sec:general}, given a general $C^2$\nexpanding map $\\Phi$, we compute the derivative of $\\Phi_\\#$ at \nits unique absolutely continuous invariant measure (Theorem \\ref{theo:expanding}). \nInstead of writting down the expression here, let us simply state the following.\n\\begin{theo}\\label{theo:generaldiff}\nIf $\\Phi$ is a $C^2$ expanding circle map, $\\Phi_\\#$ has a G\\^ateaux derivative at its\nunique invariant absolutely continuous measure $\\rho\\lambda$, whose adjoint operator \nin  $L^2_0(\\rho\\lambda)$ is $u\\mapsto \\Phi'\\,u\\circ\\Phi$.\n\\end{theo}\nIn particular this derivative is a multiple of the Perron-Fronenius\noperator (on $L^2_0(\\rho\\lambda)$) only when $\\Phi'$ is constant, \nthat is when $\\Phi$ is a model map. Using general results\nin the spectral theory of transfert operator, it is however possible to\nprove that $1$ is always an eigenvalue of infinite multiplicity,\nwith continuous eigenfunctions.\n\n\\subsection{Nearly invariant measures}\n\n\nThe spectral\nstudy of $D_\\lambda(\\Phi_\\#)$ gives us large families of nearly invariant\nmeasures, with Lipschitz para\\-metrization.\n\\begin{theo}\\label{theo:almost-invariant}\nFor all integer $n$, there is a bi-Lipschitz embedding \n$F :  B^n \\to \\mathscr{P}(\\mathbb{S}^1)$ mapping $0$ to the absolutely continuous\ninvariant measure $\\rho\\lambda$ of $\\Phi$ such that for all $a\\in B^n$,\n\\[\\mathop \\mathrm{W}\\nolimits\\big(\\Phi_\\#(F(a)),F(a)\\big) =o(|a|).\\]\nAs a consequence, for all $\\varepsilon>0$ and all integer $K$\nthere is a radius $r>0$ such that for all $k\\leqslant K$ and\nall $a\\in B^n(0,r)$ the following holds:\n\\[\\mathop \\mathrm{W}\\nolimits\\big(\\Phi_{d\\#}^k(F(a)),F(a)\\big)\\leqslant \\varepsilon |a|.\\]\n\\end{theo}\nHere $B^n$ denotes the unit Euclidean ball centered at $0$ and $\\mathop \\mathrm{W}\\nolimits$ is the quadratic\nWasserstein distance (whose definition is recalled below).\n\nIt is easy to construct invariant measures near the \nabsolutely continuous one, for example supported on a union of periodic orbits.\nOne can also\nconsider convex sums $(1-a)\\rho\\lambda+a\\mu$ where $\\mu$ is any invariant measure\nand $a\\ll 1$. But note that the curves $a\\mapsto (1-a)\\rho\\lambda+a\\mu$ need not\nbe rectifiable, let alone Lipschitz. Bernoulli measures are also examples;\nthey are singular, atomless, fully supported invariant measures of $\\Phi_d$\nthat can be arbitrary close to $\\lambda$. \n\nThe nearly invariant measures above seem of a different\nnature, and a natural question is how regular they are.\nThey are given by push-forwards of the uniform measure by continuous \nfunctions; \nfor example in the model case a one parameter family is given by\n\\[\\big(\\mathrm{Id}+t\\sum_{\\ell=0}^\\infty d^{-\\ell}\\cos(2\\pi d^\\ell \\cdot)\\big)_\\#\\lambda   \\]\nwhere $t\\in[0,\\varepsilon)$. This makes it easy to prove that almost all\nof them are atomless.\n\\begin{prop}\\label{prop:atomless}\nIf $\\mu$ is an atomless measure and $v\\in L^2(\\mu)$,\n for all but a countable number of values of $t\\in[0,1]$, the\nmeasure $\\mu+tv=(\\mathrm{Id}+tv)_{\\#}\\mu$ has no atom.\n\nIn particular, with the notation of Theorem \\ref{theo:almost-invariant},\nfor almost all $a$ the measure $F(a)$ has no atom.\n\\end{prop}\n\nThis leaves open the following, antagonist questions.\n\\begin{ques}\nIs the measure $F(a)$ absolutely continuous for most, or at least some $a\\neq 0$?\n\\end{ques}\n\n\\begin{ques}\nIs the measure $F(a)$ invariant for most, or at least some $a\\neq 0$?\n\\end{ques}\n\nThe next natural questions, not adressed at all here, concerns the \ndynamical properties of the action\non measures of higher dimensional hyperbolic dynamical systems like Anosov\nmaps or flows, or of discontinuous systems like interval exchange maps.\n\n\n\\subsection{Recalls and notations}\n\nThe most convenient point of view here is to construct the circle as\nthe quotient $\\mathbb{R}\/\\mathbb{Z}$. We shall often and without notice write\na real number $x\\in[0,1)$ to mean its image by the canonical projection. We proceed\nsimilarly for intervals of length less than $1$.\n\nRecall that the push-forward of a measure is defined by \n$\\Phi_\\#\\mu(A)=\\mu(\\Phi^{-1}A)$ for all Borelian set $A$.\n\nFor a detailled introduction on optimal transport, the interested reader can for\nexample consult \\cite{Villani}. Let us give an overview of the properties we shall need.\nGiven an exponent $p\\in[1,\\infty)$, if $(X,d)$ is a general metric space, assumed to be polish (complete \nseparable) to avoid mesurability issues and endowed with its Borel \n$\\sigma$-algebra, its $L^p$ \\emph{Wasserstein space}  is\nthe set $\\mathscr{W}_p(X)$ of probability measures $\\mu$ on $X$ whose $p$-th moment is finite:\n\\[\\int d^p(x_0,x) \\,\\mu(dx)<\\infty\\qquad\\mbox{ for some, hence all }x_0\\in X\\]\nendowed with the following metric: given $\\mu,\\nu\\in\\mathscr{W}_p(X)$ one sets\n\\[\\mathop \\mathrm{W}\\nolimits_p(\\mu,\\nu)=\\left(\\inf_\\Pi \\int_{X\\times X} d^p(x,y)\\, \n  \\Pi(dx dy)\\right)^{1\/p}\\]\nwhere the infimum is over all probability measures $\\Pi$ on $X\\times X$\nthat projects to $\\mu$ on the first factor and to $\\nu$ on the second one.\nSuch a measure is called a transport plan between $\\mu$ and $\\nu$, and is\nsaid to be optimal when it achieves the infimum. In this setting, an optimal\ntransport plan always exist. Note that when $X$ is compact, the set $\\mathscr{W}_p(X)$\nis equal to the set $\\mathscr{P}(X)$ of all probability measures on $X$.\n\nThe name ``transport plan'' is suggestive: it is a way to describe what amount of\nmass is transported from one region to another.\n\nThe function $\\mathop \\mathrm{W}\\nolimits_p$ is a metric, called the ($L^p$) Wasserstein metric, \nand when $X$ is compact it induces the weak topology. We sometimes\ndenote $\\mathop \\mathrm{W}\\nolimits_2$ simply by $\\mathop \\mathrm{W}\\nolimits$.\n\n\\section{Metric mean dimension}\\label{sec:entropy}\n\nMetric mean dimension is a metric invariant of dynamical systems introduced by\nLindenstrauss and Weiss \\cite{Lindenstrauss-Weiss}, that refines topological entropy\nfor infinite-entropy systems.\n\nLet us briefly recall the definitions. Given a\nmap $f:X\\to X$ acting on a metric space, for any\n$n\\in\\mathbb{N}$ one defines a new metric on $X$ by\n\\[d_n(x,y):= \\max\\{d(f^k(x),f^k(y));0\\leqslant k\\leqslant n\\}.\\]\nGiven $\\varepsilon>0$, one says that a subset $S$ of $X$ is\n$(n,\\varepsilon)$-separated if $d_n(x,y)\\geqslant \\varepsilon$ whenever\n$x\\neq y\\in S$. Denoting by $N(f,\\varepsilon,n)$ the maximal size of a \n$(n,\\varepsilon)$-separated set, the topological entropy of $f$ is defined as\n\\[h(f) := \\lim_{\\varepsilon\\to 0} \\limsup_{n\\to+\\infty} \n\\frac{\\log N(f,\\varepsilon,n)}{n}.\\]\nNote that this limit exists since $\\limsup_{n\\to+\\infty} \\frac1n \\log N(f,\\varepsilon,n)$\nis nonincreasing in $\\varepsilon$.\nThe adjective ``topological'' is relevant since $h(f)$ does not depend upon the\ndistance on $X$, but only on the topology it defines.\nThe topological entropy is in some sense a global measure of the dependance on initial condition\nof the considered dynamical system. \nThe map $\\Phi_d$ is a classical example, whose topological entropy is $\\log d$.\n\nNow, the metric mean dimension is\n\\[\\mathop \\mathrm{mdim}_M\\nolimits(f,d) := \\liminf_{\\varepsilon\\to 0} \\limsup_{n\\to+\\infty} \n  \\frac{\\log N(f,\\varepsilon,n)}{n|\\log\\varepsilon|}.\\]\nIt is zero as soon as topological entropy is finite. Note that this quantity\ndoes depend upon the metric; here we shall use $\\mathop \\mathrm{W}\\nolimits_p$.\nLindenstrauss and Weiss define the metric mean dimension using\ncovering sets rather than separated sets, but this does not matter since\ntheir sizes are comparable.\n\nLet us prove Theorem \\ref{theo:mdim}:\nthe metric mean dimension of $\\Phi_{d\\#}$ is at least $p(d-1)$ when\n$\\mathscr{P}(\\mathbb{S}^1)$ is endowed with the $W_p$ metric.\nIn another paper \\cite{Kloeckner2}, we prove the same kind of result,\nreplacing $\\Phi_d$ by any map having positive entropy. However\nTheorem \\ref{theo:mdim} has a better constant and its proof is simpler.\n\n\\begin{proof}[Proof of Theorem \\ref{theo:mdim}]\nTo construct\na large $(n,\\varepsilon)$-separated set, we proceed as follows: we start with the point\n$\\delta_0$, and choose a $\\varepsilon$-separated set of its inverse images. Then we inductively\nchoose $\\varepsilon$-separated sets of inverse images of each elements of the set \npreviously defined.\nDoing this, we need not control the distance between inverse images of two different elements.\n \nLet $k\\gg 1$ and $\\alpha>0$ be integers; $\\varepsilon$ will be exponential in $-k$. Let\n$A_k$ be the set all $\\mu\\in\\mathscr{P}(\\mathbb{S}^1)$ such that $\\mu((1-2^{-k},1))=0$\nand $\\mu([0,1\/d])\\geqslant 1\/2$. These conditions are designed to bound from\nbelow the distances between the antecedents to be constructed: a given amount \nof mass (second condition) will have to travel a given distance (first\ncondition).\n\nAn element $\\mu\\in A_k$ decomposes as $\\mu=\\mu_h+\\mu_t$ where\n$\\mu_h$ is supported on $[0,1-d2^{-k}]$ and $\\mu_t$ is supported\non $(1-d2^{-k},1-2^{-k})$. Let $e_1,\\ldots, e_d$ be the right inverses to\n$\\Phi$ defined onto $[0,1\/d), [1\/d,2\/d),\\ldots [(d-1)\/d,1)$ respectively.\nFor all integer tuples $\\ell=(\\ell_1,\\ldots,\\ell_d)$ such that $\\ell_1\\geqslant 2^{\\alpha k-1}$\nand $\\sum \\ell_i=2^{\\alpha k}$, define\n\\[\\mu_\\ell=e_{1\\#}(\\ell_1 2^{-\\alpha k}\\mu_h+\\mu_t)+\\sum_{i>1} e_{i\\#}(\\ell_i 2^{-\\alpha k}\\mu_h)\\]\n(see figure \\ref{fig:antecedents} that illustrates the case $d=2$).\nIt is a probability measure on $\\mathbb{S}^1$,\nlies in $A_k$ and $\\Phi_{d\\#}(\\mu_\\ell)=\\mu$. Moreover, if $\\ell'\\neq\\ell$\nthen any transport plan from $\\mu_\\ell$ to $\\mu_{\\ell'}$ has to move a \nmass at least $2^{-\\alpha k-1}$ by a distance at least $2^{-k}d^{-1}$. Therefore,\n\\[\\mathop \\mathrm{W}\\nolimits_p(\\mu_\\ell,\\mu_{\\ell'})\\geqslant d^{-1}2^{-k(\\alpha\/p+1)-1\/p}.\\]\n\n\\begin{figure}[htp]\\begin{center}\n\\input{antecedents.pstex_t}\n\\caption{Construction of separated antecedents of a given measure.}\n\\label{fig:antecedents}\n\\end{center}\\end{figure}\n\nLet $\\varepsilon=d^{-1}2^{-k(\\alpha\/p+1)-1\/p}$\nand define $S_n$ inductively as follows.\nFirst, $S_0=\\{\\delta_0\\}$. Given $S_n\\subset A_k$, $S_{n+1}$\nis the set of all $\\mu_\\ell$ constructed above, where $\\mu$ runs through\n$S_n$.\n\nBy construction, $S_{n+1}$ has at least $C2^{\\alpha k(d-1)}$ times\nhas many elements as $S_n$, for some constant $C$ depending only on $d$. \nThen $S_n$ has at least  $C^n 2^{n\\alpha k(d-1)}$ elements.\nLet $\\mu$, $\\nu$\nbe two distinct elements of $S_n$ and $m$ be the greatest index such that\n$\\Phi_{d\\#}^m\\mu\\neq \\Phi_{d\\#}^m\\nu$. Since $\\Phi_{d\\#}^n\\mu=\\delta_0=\\Phi_{d\\#}^n\\nu$,\n$m$ exists and is at most $n-1$. The measures $\\mu'=\\Phi_{d\\#}^m\\mu$ and \n$\\nu'=\\Phi_{d\\#}^m\\nu$ both lie in $S_{n-m}$ and have the same image. Therefore,\nthey are $\\varepsilon$-separated. This shows that $S_n$ is $(n,\\varepsilon)$-separated.\n\nIt follows that \n\\begin{eqnarray*}\n\\frac{\\log N(\\Phi_{d\\#},\\varepsilon,n)}{n|\\log\\varepsilon|}\n  &\\geqslant& \n  \\frac{C}{|\\log\\varepsilon|}+\\frac{\\alpha(d-1)}{\\frac{\\alpha}{p}+1}\n  \\left(\\frac{-\\frac1p-\\frac{\\log d}{\\log 2}}{|\\log\\varepsilon|}+1 \\right) \\\\\n  &\\geqslant& \\frac{\\alpha(d-1)}{\\frac{\\alpha}{p}+1}(1+o(1))+o(1).\n\\end{eqnarray*}\nIn the case of a general $\\varepsilon$, we get the same bound on\n$\\log N$ up to an additive term $n\\alpha(d-1)\\log 2$, so that\n\\[\\mathop \\mathrm{mdim}_M\\nolimits(\\Phi_{d\\#},\\mathop \\mathrm{W}\\nolimits_p) \\geqslant \\frac{\\alpha(d-1)}{\\frac{\\alpha}{p}+1}.\\]\nBy taking $\\alpha\\to\\infty$ we get $\\mathop \\mathrm{mdim}_M\\nolimits(\\Phi_{d\\#},\\mathop \\mathrm{W}\\nolimits_p)\\geqslant p(d-1)$.\n\\end{proof}\n\n\n\\section{The first-order differential structure on measures}\n\nIn this section we give a short account on the work of Gigli \\cite{Gigli}\nin the particular case of the circle.\nNote that considering the Wasserstein space of a Riemannian manifold as an\ninfinite-dimensionnal Riemannian manifold dates back to the work\nof Otto \\cite{Otto}. \nHowever, in many ways it stayed a formal view until the work of Gigli.\n\n\\subsection{Why bother with this setting?}\n\nBefore getting started, let us explain why we do not simply use the natural affine\nstructure on $\\mathscr{P}(\\mathbb{S}^1)$,\nthe tangent space at a point simply consisting on signed measures having\nzero total mass. Similarly, one could consider simpler to just\ntake the smooth functions of $\\mathbb{S}^1$ as coordinates to define a smooth structure\non $\\mathscr{P}(\\mathbb{S}^1)$. \n\nThe first argument against these points of vue is that optimal transportation is\nabout pushing mass, not (directly) about recording the variation of density at each point.\n\nMore important, these simple ideas would lead a path of the form \n$\\gamma_t=t\\delta_x+(1-t)\\delta_y$ to be smooth. However, the Wasserstein\ndistance between $\\gamma_t$ and $\\gamma_s$ has the order of $\\sqrt{|t-s|}$,\nso that $\\gamma_t$ is not rectifiable (it has infinite length)! This also holds,\nfor example, for convex sums of measures with different supports.\n\nOne could argue that the previous paths can be made Lipschitz by using $\\mathop \\mathrm{W}\\nolimits_1$\ninstead of $\\mathop \\mathrm{W}\\nolimits_2$, so let us give another argument:\nin the affine structure, the Lebesgue measure does not have a tangent space but only a \ntangent cone since $\\lambda+t\\mu$ is not a positive measure for all small\n$t$ unless $\\mu\\ll\\lambda$. If one wants to consider singular measures in the same\nsetting than regular ones, the $\\mathop \\mathrm{W}\\nolimits_2$ setting seems to be the right tool.\n\nNote that it will appear that the differential structure on $\\mathscr{P}(\\mathbb{S}^1)$ depends\nnot only on the differential structure of the circle, but also on its metric.\nThis should not be considered surprising: in finite dimension, the fact that the\ndifferential structures are defined independently of any reference to a metric comes\nfrom the equivalence of norms in Euclidean space: here, in infinite dimension, even the simple\nformula $\\mathop \\mathrm{W}\\nolimits(f(\\mu+tv),f(\\mu)+tD_xf(v)) = o(t)$ involves a metric in a crucial way.\n\nOne could also be\nsurprised that this differential structure involving the metric of the circle could\nbe preserved by expanding maps of non-constant derivative. This point shall be\ncleared in Section \\ref{sec:general}, see Proposition \\ref{prop:centering} and the\ndiscussion before it.\n\n\\subsection{The exponential map}\n\n Note that as is customary\nin these topics, by a geodesic we mean a non-constant globally minimizing geodesic segment\nor line, parametrized proportionaly to arc length.\n\nGiven $\\mu\\in\\mathscr{P}(\\mathbb{S}^1)$, there are several equivalent ways to define its\ntangent space $T_\\mu$. In fact, $T_\\mu$ has a vectorial structure only when \n$\\mu$ is atomless; otherwise it is only a tangent cone. Note that the atomless\ncondition has to be replaced by a more intricate one in higher dimension.\n\nThe most Riemannian way to construct $T_\\mu$ is to use the exponential map.\nLet $\\mathscr{P}(T\\mathbb{S}^1)_\\mu$ be the set of probability measures \non the tangent bundle\n$T\\mathbb{S}^1$ that are mapped  to $\\mu$ by the canonical projection.\n\nGiven $\\xi,\\zeta\\in \\mathscr{P}(T\\mathbb{S}^1)_\\mu$, one defines\n\\[\\mathop \\mathrm{W}\\nolimits_\\mu(\\xi,\\zeta) = \\left(\\inf_\\Pi \\int_{T\\mathbb{S}^1\\times T\\mathbb{S}^1} d^2(x,y)\n  \\,\\Pi(dx dy)\\right)^{1\/2}\\]\nwhere $d$ is any metric whose restriction to the fibers is the riemannian\ndistance (here the fibers are isometric to $\\mathbb{R}$), and the infimum \nis over transport plans $\\Pi$ that are mapped to the identity\n$(\\mathrm{Id},\\mathrm{Id})_\\#\\mu$ by the canonical projection on $\\mathbb{S}^1\\times \n\\mathbb{S}^1$. This means that we allow only to move the mass \\emph{along}\nthe fibers. Equivalently, one can desintegrate $\\xi$ and $\\zeta$ along $\\mu$,\nwriting $\\xi=\\int\\xi_x \\,\\mu(dx)$ and $\\zeta=\\int \\zeta_x \\,\\mu(dx)$, with\n$(\\xi_x)_{x\\in\\mathbb{S}^1}$ and $(\\zeta_x)_{x\\in\\mathbb{S}^1}$ two families\nof probability measures on $T_x\\mathbb{S}^1\\simeq \\mathbb{R}$ uniquely\ndefined up to sets of measure zero. Then one gets\n\\[\\mathop \\mathrm{W}\\nolimits_\\mu^2(\\xi,\\zeta)=\\int_{\\mathbb{S}^1} \\mathop \\mathrm{W}\\nolimits^2(\\xi_x,\\zeta_x) \\mu(dx)\\]\nwhere one integrates the squared Wasserstein metric defined with respect to the\nRiemannian metric, that is $|\\cdot|$.\n\nThere is a natural cone structure on $\\mathscr{P}(T\\mathbb{S}^1)_\\mu$, extending the scalar \nmultiplication on the tangent bundle: letting $D_r$ be the \ndilation of ratio $r$ along fibers, acting on $T\\mathbb{S}^1$, one defines \n$r\\cdot \\xi:=(D_r)_\\#\\xi$.\n\nThe exponential map $\\exp:T\\mathbb{S}^1\\to \\mathbb{S}^1$ now gives a map\n\\[\\exp_\\# : \\mathscr{P}(T\\mathbb{S}^1)_\\mu\\to\\mathscr{P}(\\mathbb{S}^1).\\]\nThe point is that not for all \n$\\xi\\in \\mathscr{P}(T\\mathbb{S}^1)_\\mu$, is there a $\\varepsilon>0$ such that \n$t\\mapsto \\exp_\\#(t\\cdot\\xi)$ defines a geodesic of $\\mathscr{P}(\\mathbb{S}^1)$ on \n$[0,\\varepsilon)$. Consider for example $\\mu=\\lambda$, and $\\xi$ be defined\nby $\\xi_x\\equiv1$. Then $\\exp_\\#(t\\cdot\\xi)=\\lambda$ for all $t$: one rotates all\nthe mass while letting it in place would be more efficient.\n\nThe first definition is that $T_\\mu$ is the closure in $\\mathscr{P}(T\\mathbb{S}^1)_\\mu$ of the subset\nof all $\\xi$ such that $\\exp_\\#(t\\cdot\\xi)$ defines a geodesic for small \nenough $t$.\n\n\n\\subsection{Another definition of the tangent space}\n\nLet us now give another definition, assuming $\\mu$ is atomless.\nWe denote by $|\\cdot|_{L^2(\\mu)}$ the norm defined by the measure $\\mu$, and\nby $|\\cdot|_2$ the usual $L^2$ norm defined by the Lebesgue measure\n$\\lambda$.\n\nGiven a smooth \nfunction $f:\\mathbb{S}^1\\to\\mathbb{R}$, its gradient \n$\\nabla f:\\mathbb{S}^1\\to T\\mathbb{S}^1$ can be used to push $\\mu$\nto an element $\\xi_f=(\\nabla f)_\\#\\mu$ of $\\mathscr{P}(T\\mathbb{S}^1)_\\mu$.\n This element has the \nproperty that $\\exp_\\#(t\\cdot\\xi)=(\\mathrm{Id}+t\\xi_f)_\\#\\mu$ defines a geodesic for small \nenough $t$, with a time bound depending on \n$\\nabla f$ and not on $\\mu$. More precisely,\nthe geodesicness holds as soon as no mass is moved\na distance more than $1\/2$, and no element of mass crosses another one,\nand these conditions translate to $t (\\nabla f)'(x)\\geqslant -1$ for all\n$x$. This is a particular case of Kantorovich duality, see for example\n\\cite{Villani2}, especially figure 5.2.\n\nNow, let $L^2_0(\\mu)$ be the set of all vector fields $v\\in L^2(\\mu)$\nthat are $L^2(\\mu)$-approximable by gradient of smooth functions.\nThen the image of the map $v\\mapsto (\\mathrm{Id},v)_\\#\\mu$ defined on $L^2_0(\\mu)$ \nwith value in $\\mathscr{P}(T\\mathbb{S}^1)_\\mu$ is precisely $T_\\mu$.\nIn particular, this means that as soon as $\\mu$ is atomless, the disintegration\n$(\\xi_x)_x$ of an element of $T_\\mu$ writes $\\xi_x=\\delta_{v(x)}$ for some\nfunction $v$ and $\\mu$-almost all $x$. Moreover, $v$ is $L^2(\\mu)$-approximable\nby gradient of smooth functions; note that amoung smooth vector fields,\ngradients are characterized by $\\int \\nabla f \\lambda = 0$.\nWe shall freely identify the tangent space with $L^2_0(\\mu)$ whenever $\\mu$\nhas no atom.\n\nIn the important case when $\\mu=\\rho\\lambda$ for some continuous density $\\rho$,\na vector field $v\\in L^2(\\mu)$ is approximable by gradient of smooth functions\nif and only if $\\int v\\lambda = 0$.\nWe get that in this case, $T_\\mu$ can be \nidentified with the set of functions $v:\\mathbb{S}^1\\to \\mathbb{R}$\nthat are square-integrable with respect to $\\mu$ and of mean zero \nwith respect to $\\lambda$.  When $\\mu$\nis the uniform measure, we write $L^2_0$ instead of $L^2_0(\\lambda)$.\nNote that if $v\\in L^2(\\mu)$ has neither its negative part nor its\npositive part $\\lambda$-integrable, then it can be approximated in\n$L^2(\\mu)$ norm by gradient of smooth functions, and that\nif $\\mu$ has not full support, then $L^2_0(\\mu)=L^2(\\mu)$.\n\n\nFor simplicity, given $v\\simeq \\xi\\in L^2_0(\\mu)\\simeq T_\\mu$ we shall denote\n$\\exp_\\#(t\\cdot\\xi)$ by $\\mu+tv$. In other words,\n$\\mu+tv=(\\mathrm{Id}+tv)_\\#\\mu$.\n\nThis point of view is convenient, in particular because the distance between\nexponential curves issued from $\\mu$ can be estimated easily:\n\\[\\mathop \\mathrm{W}\\nolimits(\\mu+tv,\\mu+tw)\\underset{t\\to 0}\\sim t|v-w|_{L^2(\\mu)}.\\]\n Note that when $v$ is differentiable,\nthen by geodesicness for $t$ small enough we have\n\\[\\mathop \\mathrm{W}\\nolimits(\\mu,\\mu+tv) = t |v|_{L^2(\\mu)}\\]\nand not only an equivalent. This will prove useful in the next subsection\nwhere several measures and vector fields will be involved.\n\n\n\\subsection{Two properties}\n\nWe shall prove that the exponential map can be used to construct\nbi-Lipschitz embeddings of small, finite-dimensional balls into $\\mathscr{P}(\\mathbb{S}^1)$,\nthen we shall study how the density of an absolutely continuous\nmeasure evolves when pushed by a small vector field.\n\n\nThe following natural result shall be used in the proof of Theorem \n\\ref{theo:almost-invariant}.\n\\begin{prop}\\label{prop:embedding}\nGiven $\\mu\\in \\mathscr{P}(\\mathbb{S}^1)$ and $(v_1,\\ldots,v_n)$\ncontinuous, linearly independent vector fields in $L^2_0(\\mu)$,\nthere is an $\\eta>0$ such that the map $B^n(0,\\eta)\\to\\mathscr{P}(\\mathbb{S}^1)$ defined\nby $E(a)=\\mu+\\sum a_i v_i$ is bi-Lipschitz.\n\\end{prop}\nThe difficulty is only technical: we already know that $E$ is bi-Lipschitz\nalong rays and we need some uniformity in the distance estimates to prove\nthe global bi-Lipschitzness. The continuity hypothesis is not satisfactory\nbut is all we need in the sequel.\n\nNote that we did not assume that $\\mu$ has no atom; when it has, $L^2_0(\\mu)$\n(still defined as the closure in $L^2(\\mu)$ of gradients of smooth functions)\nis not the tangent cone $T_\\mu\\mathscr{P}(\\mathbb{S}^1)$ but only a part of it. Note that\nif $v$ is a $C^1$ vector field of vanishing $\\lambda$-mean,\n$(\\mu+tv)_t$ still defines a geodesic as long as $tv'\\geqslant -1$.\n\n\\begin{proof}\nLet $a,b\\in B^n$. The plan $(\\mathrm{Id}+\\sum a_i v_i,\\mathrm{Id}+\\sum b_i v_i)_\\#\\lambda$\ntransports $E(a)$ to $E(b)$\nat a cost \n\\[\\left|\\sum (a_i-b_i)v_i\\right|_2^2 \n\\leqslant \\left(\\sum |v_i|_2^2\\right)\\, |a-b|^2\\]\nso that $E$ is Lipschitz.\n\nUp to a linear change of coordinates, we assume that the $v_i$ form\nan orthonormal family of $L^2_0(\\mu)$. To bound the distance between\n$E(a)$ and $E(b)$ from below, we shall design a vector field $\\tilde v$\nsuch that pushing $E(a)$ by $\\tilde v$ gives a measure close to \n$E(b)$.\n\nChoose $\\varepsilon>0$\nsuch that for all $i$ we have \n\\[|x-y|\\leqslant\\varepsilon \\Rightarrow |v_i(x)-v_i(y)|\\leqslant \\frac{1}{4\\sqrt{n}}.\\]\nAssume moreover $\\varepsilon<1\/8$.\n\nLet $w_i$ be gradient of smooth functions such that\n$|v_i-w_i|_\\infty\\leqslant \\varepsilon$.\nLet $\\eta>0$ be small enough to ensure $2\\sqrt{n}\\eta\\leqslant 1$ and\n$w_i'\\geqslant -(4n\\eta)^{-1}$ fo all $i$.\n\nFix $a,b\\in B^n(0,\\eta)$ and introduce two maps defined by\n$\\psi(y)=y+\\sum a_i v_i(y)$ and $\\tilde\\psi(y)=y+\\sum a_i w_i(y)$.\nNote that $\\tilde\\psi'\\geqslant 1\/2$ so that $\\tilde\\psi$ is\na diffeomorphism and $\\tilde\\psi^{-1}$ is $2$-Lipschitz. Let\n$\\tilde v = \\sum (b_i-a_i)v_i\\circ\\tilde\\psi^{-1}$.\n\nOn the first hand, given any $y\\in\\mathbb{S}^1$, we have\n\\[|\\tilde\\psi(y)-\\psi(y)|\\leqslant |a|\\left(\\sum(w_i(y)-v_i(y))^2\\right)^{1\/2} \n  \\leqslant |a|\\sqrt{n}\\varepsilon\\]\nso that\n\\[|y-\\tilde\\psi^{-1}\\psi(y)|\\leqslant 2\\sqrt{n}|a|\\varepsilon\\leqslant \\varepsilon\\]\nand\n\\[\\left|v_i(\\tilde\\psi^{-1}\\psi(y))-v_i(y)\\right|\\leqslant\\frac1{4\\sqrt{n}}.\\]\nIt follows that\n\\[\\left|\\sum(b_i-a_i)(v_i(\\tilde\\psi^{-1}\\psi(y)) -v_i(y))\\right|\\leqslant\\frac14|b-a|,\\]\nand therefore\n\\begin{equation}\n\\left|\\tilde v\\circ\\psi-\\sum(b_i-a_i)v_i\\right|_{L^2(\\nu)}\\leqslant\\frac14|b-a|\n\\label{eq:lip1}\n\\end{equation}\nwhere $\\nu$ could be any probability measure. We shall take\n$\\nu=\\mu+\\sum a_i v_i$.\n\nSimilarly,\n\\begin{eqnarray}\n|\\tilde v|_{L^2(\\nu)} &=& \\left(\\int \\tilde v^2(x) \\,(\\psi_\\#\\mu)(dx)\\right)^{1\/2} \\nonumber\\\\\n  &=& \\left(\\int \\tilde v^2(\\psi x)\\,\\mu(dx)\\right)^{1\/2} \\nonumber\\\\\n  &=& \\left|\\sum(b_i-a_i)v_i\\tilde\\psi^{-1}\\psi\\right|_{L^2(\\mu)} \\nonumber\\\\\n  &\\geqslant& \\frac34\\left|\\sum(b_i-a_i)v_i\\right|_{L^2(\\mu)} \\nonumber\\\\\n|\\tilde v|_{L^2(\\nu)}  &\\geqslant& \\frac34 |b-a|.\n\\end{eqnarray}\n\nOn the other hand, we have\n\\[ \\mathop \\mathrm{W}\\nolimits\\left(\\mu+\\sum a_i v_i, \\mu+\\sum b_i v_i\\right)\\geqslant\n  \\mathop \\mathrm{W}\\nolimits(\\nu,\\nu+\\tilde v)-\\mathop \\mathrm{W}\\nolimits\\left(\\nu+\\tilde v,\\mu+\\sum b_i v_i\\right).\\]\n\nLet $\\tilde w=\\sum(b_i-a_i)w_i\\circ\\tilde\\psi^{-1}$. We have\n$|\\tilde v-\\tilde w|_\\infty\\leqslant \\varepsilon |b-a|$.\nIn particular, $|\\tilde w|_{L^2(\\nu)}\\geqslant\\frac58|b-a|$.\nThe choice of $\\eta$ ensures that $\\tilde w'\\geqslant-1$, so that\n\\[\\mathop \\mathrm{W}\\nolimits(\\nu,\\nu+\\tilde w)=|\\tilde w|_{L^2(\\nu)}\\geqslant \\frac58|b-a|.\\]\nSince $\\mathop \\mathrm{W}\\nolimits(\\nu+\\tilde v,\\nu+\\tilde w)\\leqslant |\\tilde v-\\tilde w|_\\infty$\nwe get\n\\begin{equation}\n\\mathop \\mathrm{W}\\nolimits(\\nu,\\nu+\\tilde v)\\geqslant \\frac12|b-a|.\n\\end{equation}\nFinally, since $\\nu+\\tilde v= (\\psi+\\tilde v \\psi)_\\#\\mu$,\n\\eqref{eq:lip1} shows that \n\\[\\mathop \\mathrm{W}\\nolimits\\left(\\nu+\\tilde v,\\mu+\\sum b_i v_i\\right)\\leqslant\\frac14|b-a|\\]\nso that\n\\[ \\mathop \\mathrm{W}\\nolimits\\left(\\mu+\\sum a_i v_i, \\mu+\\sum b_i v_i\\right)\\geqslant \\frac14|b-a|.\\]\n\\end{proof}\n\n\n\\begin{prop}\\label{prop:density}\nLet $\\rho$ be a $C^1$ density and $v:\\mathbb{S}^1\\to \\mathbb{R}$\nbe a $C^1$ vector field. Then for $t\\in\\mathbb{R}$ small enough\n$\\rho\\lambda+tv$ is absolutely continuous and its density\n$\\rho_t$ is continuous and satisfy\n\\[\\rho_t(x) = \\rho(x) -t(\\rho v)'(x) + o(t)\\]\nwhere the remainder term is independent of $x$.\n\\end{prop}\n\n\\begin{proof}\nLet $t$ be small enough so that $\\mathrm{Id}+tv$ is a diffeomorphism.\nThen for all integrable function $f$, one has\n\\begin{eqnarray*}\n\\int f(x) (\\rho\\lambda+tv)(dx) &=& \\int f(x) (\\mathrm{Id}+tv)_\\#(\\rho\\lambda)(dx)\\\\\n  &=& \\int f(x+tv(x)) \\rho(x) dx\\\\\n  &=& \\int f(y) \\left(\\frac{\\rho}{1+tv'}\\right)\\circ(\\mathrm{Id}+tv)^{-1}(y) dy\n\\end{eqnarray*}\nby a change of variable. It follows that \n\\begin{eqnarray*}\n\\rho_t &=& \\frac{\\rho}{1+tv'}\\circ(\\mathrm{Id}+tv)^{-1}\\\\\n       &=& \\left(\\rho(1-tv')\\right)\\circ(\\mathrm{Id}-tv)+o(t)\\\\\n       &=& \\rho-t(\\rho'v+v'\\rho)+o(t)\n\\end{eqnarray*}\nwhere the $o(t)$ term depends upon $\\rho$\nand $v$ but is uniform in $x$.\n\\end{proof}\nNote that the $o(t)$ depends in particular on the\nmoduli of continuity of $v'$ and $\\rho'$ and need not\nbe an $O(t^2)$ unless $v$ and $\\rho$ are $C^2$.\n\n\n\\section{First-order dynamics in the model case}\\label{sec:firstorder}\n\nIn this section we show that $\\Phi_{d\\#}$ is (weakly) differentiable at the point \n$\\lambda$. Its derivative is an\nexplicit, simple endomorphism of a Hilbert space, and we shall give a brief\nstudy of its spectrum.\n\n\\begin{theo}\\label{theo:differential}\nLet $\\mathscr{L}_d:L^2_0\\to L^2_0$ be the linear operator defined by\n\\[\\mathscr{L}_d v(x)= v(x\/d)+v((x+1)\/d)+\\dots+v((x+d-1)\/d).\\]\n Then $\\mathscr{L}_d$ is the\nderivative of $\\Phi_{d\\#}$ at $\\lambda$ in the following sense:\nfor all $v\\in L^2_0\\simeq T_\\lambda$, one has\n\\[\\mathop \\mathrm{W}\\nolimits\\left(\\Phi_{d\\#}(\\lambda+tv),\\lambda+t\\mathscr{L}_d(v)\\right)=o(t).\\]\n\\end{theo}\nFirst, we recognize in $\\mathscr{L}_d$ a multiple of\nthe Perron-Frobenius operator of $\\Phi_d$,\nthat is the adjoint of the map $u\\mapsto u\\circ \\Phi$, acting on the space $L^2_0$.\nSecond, we only get a G\\^ateaux derivative, when one would prefer a Fr\\'echet one,\nthat is a formula of the kind\n\\[\\mathop \\mathrm{W}\\nolimits(\\Phi_{d\\#}(\\lambda+v),\\lambda+\\mathscr{L}_d(v))=o(|v|).\\]\nHowever, we shall see that such a uniform bound does not\nhold. \nHowever, one easily gets uniform remainder terms in restriction to any finite-dimensional\nsubspace of $L^2_0$.\n\n\\subsection{Differentiability of $\\Phi_{d\\#}$}\n\nThe main point to prove in the above theorem is the following estimate.\n\\begin{lemm}\\label{lemm:composition}\nGiven a density $\\rho$, vector fields $v_1,\\ldots,v_n\\in L^2(\\rho\\lambda)$\nand positive numbers\n$\\alpha_1,\\ldots,\\alpha_n$ adding up to $1$, one has\n\\[\\mathop \\mathrm{W}\\nolimits\\left(\\rho\\lambda+t\\sum_i \\alpha_i v_i\\,,\\,\n  \\sum_i\\alpha_i(\\rho\\lambda+tv_i)\\right)=o(t).\\]\n\\end{lemm}\nWe could deduce this result from Proposition \\ref{prop:density}\nbut for the sake of diversity let us give a different proof,\nwhich is almost contained in Figure \\ref{fig:transport}.\n\n\\begin{proof}\nWe prove the case $n=2$ since the general case can then be deduced by a straightforward\ninduction.\nLet $\\varepsilon$ be any positive number. Let $\\bar \\rho$, $\\bar v_1$ \nand $\\bar v_2$ be a piecewise constant density and two piecewise constant\nvector fields that approximate $\\rho$ in $L^1$ norm and\n$v_1$ and $v_2$ in $L^2$ norm:\n$|\\rho-\\bar\\rho|_1\\leqslant \\varepsilon^2$ and\n$|v_i-\\bar v_i|_{L^2(\\rho\\lambda)}\\leqslant\\varepsilon$.\n\nThe measure $((\\mathrm{Id}+v_i)\\times(\\mathrm{Id}+\\bar v_i))_\\#\\rho\\lambda$\nis a transport plan from \n$\\rho\\lambda+v_i$ to $\\rho\\lambda+\\bar v_i$, whose cost is \n$|v_i-\\bar v_i|_{L^2(\\rho\\lambda)}^2$.\nThis shows that $\\mathop \\mathrm{W}\\nolimits(\\rho\\lambda+v_i,\\rho\\lambda+\\bar v_i)\\leqslant \\varepsilon$.\nA transport plan $\\Pi$ from $\\rho\\lambda$ to $\\bar\\rho\\lambda$\nthat lets the common mass in place and transports the rest in any way\nmoves a mass $\\frac12|\\rho-\\bar\\rho|_1$ by a distance at most $\\frac12$,\nthus $\\mathop \\mathrm{W}\\nolimits(\\rho\\lambda,\\bar\\rho\\lambda)\\leqslant 2^{-3\/2}\\varepsilon$.\nNow $\\big(\\mathrm{Id}+\\bar v_i,\\mathrm{Id}+\\bar v_i\\big)_\\#\\Pi$ is a transport\nplan from $\\rho\\lambda+\\bar v_i$ to  $\\bar\\rho\\lambda+\\bar v_i$\nwith the same cost as $\\Pi$, so that \n$\\mathop \\mathrm{W}\\nolimits(\\rho\\lambda+\\bar v_i,\\bar\\rho\\lambda+\\bar v_i)\n\\leqslant 2^{-3\/2}\\varepsilon$. It follows that\n\\[\\mathop \\mathrm{W}\\nolimits\\left(\\sum\\alpha_i(\\rho\\lambda+ t v_i),\n  \\sum\\alpha_i(\\bar\\rho\\lambda+t \\bar v_i)\\right)\\leqslant C\\varepsilon t\\]\nfor a constant $C=2^{-3\/2}+1$, and similarly\n\\[\\mathop \\mathrm{W}\\nolimits\\left(\\rho\\lambda+\\sum\\alpha_i t v_i,\n  \\bar\\rho\\lambda+\\sum\\alpha_it \\bar v_i\\right)\\leqslant C\\varepsilon t.\\]\n\nWe can moreover assume that $\\bar\\rho$ and $\\bar v_i$ are constant on each interval\nof the form $[i\/k,(i+1)\/k)$ for some fixed $k$ (depending upon\n$\\rho$, $v_1$, $v_2$ and $\\varepsilon$).\n\nTo see what happens on such an interval $I$, temporarily denoting by\n$\\rho$, $v_1$ and $v_2$\nthe values taken by the functions $\\bar\\rho$ and $\\bar v_i$ on $I$, let us \nconstruct for $t$ small enough an economic transport plan from \n$(\\mathrm{Id}+t(\\alpha_1v_1+\\alpha_2v_2))_\\# \\rho\\lambda_{|I}$ to \n$\\alpha_1(\\mathrm{Id}+tv_1)_\\#\\rho\\lambda_{|I}+\\alpha_2(\\mathrm{Id}+tv_2)_\\#\\rho\\lambda_{|I}$. \nIf the intervals $(\\mathrm{Id}+tv_1)(I)$ and $(\\mathrm{Id}+tv_2)(I)$ meet, \none can simply let the common mass in \nplace and move at each side a mass $\\alpha_1\\alpha_2\\rho|v_1-v_2|t$\nby a distance at most $|v_1-v_2|t$ (see\nfigure \\ref{fig:transport}; this is not optimal but sufficient for our purpose).\nThis transport plan has a cost \n$t^3 \\alpha_1\\alpha_2\\rho|v_1-v_2|^3<t^3 \\rho|v_1-v_2|^3$.\n\n\\begin{figure}[htp]\\begin{center}\n\\input{transport.pstex_t}\n\\caption{The cost of this transport plan has the order of magnitude $t^3$}\n\\label{fig:transport}\n\\end{center}\\end{figure}\n\nIf the intervals $(\\mathrm{Id}+tv_1)(I)$ and $(\\mathrm{Id}+tv_2)(I)$ do not meet, then\n$t|v_1-v_2|\\geqslant 1\/k$ and\nsimple translations give a transport plan with cost at most\n\\[\\alpha_1\\rho\/k t^2|v_1-v_2|^2+\\alpha_2\\rho\/k t^2|v_1-v_2|^2\\leqslant \\rho\nt^3|v_1-v_2|^3.\\]\n\nBy adding one such plan for each interval $[i\/k,(i+1)\/k)$, we get a transport plan from\n$(\\mathrm{Id}+t(\\alpha_1\\bar v_1+\\alpha_2\\bar v_2))_\\# \\bar\\rho\\lambda$ to \n$\\alpha_1(\\mathrm{Id}+t\\bar v_1)_\\#\\bar\\rho\\lambda+\\alpha_2(\\mathrm{Id}+t\\bar v_2)_\\#\\bar\\rho\\lambda$\n whose cost is\nat most $k|\\bar v_1-\\bar v_2|_{L^3(\\bar\\rho\\lambda)}^3 t^3$. Note that even if\nthe $v_i$ are only $L^2$, $\\bar v_i$ are bounded and therefore in $L^3(\\bar\\rho\\lambda)$. \nNow we have\n\\[\\mathop \\mathrm{W}\\nolimits\\left(\\bar\\rho\\lambda+t(\\alpha_1\\bar v_1+\\alpha_2\\bar v_2),\n\\alpha_1(\\bar\\rho\\lambda+t\\bar v_1) + \\alpha_2(\\bar\\rho\\lambda+t\\bar v_2)\\right)\n  \\leqslant k^{1\/2}|\\bar v_1-\\bar v_2|_{L^3(\\bar\\rho\\lambda)}^{3\/2} t^{\\frac32}\\]\nso that, for $t$ small enough,\n\\[\\mathop \\mathrm{W}\\nolimits\\left(\\bar\\rho\\lambda+t(\\alpha_1\\bar v_1+\\alpha_2\\bar v_2),\n\\alpha_1(\\bar\\rho\\lambda+t\\bar v_1) + \\alpha_2(\\bar\\rho\\lambda+t\\bar v_2)\\right)\n  \\leqslant \\varepsilon t\\]\n\nBy triangular inequality, it follows that \n\\[\\mathop \\mathrm{W}\\nolimits\\left(\\rho\\lambda+t(\\alpha_1 v_1+\\alpha_2 v_2),\\alpha_1(\\rho\\lambda+t v_1)\n  +\\alpha_2(\\rho\\lambda+t v_2)\\right)\\leqslant C'\\varepsilon t\\]\nfor a constant $C'=2^{-1\/2}+3$.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{theo:differential}]\nRemark that \n\\begin{align*}\n\\Phi_{d\\#}(\\lambda+tv)=&\\frac1d\\big(\\lambda+dt\\,v(\\cdot\/d)\\big)\n  +\\frac1d\\big(\\lambda+dt\\,v((\\cdot+1)\/d)\\big)\\\\\n& +\\dots+ \\frac1d\\big(\\lambda+dt\\,v((\\cdot+d-1)\/d)\\big)\n\\end{align*}\nand apply the preceding lemma.\n\\end{proof}\n\nLet us prove that we cannot hope for the Fr\\'echet differentiability of \n$\\Phi_{d\\#}$. We only treat the case $d=2$ for simplicity.\n\n\\begin{prop}\nFor all positive $\\varepsilon$, there is a vector field\n$v\\in L^2_0$ that satisfies the following:\n\\begin{enumerate}\n\\item\\label{enumi:a} $|v|_2\\leqslant \\varepsilon$,\n\\item\\label{enumi:b}  $\\mathscr{L}_2 v=0$ so that $\\lambda+\\mathscr{L}_2 v=\\lambda$, and\n\\item\\label{enumi:c} $\\mathop \\mathrm{W}\\nolimits\\left(\\Phi_{2\\#}(\\lambda+v),\\lambda\\right)\\geqslant c\\varepsilon$\n\\end{enumerate}\nfor some constant $c$ independent of $\\varepsilon$ and $v$.\n\\end{prop}\n\n\\begin{proof}\nLet $k$ be a positive integer, to be precised later on. Let $v$ be the\npiecewise affine map defined as follows (see figure \\ref{fig:example}): \n$v(x)=1\/(4k)-y$ when  $x=i\/(2k) + y$ with $y\\in[0,1\/(2k))$\nand $0\\leqslant i<k$ an integer, and $v(x)=-1\/(4k)+y$ when \n$x=i\/(2k) + y$ with $y\\in[0,1\/(2k))$ and $k\\leqslant i<2k$.\nWe have $|v|_2^2=(4k)^{-2}\/3$ so that taking \n$k\\geqslant \\frac{\\sqrt{3}}4\\varepsilon^{-1}$ ensures point \\ref{enumi:a}.\nMoreover, \\ref{enumi:b} is straightforward, and we have left to prove that $k$\nchosen with the order of $\\varepsilon^{-1}$ gives \\ref{enumi:c}.\n\nOn any small enough interval $I$, if $w$ is an affine function of slope $-1$\nwith a zero at the center of $I$, then $\\lambda_{|I}+w$ is a Dirac mass at\nthe center of $I$ (each element of mass is moved to the center). If $w$\nhas slope $1$, then the mass moves in the other direction, and $\\lambda_{|I}+w$\nis uniform of density $1\/2$ on the interval $I'$ having the same center than\n$I$ and twice as long. By combining these two observations, one deduces that\n\\[\\mu:=\\Phi_{2\\#}(\\lambda+v)=1\/2\\lambda\n  +\\sum_{i=1}^k \\frac1{2k}\\delta_{\\frac{i-1\/2}{k}}.\\] \n\n\\begin{figure}[htp]\\begin{center}\n\\includegraphics{example}\n\\caption{The case $k=4$. Up: the graph of $v$; middle: $\\lambda+v$; down:\n$\\Phi_\\#(\\lambda+v)$.}\\label{fig:example}\n\\end{center}\\end{figure}\n\nEach interval of the form $I_i=[(i-5\/8)\/k,(i-3\/8)\/k)$ is given by $\\lambda$\na mass $1\/(4k)$. The discrete part of $\\mu$ consists in a Dirac mass of weight \n$1\/(2k)$ at the center of each $I_i$. Any transport plan from $\\mu$ to $\\lambda$\nmust therefore move a mass at least\n$1\/(4k)$ from each of these Dirac masses to the outside of $I_i$, so that\na total mass at least $1\/4$ has to move a distance at least $1\/(8k)$. From this\nit follows that $\\mathop \\mathrm{W}\\nolimits(\\lambda,\\mu)\\geqslant 1\/(16k)$. When $k$ is chosen with the\norder of $\\varepsilon^{-1}$, this distance has at least the order of \n$\\varepsilon$, as required. \n\\end{proof}\n\n\\subsection{Spectral study of $\\mathscr{L}_d$}\n\nLet us compute the spectrum of \n$\\mathscr{L}_d=D_\\lambda(\\Phi_{d\\#})$.  The following proposition is very \nelementary and \nnot new, but we produce a proof for the sake of completeness.\n\n\\begin{prop}\\label{prop:spec}\nA number $\\alpha$ is an eigenvalue of $\\mathscr{L}_d$ if and only if\n$|\\alpha|<d$. Moreover, each eigenvalue has an infinite-dimensional eigen\\-space.\nLast, the spectrum of $\\mathscr{L}_d$ is the closed disc of radius $2$.\n\\end{prop}\n\nThe proof of Proposition \\ref{prop:spec} consist simply in using Fourier series\nto show that (up to a multiplicative constant) $\\mathscr{L}_d$ is conjugated to a \ncountable product of the shift on $\\ell^2(\\mathbb{N})$.\n\n\\begin{proof}\nLet $c_k$ denote the function $x\\mapsto \\cos(2\\pi k x)$ defined on the\ncircle, and $s_k:x\\mapsto \\sin(2\\pi k x)$. Then it is readily checked that\n$\\mathscr{L}_d\\, c_k=\\mathscr{L}_d\\,s_k=0$ when $d$ does not divide $k$, and \n$\\mathscr{L}_d\\, c_k=dc_{k\/d}$, $\\mathscr{L}_d\\,s_k=d s_{k\/d}$ when $d|k$.\n\nLet $\\sigma$ be the shift of the Hilbert space $\\ell^2=\\ell^2(\\mathbb{N})$ of \n$\\mathbb{N}$-indexed\nsquare integrable sequences: if $\\underline{x}=(x_0,x_1,x_2,\\ldots)$ then\n$\\sigma\\underline{x}=(x_1,x_2,x_3,\\ldots)$. Let $\\sigma^{\\mathbb{N}}$\nbe the direct product of $\\sigma$, acting diagonaly on the space $(\\ell^2)^{(\\mathbb{N})}$\nof sequences $X=(\\underline{x}^0,\\underline{x}^1,\\underline{x}^2,\\ldots)$ such that\n$\\underline{x}^i\\in\\ell^2$ and $\\sum |\\underline{x}^i|_2^2<\\infty$.\nThen the map $\\Psi : (\\ell^2)^{(\\mathbb{N})} \\to L^2_0$ defined by\n\\begin{align*}\n\\Psi(X)= &\\sum_{i,j\\in\\mathbb{N}} x_j^{2(d-1)i} c_{(di+1)d^j} \n  + x_j^{2(d-1)i+1} c_{(di+2)d^j}\\\\\n  & \\qquad+\\dots+x_j^{2(d-1)i+d-2} c_{(di+d-1)d^j} \\\\\n  & + x_j^{2(d-1)i+d-1} s_{(di+1)d^j}\n  +x_j^{2(d-1)i+d} s_{(di+2)d^j} \\\\\n  & \\qquad+\\dots+x_j^{2(d-1)i+2d-3} s_{(di+d-1)d^j}\n\\end{align*}\nis an isomorphism (and even an isometry) that intertwins $\\sigma^{\\mathbb{N}}$\nand $\\frac1d\\mathscr{L}_d$. The spectral study of $\\mathscr{L}_d$ therefore reduces\nto that of $\\sigma$.\n\nA non-zero eigenvector of $\\sigma$, associated to an eigenvalue $\\alpha$,\nmust have the form $(x,\\alpha x,\\alpha^2 x,\\ldots)$ with $x\\neq 0$. Such a\nsequence is square integrable if and only if $|\\alpha|<1$. Moreover\nthe operator norm of $\\sigma$ is $1$, so that its complex spectrum is a subset\nof the closed unit disc. Since the spectrum is closed, and contains\nthe set of eigenvalues, it is equal to the closed unit disc.\n\\end{proof}\n\n\\subsection{Discussion of the non-Fr\\'echet differentiability}\n\nThe coun\\-ter-example to the Fr\\'echet differentiability of $\\Phi_\\#$\nat $\\lambda$ has high total variation, and it is likely that using a norm\nthat controls variations (e.g. a Sobolev norm) on (a subspace of) $T_\\lambda$\nshall provide a uniform error bound. \n\nMoreover, up to multiplication by $d$ the derivative $\\mathscr{L}_d$ is \nthe Perron-Frobenius operator of $\\Phi_d$,\nand such operators have\nfar more subtle spectral properties when defined over Sobolev spaces.\n\nFor these two reasons, it seems that one could search for a modification of\noptimal transport that would give a manifold structure to $\\mathscr{P}(\\mathbb{S}^1)$,\n in such a way that $T_\\lambda$ identifies with a Sobolev space. A way to \nachieve this could be to penalize not only the distance by which a transport\nplan moves mass, but also the distorsion, that is the variation of the pairwise\ndistances of the elements of mass. This should impose more regularity to optimal\ntransport plans.\n\n\n\\section{First-order dynamics for general expanding maps}\\label{sec:general}\n\nIn this section, we consider a general map $\\Phi:\\mathbb{S}^1\\to\\mathbb{S}^1$,\nassumed to be $C^2$ and expanding, {\\it i.e.} $|\\Phi'|>1$. Such a map is a self-covering,\nand has a unique absolutely continuous invariant measure \n(see e.g. \\cite{Katok-Hasselblatt})\nwhich has a positive and $C^1$ density \\cite{Krzyzewski}, denoted by $\\rho$. The measure itself is denoted\nby $\\rho\\lambda$.\nNote that as sets, $L^2(\\rho\\lambda)=L^2$, although they differ as Hilbert spaces.\nAll integrals where the variable is implicit are with respect to the Lebesgue measure $\\lambda$.\n\nThe result is as follows.\n\\begin{theo}\\label{theo:expanding}\nThe map $\\Phi_\\#$ has a G\\^ateaux derivative\n$\\mathscr{L} : L^2_0(\\rho\\lambda) \\to  L^2_0(\\rho\\lambda)$ at $\\rho\\lambda$,\ngiven by\n\\[\\mathscr{L}v(x) = \\sum_{y\\in\\Phi^{-1}(x)} \\frac{\\rho(y)}{\\rho(x)} v(y)\n              -\\frac{\\int{v\\Phi'\\frac{\\rho}{\\rho\\circ\\Phi}}}{\\rho(x)\\int1\/\\rho}\n\\]\nMoreover the adjoint operator of $\\mathscr{L}$ in $L^2_0(\\rho\\lambda)$\nis given by\n\\[\\mathscr{L}^* u = \\Phi'\\, u\\circ\\Phi.\\]\n\\end{theo}\n\n\\subsection{Proof of Theorem \\ref{theo:expanding}}\n\nFirst, as in the case of $\\Phi_{d\\#}$, Lemma \\ref{lemm:composition} shows that\nfor $v\\in L^2_0(\\rho\\lambda)$,\n\\begin{equation}\nd\\left(\\Phi_\\#(\\rho\\lambda+tv), \\rho\\lambda+t\\tilde{\\mathscr{L}}v\\right)=o(t)\n\\label{eq:tildoperator}\n\\end{equation}\nwhere \n\\[\\tilde{\\mathscr{L}}v(x) = \\sum_{y\\in\\Phi^{-1}(x)} \\frac{\\rho(y)}{\\rho(x)} v(y)\\]\nis the first term in the expression of $\\mathscr{L}$.\nIn words, each of the inverse image of $x$ gives a contribution to the local displacement\nof mass that is proportional to $v(y)$ and to $\\rho(y)$.\n\nThis seems very similar to the case of $\\Phi_\\#$, except that $\\tilde{\\mathscr{L}}$ need\nnot map $L^2_0(\\rho\\lambda)$ to itself! Let us stress, once again, that the condition\nthat $v\\in L^2_0(\\rho\\lambda)$ has mean zero is to be understood \\emph{with respect to\nthe uniform measure} $\\lambda$, since it translates the \\emph{metric} property of being (close to)\nthe gradient of a smooth function. This does not prevent Equation \\eqref{eq:tildoperator}\nto make sense, but shows that $\\tilde{\\mathscr{L}}v$ cannot be considered as the\ndirectional derivative of $\\Phi_\\#$ since it does not belong to $T_{\\rho\\lambda}=L^2_0(\\rho\\lambda)$.\nIn fact, we shall see that there is another vector field, that lies in $L^2_0(\\rho\\lambda)$ and\ngives the same pushed measure (at least at order $1$).\n\n\\begin{prop}\\label{prop:centering}\nGiven $\\tilde w\\in L^2(\\rho\\lambda)$ and assuming that $\\tilde w$ is $C^1$, there \nis a $C^1$\nvector field $w\\in L^2_0(\\rho\\lambda)$ such that\n$\\mathop \\mathrm{W}\\nolimits(\\rho\\lambda+t\\tilde w,\\rho\\lambda+tw)=o(t)$. Moreover, $w$ is given by\n\\[w=\\tilde w+\\frac{\\int \\tilde w}{\\rho\\int 1\/\\rho}.\\]\n\\end{prop}\n\n\\begin{proof}\nThis is a direct application of Proposition \\ref{prop:density}: \nwe search for a $w$ such that $(\\rho w)'=(\\rho\\tilde w)'$,\nso that the densities $\\rho_t$ and $\\tilde\\rho_t$ of $\\rho\\lambda+tw$\nand $\\rho\\lambda+t\\tilde w$ are $L^\\infty$ and therefore $L^1$ close\none to the other. This ensures that \n$\\mathop \\mathrm{W}\\nolimits(\\rho\\lambda+t\\tilde w,\\rho\\lambda+tw)\\leqslant |\\rho_t-\\tilde\\rho_t|=o(t)$.\n\nBut there exists exactly one vector field $w$ that is $C^1$, \nhas mean zero, and such that\n$(\\rho w)'=(\\rho\\tilde w)'$: it is given by the claimed formula.\n\\end{proof}\n\nNote that we did not bother to prove the unicity of $w$: Gigli's construction\nshows that the first order\nperturbation of the measure (with respect to the $L^2$ Wasserstein metric)\ncharacterizes a tangent vector \nin $T_\\mu$, see Theorem 5.5 in \\cite{Gigli}.\n\nNow if one considers the ``centering'' operator \n$\\mathscr{C}:L^2(\\rho\\lambda)\\to L^2_0(\\rho\\lambda)$\ndefined by \n\\[\\mathscr{C} v= v-\\frac{\\int v}{\\rho\\int 1\/\\rho},\\]\nthe derivative of $\\Phi_\\#$ at $\\rho\\lambda$ is given by the composition \n$\\mathscr{C}\\tilde{\\mathscr{L}}$.\nIndeed, the previous proposition shows this for a $C^1$ argument, but \n$C^1$ vector fields are dense\nin $L^2_0(\\rho\\lambda)$ and the involved operators are continuous\nin the $L^2(\\rho\\lambda)$ topology.\n\nTo get the expression of $\\mathscr{L}$ given in Theorem \\ref{theo:expanding}, one \nonly need a change of variable:\ndenoting by $\\Phi_i^{-1}$ ($i=1,2,\\ldots, d$) the right inverses to $\\Phi$ \nthat are onto intervals $[a_1=0,a_2), [a_2,a_3), \\ldots,\n[a_d,a_{d+1}=1)$ one has\n\\begin{eqnarray*}\n\\int\\tilde{\\mathscr{L}}v \n  &=& \\sum_i \\int \\frac{\\rho\\circ\\Phi_i^{-1}}{\\rho} v\\circ\\Phi_i^{-1} \\\\\n  &=& \\sum_i \\int_{a_i}^{a_{i+1}} \\frac{\\rho}{\\rho\\circ\\Phi} v \\Phi' \\\\\n  &=& \\int  v \\Phi'\\frac{\\rho}{\\rho\\circ\\Phi}.\n\\end{eqnarray*}\n\nThe computation of the adjoint is a similar change of variable that we omit. \nNote that the adjoint\nof the extension to $L^2(\\rho\\lambda)$ of $\\mathscr{L}$ (with the same expression) is\n\\[u\\mapsto \\Phi'\\, u\\circ\\Phi - \\frac{\\Phi'\\int u}{\\rho\\circ\\Phi \\int{1\/\\rho}}\\]\nand the second term vanishes when $u$ is in $L^2_0(\\rho\\lambda)$. \nThe first term is also the adjoint in $L^2(\\rho\\lambda)$\nof $\\tilde{\\mathscr{L}}$, and this adjoint preserves $L^2_0(\\rho\\lambda)$. \nIn other words,\n$\\mathscr{L}$ is the adjoint in $L^2_0(\\rho\\lambda)$ of the adjoint in\n$L^2(\\rho\\lambda)$ of $\\tilde{\\mathscr{L}}$.\nAn interesting feature of the expression of $\\mathscr{L}^*$ is that it does not \ninvolve the invariant measure.\n\n\\subsection{Spectral study}\n\nEven if $\\mathscr{L}$ is not a multiple of the Perron-Frobenius operator\nof $\\Phi$, its first term $\\tilde{\\mathscr{L}}$\nis a weighted transfert operator, with weight\n$g=\\frac{\\rho}{\\rho\\circ\\Phi}$. According to Theorem 2.5 in \\cite{Baladi},\nevery number of modulus less than $R_g=\\lim_n(\\sup\\tilde{\\mathscr{L}}^n1)^{1\/n}$\nis an eigenvalue of infinite multiplicity with continuous eigenfunctions.\n\n\\begin{prop}\nWe have $R_g\\geqslant \\min \\Phi'>1$, and in consequence\nthere is an infinite linearly independent family $(v_i)_i$\nof continuous functions in $L^2_0(\\rho\\lambda)$ such\nthat $\\mathscr{L}v_i=v_i$.\n\\end{prop}\n\n\\begin{proof}\nLet $m=\\min \\Phi'$: we have $m>1$ and, since $\\rho\\lambda$ is\ninvariant,\n\\[\\rho(x)=\\sum_{y\\in\\Phi^{-1}(x)}\\frac{\\rho(y)}{\\Phi'(y)}\n  \\leqslant \\frac1m\\sum_{y\\in\\Phi^{-1}(x)} \\rho(y) \\]\nIt follows that for all positive continuous function $f$,\n\\[\\tilde{\\mathscr{L}}f=\\sum_{y\\in\\Phi^{-1}(x)}\\frac{\\rho(y)}{\\rho(x)}f(y)\n  \\geqslant m|\\inf f|;\\]\nin particular, $R_g\\geqslant m>1$ and there is a linearly independent\ninfinite family $u_0,u_1,\\ldots,u_i\\ldots$\nof continuous $1$-eigenfunctions of $\\tilde{\\mathscr{L}}$.\nIf not all $u_i$ have mean $0$ (with respect to Lebesgue's measure\n$\\lambda$), assume the mean of $u_0$ is not zero and\nlet $v_i=u_i-\\alpha_i u_0$ where $\\alpha_i$ is chosen such that\n$\\int v_i\\lambda=0$. Otherwise, simply put $v_i=u_i$.\n\nNow, since $\\tilde{\\mathscr{L}}v_i=v_i$ and $v_i$ has mean zero,\nwe get $\\mathscr{L} v_i=\\tilde{\\mathscr{L}}v_i=v_i$.\n\\end{proof}\n\nIn the same way, we see that all numbers less than $m>1$ are eigenvalues\nof $\\mathscr{L}$ (with infinite multiplicity and continuous eigenfunctions).\n\n\\section{Nearly invariant measures}\n\nIn this section we prove Theorem \\ref{theo:almost-invariant} and Proposition\n\\ref{prop:atomless}.\n\n\\subsection{Construction}\n\nFix some positive integer $n$ and let $v_1,\\ldots,v_n$ be continuous,\nlinearly independent eigenfunctions for \n$\\mathscr{L}=D_{\\rho\\lambda}(\\Phi_\\#)$.\n\nFor all $a=(a_1,\\ldots,a_n)\\in B^n(0,\\eta)$, define\n$E(a)=\\rho\\lambda+\\sum_i a_i v_i\\in\\mathscr{P}(\\mathbb{S}^1)$ and using Proposition\n\\ref{prop:embedding}, choose $\\eta$ small\nenough to ensure that $E$ is bi-Lipschitz. Then define\n$F(a)=E(\\eta a)$ on the unit ball $B^n$.\n\n\\begin{prop}\nWe have\n\\[\\mathop \\mathrm{W}\\nolimits\\big(\\Phi_\\#(F(a)),F(a)\\big) = o(|a|)\\]\nand, as a consequence, for all $\\varepsilon>0$\nand all integer $K$, there is a radius $r$ \nsuch that for all $k\\leqslant K$\nand all $a\\in B^n(0,c)$ the following holds:\n\\[\\mathop \\mathrm{W}\\nolimits\\big(\\Phi_{d\\#}^k(F(a)),F(a)\\big)\\leqslant \\varepsilon |a|.\\]\n\\end{prop}\n\n\\begin{proof}\nSince we have restricted ourselves to a finite-dimensional space, \nwe have $\\mathop \\mathrm{W}\\nolimits\\big(\\Phi_\\#(\\rho\\lambda+\\eta\\sum a_iv_i),\n  \\rho\\lambda+\\eta\\sum a_i\\mathscr{L}(v_i)\\big) = o(|a|)$\nand, since $\\mathscr{L}(v_i)=v_i$, we get\n$\\mathop \\mathrm{W}\\nolimits\\big(\\Phi_\\#(F(a)),F(a)\\big) = o(|a|)$.\n\nThe second inequality follows easily. The map $\\Phi_\\#$ is $L$-Lipschitz for some $L>1$\n($L=d$ in the model case, $L>d$ otherwise). For all $\\varepsilon>0$\nand for all integer $K$, let $r>0$ be small enough to ensure that\n\\[|a|<\\delta\\Rightarrow \\mathop \\mathrm{W}\\nolimits\\big(\\Phi_\\#(F(a)),F(a)\\big) \n  \\leqslant \\frac{L-1}{L^{k-1}-1}\\varepsilon |a|.\\]\nThen\n\\begin{eqnarray*}\n\\mathop \\mathrm{W}\\nolimits\\big(\\Phi_{\\#}^k(F(a)),F(a)\\big) \n  &\\leqslant& \\sum_{\\ell=1}^{k-1} \\mathop \\mathrm{W}\\nolimits\\big(\\Phi_{\\#}^\\ell(F(a)),\n       \\Phi_{\\#}^{\\ell-1}(F(a))\\big)\\\\\n  &\\leqslant& \\sum_{\\ell=1}^{k-1} L^{\\ell-1} \\mathop \\mathrm{W}\\nolimits\\big(\\Phi_{d\\#}(F(a)),F(a)\\big)\\\\\n  &\\leqslant& \\varepsilon |a|.\n\\end{eqnarray*}\n\\end{proof}\n\nThis ends the proof of Theorem \\ref{theo:almost-invariant}. It would be \ninteresting to have explicit control on $r$ in terms of $\\varepsilon$,\n$n$ and $K$, and in particular to replace the $o(|a|)$\nby a $O(|a|^\\alpha)$ for some $\\alpha>1$. This seems uneasy because,\neven in the model\ncase where $v_i$ are explicit, we can approximate them by $C^\\infty$\nvector fields $w_i$ with a good control on $(-w_i')^{-1}$ and\n$w'$, but only bad bounds on $w''$ (and therefore the modulus of continuity\nof $w'$).\n\n\n\\subsection{Regularity}\n\nLet us prove that given $\\mu$ an atomless measure and\n$v\\in L^2_0(\\mu)$ (or, indifferently, $v\\in L^2(\\mu)$), for all but\ncountably many values of the parameter $t$, the measure $\\mu+tv$\nhas no atom.\n\n\\begin{proof}[Proof of Proposition \\ref{prop:atomless}]\nBy a line in $T\\mathbb{S}^1\\simeq \\mathbb{S}^1\\times \\mathbb{R}$,\nwe mean the image of a non-horizontal line of $\\mathbb{R}^2$ by\nthe quotient map $(x,y)\\mapsto (x \\mod 1,y)$. We sometimes\nrefer to a line by an equation of one of its lifts in $\\mathbb{R}^2$.\n\nThe measure $\\mu+tv$ has an atom at $s$ if and only if\nthe measure $\\Gamma=(\\mathrm{Id},v)_\\#\\mu$ defined on $T\\mathbb{S}^1$\ngives a positive mass to the line $(x+ty=s)$. Since\n$\\mu$ has no atom, neither does $\\Gamma$, and since two lines intersect in a\ncountable set, the intersection of two lines is  $\\Gamma$-negligible.\nIt follows that there can be at most $n$ different lines that are given\na mass at least $1\/n$ by $\\Gamma$. In particular, at most countably many lines\nare given a positive mass by $\\Gamma$, and the result follows.\n\\end{proof}\n\nFor a general $L^2$ vector field, we cannot hope for more.\nThe following folklore example shows a $L^2_0$ function such that\n$\\lambda +tv$ is stranger to $\\lambda$ for almost all $t$.\n\\begin{exem}\nLet $K$ be a four-corner Cantor set of $\\mathbb{R}^2$.\nMore precisely, $A,B,C,D$ are the vertices of a square,\n$S_A,S_B,S_C,S_D$ are the homotheties of coefficient $1\/4$ centered\nat these points, and $K$ is the unique fixed point of the map\ndefined on compact sets $M\\subset\\mathbb{R}^2$ by\n\\[\\mathscr{S}(M)=S_A(M)\\cup S_B(M)\\cup S_C(M)\\cup S_D(M).\\]\nThe Cantor set $K$ projects on a well-chosen line to an interval,\nsee figure \\ref{fig:Cantor}, while in almost all directions\nit projects to $\\lambda$-negligible sets, see e.g. \n\\cite{Peres-Simon-Solomyak} for a proof.\nChoose the square so that $K$ projects vertically to $[0,1]$ (identified\nto $\\mathbb{S}^1$), and for $x\\in[0,1]$ define $v(x)$ as the least $y$\nsuch that $(x,y)\\in K$. Then $v$ is $L^2$ and, up to a vertical translation,\nwe can even assume that $v\\in L^2_0$. But for almost all $t$,\nthe measure $\\lambda+tv$ is concentrated into a negligible set.\n\\end{exem}\n\n\\begin{figure}\\begin{center}\n\\includegraphics[scale=.5]{Cantor}\n\\caption{A square Cantor set that projects vertically to a segment,\n  but projects in almost all directions to negligible sets.\n  On the right, an approximation of the graph of the function $v$.}\n  \\label{fig:Cantor}\n\\end{center}\\end{figure}\n\n\n\\subsection*{Acknowledgements} I am indebted to Artur Oscar Lopes for his\nnumerous questions and comments on the various versions of this paper,\nand it is a pleasure to thank him.\n\nI also wish to thank Fr\\'ed\\'eric Faure, \n\\'Etienne Ghys, Nicola Gigli, Antoine Gournay, Nicolas Juillet\nand Herv\\'e Pajot for \ninteresting discussions and their comments on earlier versions of this paper,\nand an anonymous referee for her or his constructive criticism.\n\n\\bibliographystyle{smfalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe spatial localization of neurons in the brain plays a critical role since\ntheir connectivity patterns largely depends on their type and their position\nrelatively to nearby neurons and regions (short-range or\/and long-range\nconnections). Interestingly enough, if the neuroscience literature provides\nmany data about the spatial distribution of neurons in different areas and\nspecies (e.g. \\cite{Pasternak:1975} about the spatial distribution of neurons\nin the mouse barrel cortex, \\cite{McCormick:2000} about the neuron spatial\ndistribution and morphology in the human cortex, \\cite{Blazquez-Llorca:2014}\nabout the spatial distribution of neurons innervated by chandelier cells), the\ncomputational literature exploiting such data is rather scarce and the spatial\nlocalization is hardly taken into account in most neural network models (be it\ncomputational, cognitive or machine learning models). One reason may be the\ninherent difficulty in describing the precise topography of a population such\nthat most of the time, only the overall topology is described in term of\nlayers, structures or groups with their associated connectivity patterns (one\nto one, one to all, receptive fields, etc.). One can also argue that such\nprecise localization is not necessary because for some model, it is not\nrelevant (machine learning) while for some others, it may be subsumed into the\nnotion of cell assemblies \\cite{Hebb:1949} that represent the spatiotemporal\nstructure of a group of neurons wired and acting together. Considering cell\nassemblies as the basic computational unit, one can consider there is actually\nfew or no interaction between assemblies of the same group and consequently,\ntheir spatial position is not relevant. However, if cell assemblies allows to\ngreatly simplify models, it also brings implicit limitations whose some have\nbeen highlighted in \\cite{Nallapu:2017}. To overcome such limitations, we\nthink the spatial localization of neurons is an important criterion worth to be\nstudied because it could induces original connectivity schemes from which new\ncomputational properties can be derived as it is illustrated on figure\n\\ref{fig:diffusion}.\\\\\n\\begin{figure}[htbp]\n  \\includegraphics[width=.5\\textwidth]{.\/boots.jpg}\n  \\includegraphics[width=.5\\textwidth]{.\/boots-stipple.png}\n  \\caption{\\textbf{Stippling.} According to\n    Wikipedia\\protect\\footnotemark, {\\em Stippling is the creation of a pattern\n      simulating varying degrees of solidity or shading by using small\n      dots. Such a pattern may occur in nature and these effects are frequently\n      emulated by artists.} The pair of boots (left part) have been first\n    converted into a gray-level image and processed into a stippling figure\n    (right part) using the weighted Voronoi stippling technique by\n    \\cite{Secord:2002} and replicated in \\cite{Rougier:2017}. Image from\n    \\cite{Rougier:2017} (CC-BY license).}\n  \\label{fig:boots}\n\\end{figure}\n\nHowever, before studying the influence of the spatial localisation of neurons,\nit is necessary to design first a method for the arbitrary placement of\nneurons. This article introduces a graphical, scalable and intuitive method for\nthe placement of neurons (or any other type of cells actually) over a\ntwo-dimensional manifold and provides as well the necessary information to\nconnect neurons together using either an automatic mapping or a user-defined\nfunction. This graphical method is based on a stippling techniques originating\nfrom the computer graphics domain for non-photorealistic rendering as\nillustrated on figure \\ref{fig:boots}.\n\n\\footnotetext{\n  Stippling Wikipedia entry at {\\tt https:\/\/en.wikipedia.org\/wiki\/Stippling}}\n\n\\begin{figure}\n  \\includegraphics[width=\\textwidth]{.\/figure-diffusion.pdf}  \n  \\caption{\\textbf{Influence of spatial distribution on signal propagation.}\n    \\textbf{\\textsf{A.}} A k-nearest neighbours (k=5) connectivity pattern\n    shows mid-range connection lengths in low local density areas (left part)\n    and short-range connection lengths in high density areas (right\n    part). \\textbf{\\textsf{B.}} Shortest path from top to bottom using a\n    k-nearest neighbours connectivity pattern (k=5). The lower the density, the\n    shorter the path and the higher the density, the longer the path. On the\n    far left, the shortest path from top to bottom is only 6 connections while\n    this size triples on the far right to reach 19 connections. Said\n    differently, the left part is the fast pathway while the right part is the\n    slow pathway relatively to some input data that would feed the architecture\n    from the top. \\textbf{\\textsf{C.}} Due to the asymmetry of cells position,\n    a signal entering on the top side (materialized with small arrows) travels\n    at different speeds and will consequently reach the bottom side at\n    different times. This represents a spatialization of\n    time. \\textbf{\\textsf{D.}} Due to the asymmetry of cells position, a signal\n    entering on the left side (materialized with small arrows) slows down while\n    traveling before reaching the right side. This represents a compression of\n    time and may serve as a short-term working memory.}\n  \\label{fig:diffusion}\n\\end{figure}\n\n\n\\section{Methods}\n\nBlue noise \\cite{Ulichney:1987} is {\\em an even, isotropic yet unstructured\n  distribution of points} \\cite{Mehta:2012} and has {\\em minimal low frequency\n  components and no concentrated spikes in the power spectrum energy}\n\\cite{Zhang:2016}. Said differently, blue noise (in the spatial domain) is a\ntype of noise with intuitively good properties: points are evenly spread\nwithout visible structure (see figure \\ref{fig:CVT} for the comparison of a\nuniform distribution and a blue noise distribution). This kind of noise has been\nextensively studied in the computer graphic domain and image processing because\nit can be used for object distribution, sampling, printing, half-toning,\netc. One specific type of spatial blue noise is the Poisson disc distribution\nthat is a 2D uniform point distribution in which all points are separated from\neach other by a minimum radius (see right part of figure\n\\ref{fig:CVT}). Several methods have been proposed for the generation of such\nnoise, from the best in quality (dart throwing \\cite{Cook:1986}) to faster ones\n(rejection sampling \\cite{Bridson:2007}), see \\cite{Lagae:2008} for a\nreview. An interesting variant of the Poisson disk distribution is a non\nisotropic distribution where local variations follow a given density function\nas illustrated on figure \\ref{fig:boots} where the density function has been\nspecified using the image gray levels. On the stippling image on the right,\ndarker areas have a high concentration of dots (e.g. boots sole) while lighter\nareas such as the background display a sparse number of dots. There exist\nseveral techniques for computing such stippling density-driven pattern (optimal\ntransport \\cite{Mehta:2012}, variational approach \\cite{Chen:2012}, least\nsquares quantization \\cite{Lloyd:1982}, etc.) but the one by \\cite{Secord:2002}\nis probably the most straightforward and simple and has been recently replicated\nin \\cite{Rougier:2017}.\n\n\\begin{figure}[htbp]\n  \\includegraphics[width=\\textwidth]{.\/figure-CVT.pdf}\n  \\caption{\\textbf{Centroidal Voronoi Tesselation.}  \\textbf{\\textsf{A.}}\n    Voronoi diagram of a uniform distribution (n=256) where black dots\n    represent the uniform distribution and white circles represent the\n    centroids of each Voronoi cells. \\textbf{\\textsf{B.}} Centroidal Voronoi\n    diagram where the point distribution matches the centroid distribution.}\n  \\label{fig:CVT}\n\\end{figure}\n\n\\subsection{Distribution}\n\nThe desired distribution is given through a bitmap RGBA image that provides two\ntypes of information. The three color channels indicates the identity of a cell\n(using a simple formula of the type $identity = 256 \\times 256 \\times R + 256\n\\times G + B$ for $0 \\leq R,G,B < 256$) and the alpha channel indicates the\ndesired local density. This input bitmap has first to be resized (without\ninterpolation) such that the mean pixel area of a Voronoi cell is 500\npixels. For example, if we want a final number of 1000 cells, the input image\nneeds to be resized such that it contains at least 500x1000 pixels. For\ncomputing the weighted centroid, we apply the definition proposed in\n\\cite{Secord:2002} over the discrete representation of the domain and use a\nLLoyd relaxation scheme.\n\\[\n  {\\bf C}_i = \\frac{\\int_A {\\bf x}\\rho({\\bf x})dA}{\\int_A \\rho({\\bf x})}\n\\]\nMore precisely, each Voronoi cell is rasterized (as a set of pixels) and the\ncentroid is computed (using the optimization proposed by the author that allow\nto avoid to compute the integrals over the whole set of pixels composing the\nVoronoi cell). As noted by the author, the precision of the method is directly\nrelated to the size of the Voronoi cell. Consequently, if the original density\nimage is too small relatively to the number of cells, there might be quality\nissues. We use a fixed number of iterations ($n=50$) instead of using the\ndifference in the standard deviation of the area of the Voronoi regions as\nproposed in the original paper. Last, we added a threshold parameter that\nallows to perform a pre-processing of the density image: any pixel with an\nalpha level above the threshold is set to the threshold value before\nnormalizing the alpha channel. Figure \\ref{fig:gradient} shows the distribution\nof four populations with respective size 1000, 2500, 5000 and 10000 cells,\nusing the same linear gradient as input. It is remarkable to see that the local\ndensity is approximately independent of the total number of cells.\n\\begin{figure}\n  \\includegraphics[width=\\textwidth]{.\/figure-density.pdf}  \n  \\caption{\\textbf{Non-uniform distribution (linear gradient).} Different\n    population distribution (size of 1000, 2500, 5000 and 10000 cells) using\n    the same linear gradient as input have been computed. Each distribution has\n    been split into four equal areas and the respective proportion and number\n    of cells present in the area is indicated at the bottom of the area. The\n    proportion of cells present in each areas is approximately independent\n    ($\\pm$2.5\\%) of the overall number of cells. }\n  \\label{fig:gradient}\n\\end{figure}\n\n\n\\subsection{Connection}\n\nMost computational models need to define the connectivity between the different\npopulations that compose the model. This can be done by specifying projections\nbetween a source population and a target population. Such projections\ncorrespond to the axon of the source neuron making a synaptic contact with the\ndendritic tree of the target neuron. In order to define the overall model\nconnectivity, one can specify each individual projection if the model is small\nenough (a few neurons). However, for larger models (hundreds, thousands or\nmillions of neurons), this individual specification would be too cumbersome and\nwould hide any structure in the connectivity scheme. Instead, one can use\ngeneric connectivity description \\cite{Djurfeldt:2014} such as one-to-one,\none-to-all, convergent, divergent, receptive fields, convolutional, etc. For\nsuch connectivity scheme to be enforced, it requires either a well structured\npopulations (e.g. grid) or a simple enclosing topology \\cite{Ekkehard:2015}\nsuch as a rectangle or a disc. In the case of arbitrary shapes as shown on\nfigure \\ref{fig:mapping}, these methods cannot be used directly. However, we\ncan use an indirect mapping from a reference shape such as the unit disc and\ntake advantage of the Riemann mapping theorem that states (definition from\n\\cite{Bolt:2010}):\\\\\n\n\\textbf{Riemann mapping theorem} (from \\cite{Bolt:2010}). {\\em Let $\\Omega$ be\n  a (non empty) simply connected region in the complex plane that is not the\n  entire plane. Then, for any $z_0 \\in \\Omega$, there exists a bianalytic\n  (i.e. biholomorphic) map $f$ from $\\Omega$ to the unit disc such that\n  $f(z0)=0$ and $f'(z0)>0$.}\\\\\n\nSuch mapping is {\\em conformal}, that it, it preserves angles while {\\em\n  isometric} mapping preserves lengths (developable surfaces) and {\\em\n  equiareal} mapping preserves areas. \\citet{Kerzman:1986} introduced a method\nto compute the Riemann mapping function using the Szeg\u00f6 kernel that is\nnumerically stable while \\citet{Trefethen:1980} introduced numerical methods\nfor solving the more specific conformal Schwarz-Christoffel transformation\n(conformal transformation of the upper half-plane onto the interior of a simple\npolygon). Furthermore, a Matlab toolkit is available in \\cite{Driscoll:1996} as\nwell as a Python translation (\\url{https:\/\/github.com\/AndrewWalker\/cmtoolkit})\nthat has been used to produce the figure \\ref{fig:mapping} that shows some examples of\narbitrary shapes and the automatic mapping of the polar and Cartesian domains.\n\\begin{figure}\n  \\includegraphics[width=\\textwidth]{.\/figure-conformal-maps.png}  \n  \\caption{\\textbf{Conformal mappings.} Examples of conformal mappings on\n    arbitrary spline shapes using the conformal Riemann mapping via the Szeg\u00f6\n    kernel \\cite{Kerzman:1986}. Top line shows conformal mapping of the polar\n    domain, bottom line show conformal mapping of the Cartesian domain.}\n  \\label{fig:mapping}\n\\end{figure}\nHowever, even if automatic, this mapping can be perceived as not\nintuitive. Provided the shape are not too distorted, we'll see in the results\nsection that ad-hoc mapping can also be used.\n\n\n\\subsection{Visualization}\n\nHaving now a precise localization for each cell of each population, we have\nseveral ways of visualizing the activity within the model. The most\nstraightforward way is to simply draw the activity of a cell at its position\nusing a disc of varying color (a.k.a. colormap) or varying size, correlated\nwith cell activity. This requires the total number of cells to be not too large\nor the display would be cluttered. For a moderate number of cells, we can take\nadvantage of the dual Voronoi diagram of the cell position as illustrated on\nfigure \\ref{fig:diffusion}, using a colormap to paint the Voronoi\ncell. Finally, if the number of cells is really high, A two-dimensional\nhistogram of the mean activity (with a fixed number of bins) can be used as\nshown on figure \\ref{fig:BG}C using a bicubic interpolation filter.\n\n\n\n\\section{Results}\n\nWe'll now illustrate the use of the proposed method on three different cases.\n\n\\subsection{Case 1: Retina cells}\n\nThe human retina counts two main types of photoreceptors, namely rods, and\ncones (L-cones, M-cones and S-cones). They are distributed over the retinal\nsurface in an non uniform way, with a high concentration of cones (L-cones and\nM-cones) in the foveal region while the rods are to be found mostly in the\nperipheral region with a peak density at around 18-20$^\\circ$ of foveal\neccentricity. Furthermore, the respective size of those cells is different,\nrods being much smaller than cones. The distribution of rods and cells in the\nhuman retina has been extensively studied in the literature and is described\nprecisely in a number of work \\cite{Curcio:1990,Ahnelt:2000}. Our goal here is\nnot to fit the precise distribution of cones and rods but rather to give a\ngeneric procedure that can be eventually used to fit those figures, for a\nspecific region of the retina or the whole retina. The main difficulty is the\npresence of two types of cells having different sizes. Even though there exist\nblue-noise sampling procedures taking different size into account\n\\cite{Zhang:2016}, we'll use instead the aforementioned method using a two\nstage procedure as illustrated on figure \\ref{fig:retina}.\n\n\\begin{figure}\n  \\includegraphics[width=\\textwidth]{.\/figure-rods-cones.pdf}  \n  \\caption{\\textbf{Cones and rods distribution.}  \\textbf{\\textsf{A.}} The\n    density map for cones placement (n=25) is a circular and quadratic gradient\n    with highest density in the center. \\textbf{\\textsf{B.}} The density map\n    for rods placement (n=2500) is built using the rods distribution. Starting\n    from a linear density, ``holes'' with different sized are created at the\n    location of each cone, preventing rods to spread over these areas during\n    the stippling procedure. \\textbf{\\textsf{C.}}  Final distribution of cones\n    and rods. Cones are represented as white blobs (splines) while rods are\n    represented as Voronoi regions using random colors to better highlight the\n    covered area.}\n  \\label{fig:retina}\n\\end{figure}\n\nA first radial density map is created for the placement of 25 cones and the\nstippling procedure is applied for 15 steps to get the final positions of the 25\ncones. A linear rod density map is created where discs of varying (random)\nsizes of null density are created at the position of the cones. These discs will\nprevent the rods to spread over these areas. Finally, the stippling procedure\nis applied a second time over the built density map for 25 iterations. The\nfinal result can be seen on figure \\ref{fig:retina}C where rods are tightly\npacked on the left, loosely packed on the left and nicely circumvent the cones.\n\n\n\\subsection{Case 2: Neural field}\n\nNeural fields describe the dynamics of a large population of neurons by taking\nthe continuum limit in space, using coarse-grained properties of single neurons\nto describe the activity\n\\cite{Wilson:1972,Wilson:1973,Amari:1977,Coombes:2014}. In this example, we\nconsider a neural field with activity $u$ that is governed by an equation of\nthe type:\n\\[\n\\tau\\frac{\\partial u(x,t)}{\\partial t} = -u(x,t) + \\int_{-\\infty}^{+\\infty} w(x,y) f(u(y,t)) dy + I(x) + h\n\\]\nThe lateral connection kernel $w$ is a difference of Gaussian (DoG) with short\nrange excitation and long range inhibition and the input $I(x)$ is constant\nand noisy. In order to solve the neural field equation, the spatial domain has\nbeen discretized into $40 \\times 40$ cells and the temporal resolution has been\nset to $10ms$. On figure \\ref{fig:DNF}A, one can see the characteristic Turing\npatterns that have formed within the field. The number and size of clusters\ndepends on the lateral connection kernel. Figure \\ref{fig:DNF}B shows the\ndiscretized and homogeneous version of the DNF where each cell has been assigned\na position on the field, the connection kernel function and the parameters\nbeing the same as in the continuous version. The result of the simulation shown\non figure \\ref{fig:DNF}B is the histogram of cell activities using $40 \\times\n40$ regular bins. One can see the formation of the Turing patterns that are\nsimilar to the continuous version. On figure \\ref{fig:DNF}C however, the\nposition of the cells have been changed (using the proposed stippling method)\nsuch that there is a torus of higher density. This is the only difference with\nthe previous model. While the output can still be considered to be Turing\npatterns, one can see clearly that the activity clusters are precisely\nlocalized onto the higher density regions. Said differently, the functional\nproperties of the field have been modified by a mere change in the\nstructure. This tends to suggest that the homogeneous condition of neural fields\n(that is the standard hypothesis in most works because it facilitates the\nmathematical study) is actually quite a strong limitation that constrains the\nfunctional properties of the field.\n\n\n\\begin{figure}[htbp]\n  \\begin{center}\n  \\includegraphics[width=.32\\textwidth]{.\/figure-DNF-A.pdf}\n  \\includegraphics[width=.32\\textwidth]{.\/figure-DNF-B.pdf}\n  \\includegraphics[width=.32\\textwidth]{.\/figure-DNF-C.pdf}\n  \\end{center}\n  \\caption{\\textbf{Non-homogeneous discrete neural field}.\n    \\textbf{\\textsf{A.}}  Turing patterns resulting from a continuous and\n    homogeneous neural field with constant and noisy\n    input. \\textbf{\\textsf{B.}} Turing patterns resulting from a discrete and\n    homogeneous neural field with constant and noisy input. White dots indicate\n    the position of the cells. Mean activity is compute from the histogram of\n    cells activity using $40 \\times 40$ bins. \\textbf{\\textsf{C.}} Localized\n    Turing patterns resulting from a discrete and non-homogeneous neural field\n    with constant and noisy input. White dots indicate the position of the\n    cells. Mean activity is computed from the histogram of cells activity using\n    $40 \\times 40$ bins. }\n  \\label{fig:DNF}\n\\end{figure}\n\n\n\n\n\\subsection{Case 3: Basal ganglia}\n\nThe basal ganglia is a group of sub-cortical nuclei (striatum, globus pallidus,\nsubthamalic nucleus, subtantia nigra) associated with several functions such as\nmotor control, action selection and decision making. There exists a functional\ndissociation of the ventral and the dorsal part of the striatum (caudate,\nputamen and nucleus accumbens) that is believed to play an important role in\ndecision making \\cite{ODoherty:2004,Balleine:2007,Meer:2011} since these two\nregions do not receive input from the same structures. For a number of models,\nthis functional dissociation results in the dissociation of the striatum into\ntwo distinct neural groups even though such anatomical dissociation does not\nexist {\\em per se} (see \\cite{Humphries:2010}). Without any proper topography\nof the striatal nucleus, it is probably the most straightforward way to\nproceed. However, if each group would possess its own topography, it would become\npossible to distinguish the ventral from the dorsal part of the BG, as\nillustrated on figure \\ref{fig:BG} on a coronal view of the BG. We do not\npretend this simplified view is sufficient to give account on all the intricate\nconnections between the different nuclei composing the basal ganglia, but it\nmight nonetheless help to have better understanding of the structure because it\nbecomes possible to link external input to specific part of this or that\nstructure (eg. ventral or dorsal part of the striatum). This could lead to\ndifferential processing in different part of the striatum and may reconcile\ndifferent theories regarding the role of the ventral and the dorsal part.\n\n\\begin{figure}\n  \\includegraphics[width=\\textwidth]{.\/figure-BG.pdf}  \n  \\caption{\\textbf{Coronal view of the basal ganglia.} \\textbf{\\textsf{A.}}\n    Scalable Vector Graphic (SVG) source file defining each structure in terms\n    of border (solid black lines), major and minor axis (dashed lines), input\n    (red line) and output (blue line). Local density is given by the alpha\n    channel and structure identity is given by the color. In this coronal view\n    of the basal ganglia, the Caudate is red (RGB=(0.83,0.15,0.15)), the GPe is\n    blue (RGB=(0.12,0.46,0.70)) and the GPi is green (RGB=(0.17,0.62,0.17)).\n    \\textbf{\\textsf{B.}}  Distribution of 2500 neurons respecting the local\n    density and structural organization (Caudate: 1345 cells, GPe: 884 cells,\n    GPi: 271 cells). Neurons receiving input are drawn in red, neurons sending\n    output are drawn in blue.  Each neuron possesses two set of coordinates:\n    one global Cartesian coordinate set and a local curvilinear coordinates set\n    defined as the distances to the major and the minor axis of the structure\n    the neuron belongs to. \\textbf{\\textsf{C.}}  Mean activity histogram of the\n    different structures using 32x32 bins and a bi-cubic interpolation\n    filter. Each bin includes from zero to several neurons. \\textbf{\\textsf{D.}}\n    Cell activities represented using the dual Voronoi diagram of the cell\n    position. Each Voronoi region is painted according to the activity of the\n    corresponding centroid (i.e. neuron). }\n    \\label{fig:BG}\n\\end{figure}\n\n\n\n\n\\section{Discussion}\n\nWe've introduced a graphical, scalable and intuitive method for the placement\nand the connection of biological cells and we illustrated its use on three\nuse-cases. We believe this method, even if simple and obvious, might be\nworth to be considered in the design of a new class of model, in between\nsymbolic model and realistic model. Our intuition is that such topography may\nbe an important aspect that needs to be taken into account and studied in order\nfor the model to benefit from structural functionality. Furthermore, the\nproposed specification of the architecture as an SVG file associated with the\nscalability of the method could guarantee to some extent the scalability of the\nproperties of the model.\\\\\n\n\\textbf{Notes:} All figures were produced using the Python scientific stack,\nnamely, SciPy \\cite{Jones:2001}, Matplotlib \\cite{Hunter:2007} and NumPy\n\\cite{Walt:2011}. All sources are available on GitHub \\cite{rougier:2017b}\n\n\n\\renewcommand*{\\bibfont}{\\small}\n\\printbibliography[title=References]\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn traditional quantum physics courses at the undergraduate level, only linear Hermitian operators are discussed, keeping the conventional wisdom that a quantum observable in a measurement experiment must possess real eigenvalues and the Hermiticity property of it ensures that. However, later Bender and \nBoettcher~\\cite{bender:boettcher:prl98} showed that Hermiticity is not a necessary condition (though sufficient) for an observable (say, Hamiltonian) to have real eigenvalues. If a Hamiltonian preserves the parity ($\\pazo{P}$) and time-reversal ($T$) symmetry, it still can exhibit real eigenvalues or eigenenergies within a certain parameter regime. Such Hamiltonians are dubbed $\\pazo{PT}$ symmetric Hamiltonians. As just mentioned, beyond one or more particular points in the parameter space, the Hamiltonian starts picking up complex eigenenergies and those special points are labeled as \\emph{exceptional points} (EPs). An EP is the degeneracy point where the complex eigenenergies coalesce. However, unlike the Hermitian degeneracy point, the eigenfunctions become identical (up to a phase factor) instead of being orthogonal to each other. EPs have been interesting for the past decades as they have been the points signaling phase transitions ($\\pazo{PT}$ broken). EPs can signal several exotic phenomena such as unidirectional invisibility~\\cite{lin:etal:christo:gr:prl11,regensburger:etal:nat12,zhu:etal:ol13,feng:etal:nmat13}, loss-induced transparency~\\cite{guo:etal:prl09}, topological mode switching or energy transfer~\\cite{liu:etal:pra21,geng:etal:prsa21}, single mode lasing operation~\\cite{hodaei:etal:sc14,feng:etal:sc14}, on-chip control of light propagation~\\cite{peng:etal:nphys14}, optical sensitivity against external perturbation~\\cite{wiersig:prl14,lin:etal:christo:gr:prl11,hodaei:etal:nat17}, and dynamic phase transition in condensed matter systems~\\cite{tripathi:galda:barman:vinokur:prb16}.   \n\n\nTo demonstrate the possibility of real eigenvalues out of a non-Hermitian matrix (which turns out to be $\\pazo{PT}$\nsymmetric), let us consider a simple two-level or two-state system that can be defined by the following $2\\times 2$ matrix.\n\\begin{align}\n{\\bf H}_\\text{TLS}=\n  \\begin{bmatrix}\n   \\epsilon_1 & 0\\\\\n   0     & \\epsilon_2 \n  \\end{bmatrix}\\,.\n\\end{align}\nHere the eigenenergies $\\epsilon_1$ and $\\epsilon_2$ denote the two separate quantum states (if $\\epsilon_1\\ne \\epsilon_2$) or degenerate quantum states (if $\\epsilon_1=\\epsilon_2$).  Now if there is mixing between the separated states (say, due to photon absorption\/emission, a particle from the lower\/higher energy level reaches the higher\/lower energy level), we get a finite off-diagonal term (say, $t$). Then the Hamiltonian looks like\n\\begin{align}\n{\\bf H}_\\text{TLS}^\\text{mix}=\n  \\begin{bmatrix}\n   \\epsilon_1 & t\\\\\n   t     & \\epsilon_2 \n  \\end{bmatrix}\\,.\n\\end{align}\nThe mixing Hamiltonian is also known as the Landau-Zener Hamiltonian in the context of avoided level \ncrossing~\\cite{rubbmark:etal:pra81,shevchenko:ashhab:nori:pr10}.\nIf $\\epsilon_1$, $\\epsilon_2$, and $t$ are real, ${\\bf H}_\\text{TLS}$ and ${\\bf H}_\\text{TLS}^\\text{mix}$ are \nHermitian as they satisfy the Hermiticity condition $a_{ji}^*=a_{ij}$ where $a_{ij}$ is the matrix element at $i$-th row and $j$-th column. Now if we make the diagonal parts complex: $\\epsilon_1=\\epsilon+i\\gamma$ and $\\epsilon_2=\\epsilon-i\\gamma$ (gain term $i\\gamma$ and loss term $-i\\gamma$ added to a degenerate energy level $\\epsilon$),\nwe have\n\\begin{align}\n{\\bf H}_\\text{TLS}^1=\n  \\begin{bmatrix}\n   \\epsilon+i\\gamma & t\\\\\n   t     & \\epsilon-i\\gamma \n  \\end{bmatrix}\\,\n=\\epsilon{\\bf 1}+i\\gamma\\sigma^z+t\\sigma^x\\,.\n\\label{eq:H:TLS:loss:gain:1}\n\\end{align}\nThe Hamiltonian ${\\bf H}_\\text{TLS}^1$ fails to satisfy the Hermiticity condition and hence non-Hermitian. \nHowever, we can easily write down the following eigenvalue or characteristic equation. \n\\blgn\n(E-\\epsilon)^2+\\gamma^2-t^2=0\n\\elgn\nproviding the eigenenergies:\n\\blgn\nE_1,E_2=\\epsilon\\pm \\sqrt{t^2-\\gamma^2}\\,.\n\\elgn \nLike in the previous example, non-Hermiticity can also be introduced via asymmetry in the off-diagonal terms in the TLS matrix, for example,\n\\blgn\n{\\bf H}_{\\text{TLS}}^{2}=\n  \\begin{bmatrix}\n   \\epsilon & t+\\lambda\\\\\n   t-\\lambda     & \\epsilon \n  \\end{bmatrix}\\,\n\\elgn\nleading to the  characteristic equation:\n\\blgn\n(E-\\epsilon)^2=\\lambda^2-t^2\n\\elgn\nwhich provides the eigenenergies:\n\\blgn\nE_1,E_2=\\epsilon\\pm \\sqrt{t^2-\\lambda^2}\\,.\n\\label{eq:Es:TLS:2}\n\\elgn\nDespite ${\\bf H}_{\\text{TLS}}^{1}$ and ${\\bf H}_{\\text{TLS}}^{2}$ being non-Hermitian, their characteristic\nequations show that eigenenergies can become real within certain non-Hermiticity parameter regimes: $|\\gamma|\\le t$ and $|\\lambda|\\le t$ respectively while these parameters are real. Both these Hamiltonians preserve the $\\pazo{PT}$ symmetry~\\cite{wang:ptrsa13} and beyond the above-mentioned regimes, complex eigenenergies emerge leading to $\\pazo{PT}$ symmetry broken phases. In our paper, we shall address both these scenarios and study the nature of EPs. We dub the first kind of Hamiltonian (${\\bf H}_{\\text{TLS}}^{1}$) \\emph{diagonal or orbital} $\\pazo{PT}$-symmetric and the second kind ((${\\bf H}_{\\text{TLS}}^{2}$) \\emph{off-diagonal or kinetic}  $\\pazo{PT}$-symmetric. We construct both of these scenarios in the context of the hydrogen molecule: our testing model.\n\n \nOur paper is organized in the following way. We first discuss the non-interacting version of the hydrogen molecule\nand how the eigenenergies are obtained after constructing the basis set and the Hamiltonian matrix upon that. \nThen we introduce the asymmetry into the hopping elements keeping the $\\pazo{PT}$-symmetry reserved for the \nHamiltonian and discuss the behavior of its complex eigenenergies. We then introduce the Hubbard interaction\nterm to that and discuss the complex eigenenergies. Finally, we add complex gain and loss terms to the orbital energies (maintaining the $\\pazo{PT}$-symmetry again) and discuss the existence of multiple sets of EPs and their dependence on the interaction strength.  \n  \n\n\n\n\\section{Noninteracting hydrogen molecule}\n A hydrogen molecule consists of two hydrogen atoms where each atomic electron participates in covalent bonding with the other one. This scenario (neglecting vibrational modes and other interactions) can be modeled by a two-site electronic problem where electrons can hop from one site to another site (mimicking the orbital overlap)~\\cite{book:ashcroft:mermin76:ssp,alvarez:blanco:ejp01}. In the second quantization notation, the Hamiltonian is equivalent to the two-site tight-binding Hamiltonian:\n\\begin{align}\n\\hat H^0 = \\epsilon\\sum_\\sigma (c\\y_{1\\sigma} c\\py_{1\\sigma} + c\\y_{2\\sigma} c\\py_{2\\sigma}) + t\\sum_\\sigma (c\\y_{1\\sigma} c\\py_{2\\sigma} + c\\y_{2\\sigma} c\\py_{1\\sigma})\\,\n\\label{eq:H0}\n\\end{align}\nwhere $c\\y_{i\\sigma}$ or $c\\py_{i\\sigma}$ operator creates or annihilates an electron of spin $\\sigma$ at site $i$ ($i\\in 1,2$; $\\sigma \\in \\uparrow,\\downarrow$) $\\big[ c\\y_{i\\sigma}|0\\rangle_i=|\\sigma\\rangle_i$; $c\\py_{i\\sigma}|\\sigma\\rangle_i=|0\\rangle_i \\big]$, $\\epsilon$ is the atomic energy of a hydrogen atom, $t$ is the amplitude of hopping from site 1 to site 2 or vice versa. \n\nWe get six possible atomic states for the above Hamiltonian which form the basis $\\{\\ket{i}\\}$, $i= 1,2,3,4,5,6$, the nonzero matrix elements of the Hamiltonian are (see Appendix~\\ref{app:construct:H0})\n\\blgn\nH^0_{11}&=H^0_{22}=H_{33}=H_{44}=H_{55}=H_{66}=2\\epsilon\\\\\nH^0_{23}&=t=H^0_{32}\\\\\nH^0_{24}&=-t=H^0_{42}\\\\\nH^0_{35}&=t=H^0_{53}\\\\\nH^0_{45}&=-t=H^0_{54}\n\\elgn\nwhere $H_{ij}=\\bra{i} \\hat H \\ket{j}$ for a generic Hamiltonian matrix element. Thus the Hamiltonian \nappears in the matrix form: \n\\begin{align}\n{\\bf H^0}=\n  \\begin{bmatrix}\n  \n     2\\epsilon &0     &0      &0     &0     &0 \\\\ \n     0     &2\\epsilon &t      &-t    &0     &0 \\\\ \n     0     &t     &2\\epsilon  &0     &t     &0 \\\\ \n     0     &-t    &0      &2\\epsilon &-t    &0 \\\\ \n     0     &0     &t      &-t     &2\\epsilon &0 \\\\ \n     0     &0     &0      &0     &0     &2\\epsilon\n  \\end{bmatrix}\n  \\,.\n\\end{align}\n\nThe above matrix can be divided into three block-diagonal matrices and one can note\nthey represent three distinguished sectors of total spin $S_z=1,0,-1$ (considering each electron \nis a spin-$\\frac{1}{2}$ particle):\n\\begin{align}\n{\\bf H^0}\n&=\n\\begin{bmatrix}\n  ~\\boxed{S_z=1} &  &\\\\\n    &\\boxed{S_z=0} &\\\\\n    &  &\\boxed{S_z=-1}\n\\end{bmatrix}\n\\,.\n\\end{align}\nFor $S_z=\\pm 1$, the eigenenergies are trivial: $E=2\\epsilon$. \nFor $S_z=0$ matrix:\n\\blgn\n\\begin{bmatrix}\n  2\\epsilon &t      &-t    &0\\\\ \n  t     &2\\epsilon  &0     &t \\\\ \n   -t   &0      &2\\epsilon &-t \\\\ \n  0     &t      &-t     &2\\epsilon\\\\ \n\\end{bmatrix}\n\\,,\n\\label{eq:Sz:0:matrix}\n\\elgn\nthe characteristic equation becomes \n\\blgn\n\\begin{vmatrix}\n  2\\epsilon-E &t      &-t    &0\\\\ \n  t     &2\\epsilon-E  &0     &t \\\\ \n   -t   &0      &2\\epsilon-E &-t \\\\ \n  0     &t      &-t     &2\\epsilon-E\\\\ \n\\end{vmatrix}\n=0\\,\n\\elgn\n\\blgn\n\\Rightarrow (2\\epsilon-E)^2[(2\\epsilon-E)^2 - 4t^2]=0\n\\elgn\nsolving which we obtain the following eigenenergies:\n$2\\epsilon$ (degeneracy=4), $2(\\epsilon-t)$, and $2(\\epsilon+t)$.\nBy setting $\\epsilon$ to 0, we get: $0$, $-2t$, and $2t$ as three distinct eigenenergies.\nFor positive values of $t$, the states with eigenenergy $\\pm 2t$ correspond to \nantibonding (energy  $> \\epsilon$) and bonding states (energy $< \\epsilon$) respectively.\n \n\n\n\n\n\\section{Non-interacting hydrogen molecule with off-diagonal $\\pazo{PT}$ symmetry}\nOpen quantum systems or dissipative systems have been studied for a long time \nwhere non-Hermiticity occurs naturally as a decay term in the Hamiltonian~\\cite{frensley:rmp90,dalibard:etal92,hatano:nelson:prl96,fukui:kawakmi:prb98,bertlmann:etal:pra06}. \nIn our model Hamiltonian $H^0$, we introduce non-Hermiticity through the following dissipative current (asymmetric hopping) term $H^\\lambda$~\\cite{cabib:prb75}.\n\\blgn\n\\hat H^\\lambda = \\lambda  \\sum_\\sigma(c\\y_{1\\sigma} c\\py_{2\\sigma} - c\\y_{2\\sigma} c\\py_{1\\sigma})\\,.\n\\elgn\nOne can easily check that\n$\\hat H\\y_\\lambda=\\lambda  \\sum_\\sigma(c\\y_{2\\sigma} c\\py_{1\\sigma} - c\\y_{1\\sigma} c\\py_{2\\sigma})\\ne \\hat H^\\lambda$.\nWe rewrite our new Hamiltonian as\n\\blgn\n\\hat H^1\n&=H^0+H^\\lambda\\nonumber\\\\\n&= \\epsilon\\sum_\\sigma (c\\y_{1\\sigma} c\\py_{1\\sigma} + c\\y_{2\\sigma} c\\py_{2\\sigma}) + \\sum_\\sigma[t^+ c\\y_{1\\sigma} c\\py_{2\\sigma} + t^{-}c\\y_{2\\sigma} c\\py_{1\\sigma}]\\,\n\\label{eq:H:primed}\n\\elgn \nwhere $t^+\\equiv t+\\lambda;\\quad t^-\\equiv t-\\lambda$.\n\n\\subsection*{$\\pazo{PT}$ symmetry:} \nSince $\\hat H$ is already Hermitian and hence also $\\pazo{PT}$ symmetric, to prove that $\\hat H^1$ \nis $\\pazo{PT}$ symmetric as well, we only need to show that $\\hat H^\\lambda$ is $\\pazo{PT}$ symmetric.\n$\\lambda$ is equivalent to a hopping amplitude and hence it changes sign under time-reversal:  \n\\blgn\n\\pazo{T} \\hat H^\\lambda \\pazo{T}^{-1} = -\\lambda \\sum_\\sigma(c\\y_{1\\sigma} c\\py_{2\\sigma} - c\\y_{2\\sigma} c\\py_{1\\sigma})\\,. \n\\elgn\nNow under parity ($\\pazo{P}$) operation, site 1 and 2 get interchanged and we finally obtain \n\\blgn\n\\pazo{P}\\pazo{T} \\hat H^\\lambda\\pazo{T}^{-1}\\pazo{P}^{-1} = -\\lambda \\sum_\\sigma(c\\y_{2\\sigma} c\\py_{1\\sigma} - c\\y_{1\\sigma} c\\py_{2\\sigma})  \n= \\hat H^\\lambda\\,. \n\\elgn \nHence $\\hat H^\\lambda$ is invariant under $\\pazo{PT}$ symmetry operation and \nthe Hamiltonian in matrix form: \n\\begin{align}\n{\\bf H^1}=\n  \\begin{bmatrix}\n  \n     2\\epsilon &0     &0      &0     &0     &0 \\\\ \n     0     &2\\epsilon &t^-      &-t^-    &0     &0 \\\\ \n     0     &t^+     &2\\epsilon  &0     &t^-     &0 \\\\ \n     0     &-t^+    &0      &2\\epsilon &-t^-    &0 \\\\ \n     0     &0     &t^+      &-t^+     &2\\epsilon &0 \\\\ \n     0     &0     &0      &0     &0     &2\\epsilon\n  \\end{bmatrix}\n  \\,.\n\\label{eq:H1:matrix}\n\\end{align}\nLike in the earlier case, we find this matrix also bears a block-diagonal form where the blocks represent three distinguished sectors of total spin ($S_z$) 1, 0 and -1 respectively.\nThe characteristic equation of the $S_z=0$ block is \n\\blgn\n\\begin{vmatrix}\n  2\\epsilon-E &t_-      &-t_-    &0\\\\ \n  t_+     &2\\epsilon-E  &0     &t_- \\\\ \n   -t_+   &0      &2\\epsilon-E &-t_- \\\\ \n  0     &t_+      &-t_+     &2\\epsilon-E\\\\ \n\\end{vmatrix}\n= 0\\,.\n\\elgn\n\\begin{comment}\n\\blgn\n\\Rightarrow\n(2\\epsilon-E)\\, \n\\begin{vmatrix}\n2\\epsilon-E & 0 & t_- \\\\ \n0 & 2\\epsilon-E & -t_- \\\\\nt_+ & -t_+ & 2\\epsilon-E\n\\end{vmatrix}\n-t_-\n\\begin{vmatrix}\nt_+ & 0 & t_- \\\\ \n-t_+ & 2\\epsilon-E & -t_- \\\\\n0 & -t_+ & 2\\epsilon-E \n\\end{vmatrix}\n\\\\-t_-\n\\begin{vmatrix}\nt_+ & 2\\epsilon-E & t_- \\\\ \n-t_+ & 0 & -t_- \\\\\n0 & t_+ & 2\\epsilon-E \n\\end{vmatrix}\n=0\n\\elgn\n\\blgn\n&\\Rightarrow (2\\epsilon-E) \\bigg[\n    (2\\epsilon-E)\\big\\{(2\\epsilon-E)^2-t_+t_-\\big\\}\n    +t_-\\big\\{-t_+(2\\epsilon-E)\\big\\}\n\\bigg]\\nonumber\\\\\n&\\quad -t_-\\bigg[\nt_+\\big\\{(2\\epsilon-E)^2-t_+t_-\\big\\}+t_+(t_+t_-)\n\\bigg]\\nonumber\\\\\n&\\quad -t_-\\bigg[\nt_+(+t_+t_-)-(2\\epsilon-E)\\big\\{-t_+(2\\epsilon-E)\\big\\}\n+t_-(-t_+^2)\n\\bigg]=0\\\\\n\\elgn\n\\end{comment}\n\\blgn\n&\\Rightarrow (2\\epsilon-E)^2[(2\\epsilon-E)^2-2t_+t_-]\n-2\\tmt_+(2\\epsilon-E)^2=0\\nonumber\\\\\n&\\Rightarrow (2\\epsilon-E)^2[(2\\epsilon-E)^2-4t_+t_-]=0\\,.\n\\label{eq:sing:diss:nonint}\n\\elgn\nThus the eigenenergies of $\\hat H^1$ are $2\\epsilon$ (degeneracy 4), $2(\\epsilon\\pm \\sqrt{t^2-\\lambda^2})$. \nWhen $|\\lambda|>t$ situation occurs, the last two eigenenergies (we name this pair as $E^\\pm$) \nbecome complex: $E^\\pm=2(\\epsilon\\pm i\\sqrt{\\lambda^2-t^2})$.\nThus symmetrically around $\\lambda=0$, a pair of EPs arise at $\\lambda_e=\\pm t$ in the parameter space of $\\lambda$. \nIn \\fref{fig:ImE:vs:lambda:nonint}, we plot the real and imaginary parts of $E^\\pm$ as functions of $\\lambda$. For our parameter choice $t=1$ and $\\epsilon=0.5$, we find at $|\\lambda|\\ge t$, the real parts become zero and the imaginary parts become finite, signifying EPs at $\\lambda_e=\\pm t=\\pm 1$. The eigenenergies are very similar to that of the typical TLS Hamiltonian in ~\\eref{eq:Es:TLS:2} discussed in the Introduction. \n\\begin{figure}[tph!]\n\\subfigure[]{\n\\centering\\includegraphics[totalheight=6cm]{.\/ImE_vs_lambda_nonint.eps}\n\\label{fig:ImE:vs:lambda:nonint}\n}\n\\subfigure[]{\n\\centering\\includegraphics[totalheight=6cm]{.\/ReE_vs_lambda_nonint.eps}\n\\label{fig:ReE:vs:lambda:nonint}\n}\n\\caption{(a) Imaginary and (b) real parts of the two complex eigenenergies of the Hamiltonina $H^1$ plotted as functions of $\\lambda$ for $t=1.0$, $\\epsilon=0.5$.}\n\\label{fig:E:vs:lambda:nonint}\n\\end{figure}\n\n\n\n\\section{Hubbard hydrogen molecule with off-diagonal $\\pazo{PT}$ symmetry}\nWe turn on the Coulomb interaction between the atoms in the hydrogen molecule and for simplicity, we consider it be the on-site Hubbard interaction ($H^U$) which is routinely used in studies of correlated materials~\\cite{book:gebhard10:mott:mit,book:ashcroft:mermin76:ssp}.  The Hubbard interaction term is expressed as\n\\blgn\nH^U \\equiv U(\\hat{n}_{1\\uparrow}\\hat{n}_{1\\downarrow}+\\hat{n}_{2\\uparrow}\\hat{n}_{2\\downarrow})\n\\elgn\nwhere $\\hat n_{i\\sigma}$ is the occupation number operator (${\\hat n}_{i\\sigma}=c\\y_{i\\sigma} c_{i\\sigma}$) and $U$ \namounts to the Coulomb energy one must pay to bring two electrons of opposite spins together. \nThe full interacting Hamiltonian then becomes\n\\blgn\nH^2 = H^0+ H^\\lambda + H^U = H^1 + H^U\\,.\n\\elgn\nSince $\\hat n_{i\\sigma}$ is the occupation number operator, we can easily notice\n\\blgn\nH^2\\ket{1}&=0\\\\\nH^2\\ket{2}&=U\\ket{2}\\\\\nH^2\\ket{3}&=0\\\\\nH^2\\ket{4}&=0\\\\\nH^2\\ket{5}&=U\\ket{5}\\\\\nH^2\\ket{6}&=0\\\\\n\\elgn\nWorking with the same basis states as before, the total Hamiltonian in matrix form can be written as the sum of the respective matrices for $H^U$ and $H^1$:\n\\begin{align}\n{\\bf H^2}\n=\n  \\begin{bmatrix}\n  \n     2\\epsilon &0     &0      &0     &0     &0 \\\\ \n     0     &2\\epsilon+U &t^-      &-t^-    &0     &0 \\\\ \n     0     &t^+     &2\\epsilon  &0     &t^-     &0 \\\\ \n     0     &-t^+    &0      &2\\epsilon &-t^-    &0 \\\\ \n     0     &0     &t^+      &-t^+     &2\\epsilon+U &0 \\\\ \n     0     &0     &0      &0     &0     &2\\epsilon\n  \\end{bmatrix}\n  \\,.\n\\label{eq:H2:Sz:0:matrix}\n\\end{align}\n The characteristic equation for the $S_z=0$ sector of \\eref{eq:H2:Sz:0:matrix} is \n\\blgn\n\\begin{vmatrix}\n  2\\epsilon+U-E &t_-      &-t_-    &0\\\\ \n  t_+     &2\\epsilon-E  &0     &t_- \\\\ \n   -t_+   &0      &2\\epsilon-E &-t_- \\\\ \n  0     &t_+      &-t_+     &2\\epsilon+U-E\\\\ \n\\end{vmatrix}\n=0\n\\nonumber\n\\elgn\n\\begin{figure}[tph!]\n\\subfigure[]{\n\\centering\\includegraphics[totalheight=6cm]{.\/ImE_vs_lambda_int.eps}\n\\label{fig:ImE:vs:lambda:int}\n}\n\\subfigure[]{\n\\centering\\includegraphics[totalheight=6cm]{.\/ReE_vs_lambda_int.eps}\n\\label{fig:ReE:vs:lambda:int}\n}\n\\subfigure[]{\n\\centering\\includegraphics[totalheight=6cm]{.\/lambda_e_vs_U.eps}\n\\label{fig:le:vs:U}\n}\n\\caption{(a) Imaginary  and (b) real parts of the two complex eigenenergies plotted as functions of $\\lambda$ for $t=1.0$, $\\epsilon=0.5$, and $U=2.0$. (c) The exceptional points positions $|\\lambda_e|$ varying with \nHubbard interaction strength $U$ marks the boundary between $\\pazo{PT}$ broken and unbroken phases.}\n\\label{fig:eig:int}\n\\end{figure}\n\n\\blgn\n&\\Rightarrow (2\\epsilon-E)(2\\epsilon+U-E)\\nonumber\\\\\n&\\qquad\\times\\bigg[(2\\epsilon-E)(2\\epsilon+U-E)-4t_+t_-\\bigg]=0\n\\label{eq:sing:diss:Hubb:form0}\\\\\n&\\Rightarrow (2\\epsilon-E)(2\\epsilon+U-E)\\nonumber\\\\\n&\\qquad\\times\\bigg[(2\\epsilon-E+U\/2)^2-U^2\/4-4t_+t_-\\bigg]=0\\,.\n\\label{eq:sing:diss:Hubb}\n\\elgn\nThus the eigenenergies of $\\hat H^2$ are $2\\epsilon$ (degeneracy 3), $2\\epsilon+U$, $\\frac{1}{2}(4\\epsilon\\pm \\sqrt{16t_+t_- + U^2}+U)=\\frac{1}{2}(4\\epsilon\\pm \\sqrt{16(t^2-\\lambda^2) + U^2}+U)$. We can check that by setting $U=0$ in \\eref{eq:sing:diss:Hubb}, we get back the non-interacting limit (\\eref{eq:sing:diss:nonint}). \nWe have complex eigenenergies when the discriminant  (term inside the square root) becomes negative, i.e. when $|\\lambda|> \\sqrt{t^2+U^2\/16}$. \nThus presence of interaction shifts the positions of the EPs and we have $\\lambda_e=\\pm \\sqrt{t^2+U^2\/16}$. For our choice of parameters: $t=1$, $U=2$, $\\epsilon=0.5$, we find $\\lambda_e\\simeq \\pm 1.118$ (see \\fref{fig:ImE:vs:lambda:int} and \\fref{fig:ReE:vs:lambda:int} for the imaginary and real parts of $E^\\pm$).\n\\fref{fig:le:vs:U} shows $\\lambda_e$ symmetrically shifts from the non-interacting limit ($\\lambda_e(U=0)=1$) as $U$ moves both in positive and negative directions. The parabolic curve for $|\\lambda_e|$ marks the boundary between $\\pazo{PT}$ broken and unbroken phases on the $|\\lambda|-U$ plane.\n\n\n\\section{Hubbard hydrogen molecule with diagonal $\\pazo{PT}$ symmetry}\nWe now consider the case when the orbital energies of the hydrogen atoms get tuned to different \nenergy levels by addition of complex loss and gain terms. For simplicity, let $\\eps_+=\\epsilon+i\\gamma$, $\\eps_-=\\epsilon-i\\gamma$ be the energies, i.e. there are equal amounts of loss and gain terms added to the orbital energies. Hence the orbital part of our Hamiltonian becomes  \n\\blgn\n\\hat H^\\gamma= \\eps_+\\sum_\\sigma c\\y_{1\\sigma} c\\py_{1\\sigma} +\\eps_-\\sum_\\sigma c\\y_{2\\sigma} c\\py_{2\\sigma}\\,.\n\\elgn\nTwo-level or two-band systems with loss and gain terms have been successfully realized in several photonic and optical setups~\\cite{person:rotter:stockmann:barth:prl00,makris:elganainy:christodoulides:musslimani:prl08,guo:etal:prl09,feng:elganainy:ge:nphoton17}. Considering both diagonal and off-diagonal non-Hermiticity, our most generic $\\pazo{PT}$ symmetric Hamiltonian reads\n\\blgn\n\\hat H^3 \n&= H^\\lambda + H^\\gamma + H^U\\nonumber\\\\\n&= \\sum_\\sigma\\big[\\eps_+ c\\y_{1\\sigma} c\\py_{1\\sigma} + \\eps_-  c\\y_{2\\sigma} c\\py_{2\\sigma} \n+  t_+ c\\y_{1\\sigma} c\\py_{2\\sigma} + t_- c\\y_{2\\sigma} c\\py_{1\\sigma}\\big]\\nonumber\\\\ \n&\\quad+ U(\\hat{n}_{1\\uparrow}\\hat{n}_{1\\downarrow}+\\hat{n}_{2\\uparrow}\\hat{n}_{2\\downarrow})\\,. \n\\label{eq:H3}\n\\elgn \n\\subsection*{$\\pazo{PT}$ symmetry:}\n$H^\\gamma$ is $\\pazo{PT}$ symmetric as we can check:\nUnder $\\pazo{T}$ operation\n\\blgn\n\\pazo{T} H^\\gamma \\pazo{T}^{-1}= \\sum_\\sigma\\big[\\eps_- c\\y_{1\\sigma} c\\py_{1\\sigma} + \\eps_+ c\\y_{2\\sigma} c\\py_{2\\sigma}\\big]   \n\\elgn\nand under $\\pazo{PT}$ operation\n\\blgn\n\\pazo{P}\\pazo{T} H^\\gamma\\pazo{T}^{-1} \\pazo{P}^{-1} = \\sum_\\sigma\\big[\\eps_- c\\y_{2\\sigma} c\\py_{2\\sigma} + \\eps_+ c\\y_{1\\sigma} c\\py_{1\\sigma}\\big]\n= H^\\gamma\\,.   \n\\elgn\nFollowing the same basis formulation, we get the Hamiltonian in matrix form:\n\\begin{align}\n&{\\bf H^3}\\nonumber\\\\\n&=\n  \\begin{bmatrix}\n  \n     \\eps_++\\eps_- &0     &0      &0     &0     &0 \\\\ \n     0     &2\\eps_-+U &t_-      &-t_-    &0     &0 \\\\ \n     0     &t_+     &\\eps_++\\eps_-  &0     &t_-     &0 \\\\ \n     0     &-t_+    &0      &\\eps_++\\eps_- &-t_-    &0 \\\\ \n     0     &0     &t_+      &-t_+     &2\\eps_++U &0 \\\\ \n     0     &0     &0      &0     &0     &\\eps_++\\eps_-\n  \\end{bmatrix}\n  \\,.\n\\label{eq:H3:Sz:0:matrix}\n\\end{align}\nAgain like in the earlier cases, the $S_z=0$ sector of the block-diagonal form\nyields the characteristic equation:\n\\begin{comment} \n\\blgn\n\\begin{vmatrix}\n  2\\eps_-+U-E &t_-      &-t_-    &0\\\\ \n  t_+     &\\eps_++\\eps_--E  &0     &t_- \\\\ \n   -t_+   &0      &\\eps_++\\eps_--E &-t_- \\\\ \n  0     &t_+      &-t_+     &2\\eps_++U-E\\\\ \n\\end{vmatrix}\\nonumber\\\\\n=0\n\\nonumber\n\\elgn\n\\end{comment} \n\\begin{comment}\n\\blgn\n\\Rightarrow\n&(2\\eps_-+U-E)\\, \n\\begin{vmatrix}\n\\eps_++\\eps_--E & 0 & t_- \\\\ \n0 & \\eps_++\\eps_--E & -t_- \\\\\nt_+ & -t_+ & 2\\eps_++U-E\n\\end{vmatrix}\n-t_-\n\\begin{vmatrix}\nt_+ & 0 & t_- \\\\ \n-t_+ & \\eps_++\\eps_--E & -t_- \\\\\n0 & -t_+ & 2\\eps_++U-E \n\\end{vmatrix}\n\\nonumber\\\\\n&\\qquad\\qquad\n-t_-\n\\begin{vmatrix}\nt_+ & \\eps_++\\eps_--E & t_- \\\\ \n-t_+ & 0 & -t_- \\\\\n0 & t_+ & 2\\eps_++U-E \n\\end{vmatrix}\n=0\\nonumber\\\\\n&\\Rightarrow \n(2\\eps_-+U-E) \\bigg[\n    (\\eps_++\\eps_--E)\\big\\{(\\eps_++\\eps_--E)(2\\eps_++U-E)-t_+t_-\\big\\}\n    +t_-\\big\\{-t_+(\\eps_++\\eps_--E)\\big\\}\n     \\bigg]\\nonumber\\\\\n&\\quad \n-t_-\\bigg[t_+\\big\\{(\\eps_++\\eps_--E)(2\\eps_++U-E)-t_+t_-\\big\\}+\\tmt_+^2\n\\bigg]\\nonumber\\\\\n&\\quad \n-t_-\\bigg[t_+(+t_+t_-)-(\\eps_++\\eps_--E)\\big\\{-t_+(2\\eps_++U-E)\\big\\}\n+t_-(-t_+^2)\n\\bigg]=0\\nonumber\\\\\n\\Rightarrow\n&(2\\eps_-+U-E)\\bigg[(\\eps_++\\eps_--E)\\,\\big\\{(\\eps_++\\eps_--E)(2\\eps_++U-E)-t_+t_-\\big\\}\n-\\tmt_+(\\eps_++\\eps_--E)\\bigg]\\nonumber\\\\\n&\\quad-t_+t_-\\bigg[(\\eps_++\\eps_--E)(2\\eps_++U-E)\\bigg]\\nonumber\\\\\n&\\quad-t_+t_-\\bigg[(\\eps_++\\eps_--E)(2\\eps_++U-E)\\bigg]=0\\nonumber\\\\\n\\elgn\n\\blgn\n\\Rightarrow \n&(2\\eps_-+U-E)(\\eps_++\\eps_--E)\\nonumber\\\\\n&\\quad\\times\\bigg[(\\eps_++\\eps_--E)(2\\eps_++U-E)-2t_+t_-\\bigg]\\nonumber\\\\\n&\\quad -2t_+t_-(\\eps_++\\eps_--E)(2\\eps_++U-E)=0\\nonumber\\\\\n\\end{comment}\n\\blgn\n&(\\eps_++\\eps_--E)\\nonumber\\\\\n&\\quad\\times\\bigg[(2\\eps_-+U-E)(\\eps_++\\eps_--E)(2\\eps_++U-E)\\nonumber\\\\\n&\\quad-4t_+t_-(\\eps_++\\eps_-+U-E)\\bigg]=0\n\\label{eq:doub:diss:Hubb:form0}\\\\\n&\\Rightarrow (\\eps_++\\eps_--E)\\nonumber\\\\\n&\\quad\\times\\bigg[(2\\eps_-+U-E)(\\eps_++\\eps_--E)(2\\eps_++U-E)\\nonumber\\\\\n&\\quad-4t_+t_-(\\eps_++\\eps_--E)-4t_+t_- U\\bigg]=0\\,.\n\\label{eq:doub:diss:Hubb}\n\\elgn\n\\eref{eq:doub:diss:Hubb:form0} reproduces \\eref{eq:sing:diss:Hubb:form0} once we set $\\gamma=0$ (then we have $\\eps_+=\\eps_-=\\epsilon$). The eigenenergies of $\\hat H^3$ are\n$2\\epsilon$ (degeneracy 3), and the three roots of the cubic equation inside the \nbracket of \\eref{eq:doub:diss:Hubb}:\n\\defS{S}\n\\defD{D}\n\\blgn\n&(2\\eps_-+U-E)(\\eps_++\\eps_--E)(2\\eps_++U-E)\\nonumber\\\\\n&\\quad-4t_+t_-(\\eps_++\\eps_--E)-4t_+t_- U = 0\\,\n\\label{eq:cubic:doub:diss:Hubb}\n\\elgn\nwhich can be simplified as (see Appendix~\\ref{app:cubic})\n\\blgn\nX^3-U X^2-K X - L=0\n\\label{eq:doub:diss:Hubb:cubic:form}\n\\elgn\nwith $X\\equiv x+U$; $x\\equiv \\eps_++\\eps_--E$; $K\\equiv 4(t^2-\\gamma^2-\\lambda^2)$; $L\\equiv 4\\gamma^2 U$. \n\n\n\n\\begin{figure}[htp!]\n\\subfigure[]{\n\\includegraphics[totalheight=4.5cm]{.\/ImE_vs_lambda_fixed_gamma.eps}\n\\label{fig:ImE:vs:lambda:fixed:gamma}\n}\n\\subfigure[]{\n\\includegraphics[totalheight=4.5cm]{.\/ReE_vs_lambda_fixed_gamma.eps}\n\\label{fig:ReE:vs:lambda:fixed:gamma}\n}\n\\caption{(a) Imaginary and (b) real parts of the complex eigenenergy pair plotted as a function of  \ndissipative parameter $\\lambda$ for $t=1.0$, $\\epsilon=0.5$, and $U=2.0$ at $\\gamma=0.1$.}\n\\label{fig:E:vs:lambda:fixed:gamma}\n\\end{figure}\nThus once we solve for $X$ in \\eref{eq:doub:diss:Hubb:cubic:form} by typical Cardano's method~\\cite{book:tignol01:galois} or numerically~\\cite{book:press:etal02:nrecipe:inC}, we\nexpect to have at least one real root all the time, the other two roots become complex\nconjugates of each other (since the coefficients of $X$ are real) beyond a certain parameter space. This pair of complex conjugate roots give rise to EPs at the parameter space when the complex roots just become real. \nSince we introduce two kinds of non-Hermiticity via the orbital energy and the hopping \nterms, it may be natural to expect observing additional EPs. These EPs are different from higher order EPs~\\cite{hodaei:etal:nat17}, since we are focusing always\non the pair of energy levels that can become complex in certain parameter regimes, while the other levels always promise to be real. \nWe notice, for a fixed \n$\\gamma$, as we shift $\\lambda$ from zero, $\\text{Im}\\,E^\\pm$ start becoming finite beyond a point $\\lambda_{e1}$, then again disappear at $\\lambda_{e2}$, and then become finite above $\\lambda_{e3}$.   \n (see \\fref{fig:ImE:vs:lambda:fixed:gamma}). $\\lambda_{e1}$, $\\lambda_{e2}$, and $\\lambda_{e3}$: all these are EPs as they are degenerate onset points of imaginary eigenenergies and like in the previous cases, they appear symmetrically around $\\lambda=0$. Though presence of additional EPs can be anticipated due to double non-Hermitian terms in the Hamiltonian and cubic nature of the characteristic equation (\\eref{eq:doub:diss:Hubb:cubic:form}), the behavior of all of them are not alike.\nUnlike the previous cases, the additional EPs break the mirror symmetry between $\\text{Re}\\,E^\\pm$ \nseen in the earlier case: the energy levels are not equally distributed around the EPs (see \\fref{fig:ReE:vs:lambda:fixed:gamma}). These additional EPs are different because the eigenenergies generate from complex conjugate pairs of root of a cubic equation, where the discriminant depends on an additional coefficient compared to the quadratic equation's case.%\nThe asymmetry in the real parts of $E^\\pm$ gets reversed once we change of the sign of $U$. \nThe asymmetry becomes more evident when we plot them against $\\gamma$ for fixed\n$\\lambda$ or even when $H^\\lambda$ is turned off (see \\fref{fig:ReE:vs:gamma:fixed:lambda}). However, when we set $U=0$,\nwe get back symmetric real eigenenergy pair just like a typical TLS (see \\fref{fig:ReE:vs:gamma:fixed:lambda:U0}).\nThis can be easily understood by noticing that \\eref{eq:doub:diss:Hubb:cubic:form} reduces to effectively quadratic equation $x^2-4(t^2-\\gamma^2-\\lambda^2)=0$ (for $t^2\\ne \\gamma^2+\\lambda^2$) which produces typical square-root EPs at $\\gamma_e=\\pm 2\\sqrt{t^2-\\lambda^2}$ and in $\\lambda_e=\\pm 2\\sqrt{t^2-\\gamma^2}$ in $\\gamma$ and $\\lambda$ parameter spaces respectively, similar to the form $\\lambda_e$ has for $H^1$ and $H^2$.\n\\begin{figure}[htp!]\n\\subfigure[]{\n\\centering\\includegraphics[totalheight=4.3cm,clip]{.\/ImE_vs_gamma_fixed_lambda.eps}\n\\label{fig:ImE:vs:gamma:fixed:lambda}\n}\n\\subfigure[]{\n\\centering\\includegraphics[totalheight=4.3cm,clip]{.\/ReE_vs_gamma_fixed_lambda.eps}\n\\label{fig:ReE:vs:gamma:fixed:lambda}\n}\n\\subfigure[]{\n\\centering\\includegraphics[totalheight=4.3cm,clip]{.\/ImE_vs_gamma_fixed_lambda_U0.eps}\n\\label{fig:ImE:vs:gamma:fixed:lambda:U0}\n}\n\\subfigure[]{\n\\centering\\includegraphics[totalheight=4.3cm,clip]{.\/ReE_vs_gamma_fixed_lambda_U0.eps}\n\\label{fig:ReE:vs:gamma:fixed:lambda:U0}\n}\n\\caption{(a) Imaginary and (b) real parts of the complex eigenenergy pair plotted as a function of  \nloss\/gain parameter $\\gamma$ for $t=1.0$, $\\epsilon=0.5$, and $U=2.0$ at $\\lambda=0$. (c) Imaginary and (d) real parts of the complex eigenenergy pair plotted against $\\gamma$ for the non-interacting case ($U=0$) at $\\lambda=0.6$ while other parameters remain the same. In the non-interacting situation, the TLS eigenenergy symmetry is recovered.}\n\\label{fig:E:vs:gamma:fixed:lambda}\n\\end{figure}\nThe $\\pazo{PT}$ broken and unbroken phase diagrams are shown in \\fref{fig:gammae:vs:U}. For no other non-Hermiticity parameter, the phase boundary hits unity in the non-interacting limit ($U=0$) at $t=1$, agreeing with the result recently obtained by Pan {\\it et al.}~\\cite{pan:wang:cui:chen:pra20}. However, as soon as the off-diagonal non-Hermiticity parameter is turned on (e.g. $\\lambda=0.5$ case shown \\fref{fig:gammae:vs:U}), the boundary diminishes implying $PT$-symmetry breaking at lower values of $\\gamma$. \n\\begin{figure}[htp!]\n\\centering\\includegraphics[height=6cm,clip]{.\/gamma_e_vs_U.eps}\n\\caption{$\\pazo{PT}$ broken and unkbroken phases on $\\gamma$-$U$ plane for $t=1$, $\\epsilon=0.5$.\nThe upper and lower curves show the phase boundary for zero and finite ($\\lambda=0$) off-diagonal non-Hermiticity parameters.}\n\\label{fig:gammae:vs:U}\n\\end{figure}\n\n\\subsection*{Dependence of execeptional points on the Hubbard interaction $U$:}\nAs we notice that the presence of three sets of EPs and interaction plays a role in creating an asymmetry in the real eigenvalues, we decide to plot their positions $\\lambda_{e1}$, $\\lambda_{e2}$, and $\\lambda_{e3}$ against the interaction strength. \\fref{fig:lambdae1:vs:U} shows that  $\\lambda_{e1}$ always exists (even when $U=0$) and it decreases as $U$ is increased. On the other hand, \\fref{fig:lambdae2:vs:U} and  \\fref{fig:lambdae3:vs:U} clearly show that both  $\\lambda_{e2}$ and $\\lambda_{e3}$ arise only at a finite value of $U$ and depending on the value of loss-gain parameter $\\gamma$, it monotonically increases with $U$. $\\lambda_{e3}$'s positions do not vary as significantly as $\\lambda_{e2}$'s do for different $\\gamma$ values (e.g. $\\gamma=0.1$ and $\\gamma=0.2$) shown in the figures). In the non-interacting case, the loop structures in $\\text{Im}\\,E^\\pm$ (hence $\\lambda_{e2}$ and $\\lambda_{e3}$) disappear and we only obtain $\\lambda_{e1}$.  \nThus we can categorize two distinguishable kinds of EPs: (A) \\emph{interaction generated} ($\\lambda_{e2}$ and $\\lambda_{e3}$) and (B) \\emph{self-generated}. These interaction generated EPs are different from traditional EPs often discussed in the literature and deserve special attention and further theoretical and experimental research.\n\\begin{figure}[htp!]\n\\subfigure[]{\n\\centering\\includegraphics[totalheight=5cm]{.\/lambdae1_vs_U.eps}\n\\label{fig:lambdae1:vs:U}\n}\n\\subfigure[]{\n\\centering\\includegraphics[totalheight=5cm]{.\/lambdae2_vs_U.eps}\n\\label{fig:lambdae2:vs:U}\n}\n\\subfigure[]{\n\\centering\\includegraphics[totalheight=5cm]{.\/lambdae3_vs_U.eps}\n\\label{fig:lambdae3:vs:U}\n}\n\\caption{Positions of exceptional points (a) $\\lambda_{e1}$, (b) $\\lambda_{e2}$, and (c) $\\lambda_{e3}$ as Hubbard interaction strength $U$ is varied for $\\gamma=0.1$ and $\\gamma=0.2$ keeping $t=1$, $\\epsilon=0.5$.}\n\\label{fig:lambdae1:lambdae2:lambdae3:vs:U}\n\\end{figure}\n\\section{Conclusion}\n$\\pazo{PT}$ symmetric non-Hermitian physics have been successfully observed in several two level photonic and optical systems. \nOne particular feature of such Hamiltonians is the existence of exceptional points (EPs) beyond which complex eigenenergies emerge signaling breaking of the symmetry in the eigenfunctions. As a simplistic model, we consider a hydrogen molecule with Hubbard interaction acting between its atoms' electrons. We then introduce both diagonal and off-diagonal $\\pazo{PT}$ symmetries and notice that\ninteraction plays differently with different kinds of EPs generated by the parameters of the Hamiltonian. Changing the position of one kind of EPs in the increasing direction and the other kind in decreasing direction by varying interaction strength can offer flexibility in fine tuning EPs and more control over their potential applications. In a realistic hydrogen molecule, non-Hermitian loss-gain terms might be introduced through laser induced molecular ionization and dissociation~\\cite{lefebvre:etal:prl09,wrona:etal:srep20}. Besides this, a more precise two-site Hubbard model could be emulated in an ultracold double well system~\\cite{murmann:etal:prl15} or via  NMR~\\cite{melo:etal:nmr:po21}. \nThe role of interaction on the EPs has been studied recently~\\cite{pan:wang:cui:chen:pra20} for the Hubbard interaction. However, the interplay of the diagonal and off-diagonal $\\pazo{PT}$-symmetries and the role\nof interaction on them have not been studied ever to the best of our knowledge. Such interplay might be extended to the fermionic or bosonic lattice Hubbard models and effect on interesting physics such as closure of Mott gap~\\cite{tripathi:galda:barman:vinokur:prb16,tripathi:vinokur:srep20} or multiple $\\pazo{PT}$-broken phases~\\cite{jin:song:ap13} can be studied.\n\n\\section{Acknowledgement and announcement}\nThe authors thank the HBCSE, Mumbai for providing an opportunity to collaborate through their NIUS Physics 15.3 camp.\n\nOur codes are available on the Github repository: \\newline\\url{https:\/\/github.com\/hbaromega\/PT-symmetric-2-site-Hubbard-hydrogen}, \\newline under GNU General Public License.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWe consider periodic adiabatic processes of spinless short-range entangled\nphases with period $T$ at zero temperature. The ultimate goal when attacking\nthis type of time-dependent problems on the general ground would be to obtain\nan expression for the physical observables induced by the time evolution in\nterms of \\textit{instantaneous} eigenstates and eigenenergies of the system.\n\nIn their pioneering works, Niu and Thouless~\\cite{thouless1983,niu1984} found\nsuch an expression for the current operator uniformly averaged over the entire\nspace. In the formulation, they assumed the periodic boundary conditions with\nthe period $L_i$ for $i=x,y$ and introduced the solenoidal flux\n$\\bm{\\phi}=(\\phi_x,\\phi_y)$ as illustrated in Fig.~\\ref{fig:current}. \nFor concreteness we work in two spatial dimensions throughout this work. Then\nthe current operator can be expressed as\n\\begin{equation}\n\\hat{j}_{t\\bm{\\phi}}^i\\equiv\\frac{1}{L_i}\\int d^2x\\hat{j}_{t\\bm{\\phi}}^i(\\bm{x})=\\partial_{\\phi_i}\\hat{H}_{t\\bm{\\phi}},\\quad i=x,y.\\label{eq:currentphi}\n\\end{equation}\n(For brevity we show the dependence on time $t$, flux $\\bm{\\phi}$, and etc., in\nthe subscript.) Further taking an average over all values of $\\bm{\\phi}$, the\nexpectation value of the current operator induced by the adiabatic\ntime-evolution can be expressed as the time derivative of the many-body Berry\nphase for varying $\\bm{\\phi}$\n\\begin{eqnarray}\n\\int\\frac{d^2\\phi}{(2\\pi)^2}\\langle\\hat{\\bm{j}}_{t\\bm{\\phi}}\\rangle=\\partial_t\\left(\\int\\frac{d^2\\phi}{(2\\pi)^2}\\langle\\Phi_{t\\bm{\\phi}}\\vert i\\partial_{\\bm{\\phi}}\\vert\\Phi_{t\\bm{\\phi}}\\rangle\\right),\t\n\t\\label{eq:jx}\n\\end{eqnarray}\nwhere $\\vert\\Phi_{t\\bm{\\phi}}\\rangle$ is the instantaneous ground state of the\nHamiltonian $\\hat{H}_{t\\bm{\\phi}}$.  This expression assumes the periodicity in\n$\\bm{\\phi}$ (see Eqs.~\\eqref{eq:p1}, \\eqref{eq:p2} below). It is \\textit{not}\npossible to further impose the periodicity in time simultaneously. Instead we\nhave $\\vert\\Phi_{T\\bm{\\phi}}\\rangle\n=e^{-i\\bm{\\phi}\\cdot\\bm{Q}}\\vert\\Phi_{0\\bm{\\phi}}\\rangle$, where\n$\\bm{Q}=\\int_0^Tdt\\int\\frac{d^2\\phi}{(2\\pi)^2}\\langle\\hat{\\bm{j}}_{t\\bm{\\phi}}\\rangle\\in\\mathbb{Z}^2$ is the pumped\ncharge during in one cycle.\n\nThe result~(\\ref{eq:jx}) is formally similar to the constitutive relation for\nMaxwell's equations\n\\begin{align}\n\t\\bm j(t, \\bm x)&=\\partial_t\\bm p(t, \\bm x)+\\bm\\nabla\\times\\bm m(t, \\bm x),\\label{eq:jmeso}\n\\end{align}\nwhere $\\bm p$ and $\\bm m$ are the bulk polarization and the bulk magnetization.\nLater it was shown~\\cite{king-smith1993,vanderbilt1993,resta1994,resta2007}\nthat Thouless result~(\\ref{eq:jx}) combined with constitutive\nrelation~(\\ref{eq:jmeso}) gives a useful formula for the bulk polarization ---\nthis development marked the birth of ``the modern theory'' of electric\npolarization. The bulk polarization is given by the integral of the Berry\nconnection  (the \\textit{first} Chern-Simons form) $P_1$ [see\nEq.~\\eqref{eq:modthp} for the formula for band insulators].\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=0.6\\columnwidth]{torus.pdf}\t\t\n\t\t\\caption{\\label{fig:current} Two dimensional system with\n\t\tperiodic boundary conditions viewed as toroidal topology. The\n\t\ttwo solenoidal fluxes are denoted by $\\phi_x$ and $\\phi_y$. \n\t\tThe averaged current operator can be expressed as the\n\tderivative of the Hamiltonian with respect to the fluxes [Eq.~\\eqref{eq:currentphi}].}\n\t\\end{center}\n\\end{figure}\n\nAlternatively, let us impose \\textit{the periodicity in time}\n$\\vert\\Phi_{T\\bm{\\phi}}\\rangle=\\vert\\Phi_{0\\bm{\\phi}}\\rangle$ instead.  In this\nsetting it is useful to integrate over time rather than the solenoidal flux.\nThen the Thouless result reads~\\cite{niu1984}\n\\begin{align}\n&\\int_0^Tdt\\langle \\hat{\\bm{j}}_{t\\bm{\\phi}}\\rangle=-\\partial_{\\bm{\\phi}}\\varphi_{\\bm{\\phi}},\n\t\\label{eq:jTh}\\\\\n&\\varphi_{\\bm{\\phi}}\\equiv \\int_0^Tdt\\langle\\Phi_{t\\bm{\\phi}}\\vert i\\partial_t\\vert\\Phi_{t\\bm{\\phi}}\\rangle,\n\\label{eq:Berry}\n\\end{align}\nwhere $\\varphi_{\\bm{\\phi}}$ is the many-body Berry phase associated with the\nadiabatic time-evolution. In the thermodynamic limit,\n$\\partial_{\\bm{\\phi}}\\varphi_{\\bm{\\phi}}$ is independent of\n$\\bm{\\phi}$~\\cite{niu1984,PhysRevB.98.155137} and one can set\n$\\bm{\\phi}=\\bm{0}$ for instance. There is also a contribution from\n$\\partial_{\\bm{\\phi}}E_{t\\bm{\\phi}}$ in Eqs.~\\eqref{eq:jx}, \\eqref{eq:jTh} but\nit is negligibly small for the same reason. We find this formulation of the\nThouless pump more useful because it can be generalized to wider class of\nphysical observables as we discuss below.\n\nThe persistent current associated with a part of the orbital magnetization can also\nbe expressed using the instantaneous eigenstates and eigenenergies of the\nHamiltonian.  For band insulators, it can be written as the curl of a vector\n(see Fig.~\\ref{fig:magnetization}a), which together with the constitutive\nrelation~(\\ref{eq:jmeso}), allows one to define the orbital magnetization\n$\\bm{m}_{\\text{pers}}$. (The subscript refers to the contribution associated\nwith the persistent current.) Alternatively, one can evaluate the change of the\ninstantaneous ground state energy with respect to the external magnetic field.\nThis recent development~\\cite{xiao2005,thonhauser2005,ceresoli2006,shi2007}\ngoes under the name of ``the modern theory'' of the orbital magnetization [see\nEq.~\\eqref{eq:modthm}]. Unlike the bulk polarization, $\\bm{m}_{\\text{pers}}$ is\nnot related to topological response.\n\nIn this work we develop a general formulation of the remaining contribution to\nthe electric current in the constitutive relation ~\\eqref{eq:jmeso} that are\nneither captured by the averaged current in Eqs.~\\eqref{eq:jx}, \\eqref{eq:jTh}\nnor by the persistent current\n$\\bm{\\nabla}\\times\\bm{m}_{\\text{pers}}(t,\\bm{x})$.  We find that, after\ncoarse-graining in time, this contribution can be expressed as the curl of an\nadditional term $\\mathbcal{m}$ to the orbital magnetization so that $\\bm{m}$ in\nEq.~\\eqref{eq:jmeso} is given by\n\\begin{equation}\n\\bm{m}=\\bm{m}_{\\text{pers}}+\\mathbcal{m}.\n\\end{equation}\nOur main result is that $\\mathbcal{m}$ can be obtained as a derivative of the\nmany-body Berry phase with respect to an external magnetic field $B_z$ applied\nin $z$ direction\n\\begin{align}\n\tTV\\mathcal{m}_z&=\\partial_{B_z}\\varphi_{B_z}\\rvert_{B_z=0},\n\t\\label{eq:calMz}\n\\end{align}\nwhere $V$ represents the system size and $\\varphi_{B_z}$ is defined by\nEq.~(\\ref{eq:Berry}) upon substitution $\\bm\\phi\\rightarrow B_z$. This\nexpression is well-defined in two-dimensional systems with the open boundary\ncondition at least in one direction. There are known subtleties when applying\nuniform magnetic field to periodic systems. See Sec.~\\ref{subsec:general} for\nthe detailed discussion. In the following we assume vanishing Chern numbers in\n$(\\phi_x,\\phi_y)$, $(t,\\phi_x)$ and $(t,\\phi_y)$ spaces.\n\nAs comparison, in the presence of an external uniform magnetic field $\\bm\nB=(0,0,B_z)^{\\rm T}$, the instantaneous orbital magnetization\n$\\bm{m}_{\\text{pers}}$ gives an energy shift\n$\\int_0^Tdt(E_{tB_z}-E_{t0})\/T=-V\\bm{m}_{\\text{pers}}\\cdot\\bm B+O(\\bm{B}^2)$ of\nthe many-body ground state. (Throughout this work we set $\\hbar=1$.)\nAccordingly, after the period $T$, the ground state acquires an additional\nphase proportional $TV\\bm{m}_{\\text{pers}}\\cdot\\bm B$.  On the other hand, a\nnon-zero value of $\\mathbcal{m}$ shows up as Berry phase\n$TV\\mathbcal{m}\\cdot\\bm B+O(\\bm{B}^2)$ acquired by the many-body ground state.\nFor this reason, we name $\\mathbcal{m}$ \\textit{geometric orbital\nmagnetization}. The bulk quantity $T\\mathbcal{m}$ is independent of the period\n$T$ and is defined ``mod $e$''. This ambiguity is reflecting the possibility of\ndecorating the boundary by one-dimensional Thouless pump.\n\nFor band insulators, we perform the perturbation theory with respect to the\napplied magnetic field following Refs.~\\onlinecite{shi2007,essin2010} and find\nthat geometric orbital magnetization consists of two contributions \n\\begin{equation}\n\\mathbcal{m}=\\mathbcal{m}^{\\text{top}}+\\mathbcal{m}^{\\text{non-top}},\n\\label{eq:calm2}\n\\end{equation}\nthe topological contribution $\\mathbcal{m}^{\\text{top}}$ is expressed as\nintegral  of the \\textit{third} Chern-Simons form $P_3$ in $(t,k_x,k_y)$ space\n[see Eq.~\\eqref{eq:3Dtopo2}], while the non-topological contribution\n$\\mathbcal{m}^{\\text{non-top}}$  is written in terms of instantaneous Bloch\nstates and energies [Eq.~\\eqref{eq:nontopcalM}].  The obtained expression for\n$\\mathbcal{m}$ of band insulators has a formal similarity with the expression\nfor the magnetoelectric polarizability of three-dimensional band\ninsulators~\\cite{essin2010,Malashevich2010,Chen2011} upon identification\n$t\/T\\leftrightarrow k_z\/2\\pi$. It is worth mentioning that due to relatively\nlarge gap (order of electronvolts) of band insulators, the adiabaticity\nconditions is not particularly restrictive, the period $T$ can be as small as\nseveral femtoseconds.\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=1.0\\columnwidth]{magnetization2.pdf}\t\t\n\t\t\\caption{\\label{fig:magnetization}\n\t\tDifferent contributions to orbital magnetization of\n\t\ttwo-dimensional periodic adiabatic process with period $T$.  a): Persistent current $\\bm\n\t\tj_{\\text{pers}}$ within each unit cell produces the\n\t\tinstantaneous orbital magnetization $\\bm m$.  b): An adiabatic\n\t\tprocess where an electron trapped in a potential well whose\n\t\tcenter $\\bm{x}=\\bm{r}(t)$ is moving along dashed curve. In the\n\t\tpresence of an externally applied magnetic field, the many-body\n\t\tBerry phase $\\varphi_{B_z}$ is given by the Aharonov-Bohm flux\n\t\t(the hatched area).  c): Periodic boundary conditions are necessary\n\t\twhen each of two potential wells comes back to its initial\n\t\tposition after time $T$ by passing though the seam. Two\n\t\tpossible areas to define Aharonov-Bohm flux (the hatched one and\n\t\tthe non-hatches one) differ by an integer flux quanta.  d): Unit\n\t\tcell consists of a single anisotropic potential well that is\n\t\tspinning during adiabatic process. e): Two identical potential\n\t\twells that exchange their positions after single adiabatic\n\t\tcycle. All adiabatic processes shown here have vanishing\n\t\tintegrated current~(\\ref{eq:jTh}).}\n\t\\end{center}\n\\end{figure}\n\nTo gain intuitive understanding of the two contributions~(\\ref{eq:calm2}), one\ncan think of the topological piece $\\mathbcal{m}^{\\text{top}}$ to be\noriginating from the Aharanov-Bohm contribution to the many-body Berry phase in\nthe magnetic field. Thus $\\mathbcal{m}^{\\text{top}}$ describes the\nmagnetization from electrons, whose positions are moving during the adiabatic\nprocess, as depicted with dashed lines and arrows in\nFig.~\\ref{fig:magnetization}b-c and e. Although Fig.~\\ref{fig:magnetization}a\nmay look similar, $\\bm j_{\\text{pers}}$ in Fig.~\\ref{fig:magnetization}a\nrepresents a \\emph{static} persistent current that is uniformly distributed on\nthe ring.  In contrast, the current density in Fig.~\\ref{fig:magnetization}b at\neach time is localized to the position of the potential well and it becomes\ndivergence-free only after averaging over the period $T$. Similarly, the\nnon-topological piece $\\mathbcal{m}^{\\text{non-top}}$ can be understood to be\noriginating from ``spinning'' of anisotropic crystalline potentials, see\nFig~\\ref{fig:magnetization}d.\n\nLet us mention at this point several related works. Adiabatic\ndynamics can be induced by time-dependent lattice deformations (phonons), which\nis the subject of studies on dynamical deformations of\ncrystals.~\\cite{ceresoli2002,juraschek2017,juraschek2018,dong2018,stengel2018}\nIn Refs.~\\onlinecite{juraschek2017,juraschek2018} it was shown that\na time-varying polarization gives rise to a contribution to the orbital\nmagnetization, and the semi-classical description developed in\nRef.~\\onlinecite{dong2018} found the same effect within their framework. The\ntime-varying polarizations in these works correspond to the situation depicted\nin Fig.~\\ref{fig:magnetization}b. In the case of band insulators, we find that\nthey are captured by the \\textit{abelian} third Chern-Simons form. Furthermore,\nRefs.~\\onlinecite{ceresoli2002,stengel2018} showed that rotation of molecules,\nas in Fig.~\\ref{fig:magnetization}d, gives rise to an orbital magnetization\ncontribution that can be captured by relation~(\\ref{eq:calMz}). The present\napproach gives unified description of the above-mentioned effects. More\nimportantly, it properly describes the orbital magnetization in adiabatic\nprocesses that have not been previously considered: the process in\nFig.~\\ref{fig:magnetization}e has inversion symmetry at all times, thus\npolarization is time-independent, yet it gives rise to non-zero $\\mathbcal{m}$,\nwhich for the case of band insulators is captured by \\textit{non-abelian} third\nChern-Simons form, see Sec.~\\ref{subsec:tgeoM}.\n\nAnalogous to the bulk polarization, crystalline symmetries can quantize topological\ngeometric orbital magnetization $\\mathbcal{m}^{\\text{top}}$. We show that,\nunder certain crystalline symmetries, $\\mathbcal{m}^{\\text{top}}$ is related to\nrecently discussed higher-order topological\nphases.~\\cite{parameswaran2017,schindler2018,peng2017,langbehn2017,song2017,fang2018,ezawa2018,shapourian2017,zhu2018,yan2018,wang2018,wang2018b,khalaf2018,khalaf2018b,trifunovic2019,nobuyuki2018}\nAmong them, the topological insulators that exhibit quantized corner charges in\nthe presence of certain crystalline symmetries attracted recently a lot of\ntheoretical~\\cite{benalcazar2017,benalcazar2017,benalcazar2018} and\nexperimental~\\cite{serra-garcia2018,peterson2018} attention. Although, due to\ncrystalline symmetries, the bulk quadrupole moment is well defined in these\nsystems (see Fig.~\\ref{fig:C4}a), it is still disputed in the literature\nwhether such definition is possible in the absence of any quantizing crystalline\nsymmetries.~\\cite{kang2018,metthew2018,ono2019} Recent work in\nRef.~\\onlinecite{vanmiert2018b} revealed a connection between higher-order\ntopological\ninsulators~\\cite{parameswaran2017,schindler2018,peng2017,langbehn2017,song2017,fang2018,ezawa2018,shapourian2017,zhu2018,yan2018,wang2018,wang2018b,khalaf2018,khalaf2018b,trifunovic2019,nobuyuki2018}\nprotected by roto-inversion symmetries and adiabatic processes that involve\ntopological insulators with quantized corner charges. In\nSec.~\\ref{sec:symmetries} we show that the adiabatic processes discussed by van\nMiert and Ortix~\\cite{vanmiert2018b} are characterized by quantized geometric\norbital magnetization, and we relate the value of $T\\mathcal{m}_z^{\\text{top}}$ to\nthe quantized corner charge.\n\nThe remaining of this article is organized as follows: in Sec.~\\ref{sec:prelim}\nwe review the modern theory of the polarization and the orbital\nmagnetization, Sec.~\\ref{sec:noninteracting} contains derivation of our main\nresults, Sec.~\\ref{sec:symmetries} discusses the role of symmetries in the\nadiabatic process, and Sec.~\\ref{sec:examples} presents various non-interacting\nexamples that illustrate difference between instantaneous orbital\nmagnetization, topological and non-topological geometric orbital magnetization.\nMore precisely, we consider toy models illustrating systems depicted in\nFig.~\\ref{fig:magnetization}. As a more realistic application of physics\nconsidered in this work, we present in Sec.~{\\ref{subsec:rotoM}} calculation of\nmagnetization induced by rotation of an insulator. A long time\nago,~\\cite{barnett1915,barnett1935} Barnett considered magnetization of an\nuncharged paramagnetic material when spun on its axis. Modeling paramagnetic\nmaterial as collection of local magnetic moments that are randomly oriented,\nBarnett~\\cite{barnett1915} argued that rotation creates a torque that acts to\nalign local magnetic moments with rotation axis. This torque gives rise to\nmagnetization $M=\\chi\\Omega\/\\gamma$, where $\\chi$ is paramagnetic\nsusceptibility, $\\Omega$ is rotation frequency and $\\gamma$ is electron\ngyromagnetic ratio. Barnett's measurement of this\neffect~\\cite{barnett1915} provided first accurate measurement of electron\ngyromagnetic ratio. We calculate $\\mathbcal{m}$ for this model, which, as seen from\nEq.~(\\ref{eq:calMz}), is also proportional to rotational frequency\n$\\Omega=2\\pi\/T$ and estimate quantum correction to Barnett effect. Electron\ncontribution to $\\mathbcal{m}$ has both topological and non-topological piece,\nbut since the system is uncharged, we find that electron contribution to\n$\\mathbcal{m}^{\\text{top}}$ is canceled by corresponding ionic contribution. Thus\nresulting $\\mathbcal{m}$ is solely due to anisotropy of crystalline potential,\nanalogous to toy model in Fig.~\\ref{fig:magnetization}d. In\nSec.~\\ref{sec:examples2} we consider examples of general interacting systems\nwhere periodic adiabatic process consists of\n``spinning''~\\cite{ceresoli2002,stengel2018} or\n``shaking''~\\cite{juraschek2017,juraschek2018,dong2018} of the whole\nsystem.~\\cite{goldman2014} Our conclusions and outlook can be found in\nSec.~\\ref{sec:conclusions}.\n\n\\section{Preliminaries}\\label{sec:prelim}\nHere we review the formulation of the polarization and the orbital\nmagnetization for band insulators in $2+1$ dimensions developed in\nRefs.~\\onlinecite{king-smith1993,vanderbilt1993,resta1994,resta2007,xiao2005,thonhauser2005,ceresoli2006,shi2007}.\nTo simplify notations, we assume primitive lattice vectors of the square\nlattice type, but this general framework is \\textit{not} restricted to this\nspecial choice. \n\n\\subsection{Modern theory}\nLet us denote by\n$\\psi_{t\\bm{k}n}(\\bm{x})=(a\/\\sqrt{V})e^{i\\bm{k}\\cdot\\bm{x}}u_{t\\bm{k}n}(\\bm{x})$\nthe instantaneous Bloch function of $n$-th occupied band, satisfying\n$h_{t}|\\psi_{t\\bm{k}n}\\rangle=\\varepsilon_{t\\bm{k}n}|\\psi_{t\\bm{k}n}\\rangle$.\nHere $h_t$ is the single-particle Hamiltonian with a periodic potential, $V=L_xL_y$ is the\nsystem size and $a$ is the lattice constant.  We choose the cell-periodic gauge\nso that they obey the following conditions for any lattice vector $\\bm{R}$ and\nreciprocal lattice vector $\\bm{G}$~\\cite{vanderbilt2018}\n\\begin{align}\n&u_{t\\bm{k}n}(\\bm{x}+\\bm{R})=u_{t\\bm{k}n}(\\bm{x}),\\\\\n&u_{t\\bm{k}+\\bm Gn}(\\bm{x})=e^{-i\\bm  G\\cdot\\bm{x}}u_{t\\bm{k}n}(\\bm{x}).\\label{eq:uk_b1}\n\\end{align}\n\nAccording to the modern theory, the bulk polarization density $\\bm{p}(t)$ is\ngiven by\n\\begin{equation}\n\\bm{p}(t)=\\frac{ei}{V}\\sum_{\\bm{k}n\\in\\text{occ}}\\langle u_{t\\bm{k}n}|\\bm{\\nabla}_{\\bm{k}}u_{t\\bm{k}n}\\rangle\\,\\,\\text{ mod }\\,\\,\\frac{e}{a}.\\label{eq:modthp}\n\\end{equation}\nwhere $e$ $(<0)$ is the electric charge. The sum over $\\bm{k}$ can be replaced\nwith the integral $V\\int \\frac{d^2k}{(2\\pi)^2}$ over the first Brillouin zone.\nSimilarly, the orbital magnetization density $\\bm{m}_{\\text{pers}}(t)$ is given\nby\n\\begin{equation}\n\\bm{m}_{\\text{pers}}(t)=\\frac{ei}{2V}\\sum_{\\bm{k}n\\in\\text{occ}}\\langle\\bm{\\nabla}_{\\bm{k}}u_{t\\bm{k}n}|\\times(h_{t\\bm{k}}+\\varepsilon_{t\\bm{k}n})|\\bm{\\nabla}_{\\bm{k}}u_{t\\bm{k}n}\\rangle,\\label{eq:modthm}\n\\end{equation}\nwhere $h_{t\\bm{k}}\\equiv e^{-i\\bm{k}\\cdot\\bm{x}}h_{t}e^{i\\bm{k}\\cdot\\bm{x}}$. The ambiguity in\nEq.~\\eqref{eq:modthp} can be seen by a smooth gauge transformation\n$|u_{t\\bm{k}n}\\rangle'= \\sum_m|u_{t\\bm{k}m}\\rangle(U_{\\bm{k}})_{m,n}$ that\nchanges the integral in Eq.~\\eqref{eq:modthp} by an integer multiple of $e\/a$,\nwhile the integral in~\\eqref{eq:modthm} remains unchanged. \n\n\nIn addition to the derivation via the Thouless pump as we described in the\nintroduction, the formula \\eqref{eq:modthp} was also verified in terms of the\nWannier state localized around the unit cell $\\bm{R}$.\n\\begin{equation}\n|w_{tn\\bm{R}}\\rangle\\equiv\\frac{a}{\\sqrt{V}}\\sum_{\\bm k}e^{-i\\bm k\\cdot\\bm R}|\\psi_{tn\\bm k}\\rangle.\\label{eq:Wannier}\n\\end{equation}\nIn terms of the Wannier function, $\\bm{p}(t)$ is the deviation of the Wannier center from $\\bm{R}$, i.e.,  $\\bm{p}(t)=\\frac{e}{a^2}\\int d^2x(\\bm{x}-\\bm{R})|w_{t n\\bm{R}}(\\bm{x})|^2$.  \n\n\nWhen the origin of unit cell is changed by $\\bm{\\delta}$, we find\n\\begin{align}\n&|u_{t\\bm{k}n}\\rangle'=e^{i\\bm  k\\cdot \\bm\\delta}|u_{t\\bm{k}n}\\rangle,\\label{origin}\\\\\n&\\bm{p}'(t)=\\bm{p}(t)-\\frac{e\\bm{\\delta}}{a^2},\\\\\n&\\bm{m}_{\\text{pers}}'(t)=\\bm{m}_{\\text{pers}}(t).\n\\end{align}\nNamely, $\\bm{p}(t)$ depends on the specific choice of the origin, while $\\bm{m}_{\\text{pers}}(t)$ does\nnot. Therefore, it is not $\\bm{p}(t)$ itself but rather the change $\\Delta \\bm{p}(t)$\nthat is of physical interest. It also follows that for an periodic adiabatic\nprocess, where the system is translated by certain number of unit\ncells during the period $T$, the orbital magnetization is periodic in time\nwhile the polarization is not.\n\nFor interacting systems under the periodic boundary condition, the combination\nin the parenthesis in Eq.~\\eqref{eq:jx} replaces Eq.~\\eqref{eq:modthp}.  The\nperiodicity in $\\phi_i$ in this formulation is encoded in the relation\n\\begin{align}\n&\\vert\\Phi_{t2\\pi\\phi_y}\\rangle =e^{-2\\pi i\\hat{P}_x}\\vert\\Phi_{t0\\phi_y}\\rangle,\\label{eq:p1}\\\\\n&\\vert\\Phi_{t\\phi_x2\\pi}\\rangle =e^{-2\\pi i\\hat{P}_y}\\vert\\Phi_{t\\phi_x0}\\rangle,\\label{eq:p2}\n\\end{align}\nwhere $\\hat{P}_i$ is the polarization operator (see\nRef.~\\onlinecite{watanabe2018} for example). \n\n\\subsection{Topological response}\\label{sec:top_res}\nTo discuss the physical consequence of $\\Delta \\bm{p}(t)$, let us recall first the\ntopological linear response in $(1+1)$ dimension~\\cite{thouless1983} that holds\nat a mesoscopic scale after coarse-graining\n\\begin{align}\n&j^\\mu(t,x)=-\\sum_\\nu\\varepsilon^{\\mu\\nu}\\partial_\\theta P_1(\\theta)\\partial_\\nu\\theta,\\label{eq:1Dtopo}\\\\\n&P_1(\\theta)\\equiv-e\\int\\frac{dk}{2\\pi} \\tr A_{\\theta k},\\label{eq:1Dtopo2}\\\\\n&(A_{\\theta k})_{n,m}\\equiv-i\\langle u_{\\theta kn}|\\partial_{k}u_{\\theta km}\\rangle.\n\\end{align}\nHere, $x^\\mu$ ($\\mu=0,1$) represents $(t,x)$, and $j^\\mu$ corresponds to\n$(n,j)$. $A_{\\theta k}$ is a (finite dimensional) matrix constructed by\n\\textit{occpied} Bloch states and the trace in Eq.~\\eqref{eq:1Dtopo2} is the\nmatrix trace.  Comparing with above equations, we see that $P_1(\\theta(t))$ is\nthe 1D version of $\\bm{p}(t)$ in Eq.~\\eqref{eq:modthp}. This response is\nderived starting from the Chern-Simons theory\n$j^\\mu=\\frac{C_1}{2\\pi}\\sum_{\\nu\\lambda}\\varepsilon^{\\mu\\nu\\lambda}\\partial_\\nu\nA_\\lambda^{\\text{ex}}$ in $(2+1)$ dimensions that describes the response toward\nan external field $A^{\\text{ex}}$ and reducing the dimension to $(1+1)$\ndimensions.\n\nThe parameter $\\theta$ in Eq.~\\eqref{eq:1Dtopo} is a slowly varying field\ninterpolating between two different systems.  For example, an adiabatic time\nevolution $\\theta(t)$ induces the bulk current $j(t,x)=\\partial_t\nP_1(\\theta(t))$. The bulk charge transfer from $t=0$ to $t=T$ is thus given\nby $\\int_{0}^{T}dt j(t,x)=P_1(\\theta(T))-P_1(\\theta(0))$.\nSimilarly, a transition of one 1D system to another can be described by\n$\\theta(x)$, giving rise to a charge density $n(t,x)=-\\partial_x\nP_1(\\theta(x))$. Therefore, the total charge $Q^{\\text{edge}}$ accumulated to\nthe boundary is $Q^{\\text{edge}}=\\int_{x_0}^{x_1}dx\nn(t,x)=P_1(\\theta(x_0))-P_1(\\theta(x_1))$.  For a given $\\theta$ that specifies\n$P_1(\\theta)$ as a continuous function of $t$ and $x$, even the integer part of\n$Q^{\\text{edge}}$ is well-defined. However, only the fractional part of\n$Q^{\\text{edge}}$ is independent of the detailed choice of the interpolation\n--- the fractional part depends only on the initial and the final\nvalues of $P_1$ that can be individually computed by Eq.~\\eqref{eq:1Dtopo2}.\nWhat we described here can be straightforwardly translated to 2D systems.  The\npumped charge through the bulk per unit length along $\\bm n$ is given by\n$\\bm{Q}\\cdot\\bm n$, where\n\\begin{equation}\n\t\\bm{Q}\\equiv\\int_{0}^{T}dt\\,\\partial_t\\bm{p}(t)=\\frac{1}{T}[\\bm{p}(T)-\\bm{p}(0)].\\label{eq:Jb}\n\\end{equation}\n\nThe analog of Eq.~\\eqref{eq:1Dtopo} in $(3+1)$ dimensions reads~\\cite{qi2008}\n\\begin{align}\n&j^\\mu(t,\\bm x)=-\\frac{1}{2\\pi}\\sum_{\\nu,\\lambda,\\rho}\\varepsilon^{\\mu\\nu\\lambda\\rho}\\partial_\\theta P_3(\\theta)\\partial_\\nu\\theta \\partial_\\lambda A_\\rho^{\\text{ex}},\\label{eq:3Dtopo}\\\\\n&P_3(\\theta)\\equiv -e\\int\\frac{d^3k}{8\\pi^2}\\tr \\bm A_{\\theta\\bm{k}}\\cdot(\\bm\\nabla_{\\bm k}+\\tfrac{2i}{3}\\bm A_{\\theta\\bm{k}})\\times\\bm A_{\\theta\\bm{k}},\\\\\n&(\\bm A_{\\theta\\bm{k}})_{n,m}\\equiv-i\\langle u_{\\theta\\bm{k}n}|\\bm\\nabla_{\\bm k}u_{\\theta\\bm{k}m}\\rangle.\n\\end{align}\nHere, $\\mu,\\nu,\\rho,\\lambda=0,1,2,3$. Again, $\\bm A_{\\theta\\bm{k}}$ is defined\nby occupied Bloch states. This response is derived from the\nChern-Simons theory\n$j^\\mu=\\frac{C_2}{8\\pi^2}\\varepsilon^{\\mu\\nu\\lambda\\rho\\sigma}\\partial_\\nu\nA_\\lambda^{\\text{ex}}\\partial_\\rho A_\\sigma^{\\text{ex}}$ in $(4+1)$ dimensions by a\ndimensional reduction. This topological response implies, for example, the\nmagnetoelectric effect~\\cite{qi2008,essin2009,essin2010}\n$\\rho(z)=-\\frac{1}{2\\pi} \\partial_zP_3(\\theta(z))B_z^{\\text{ex}}$.  \n\n\\section{Geometric orbital magnetization}\\label{sec:noninteracting}\nIn this section we present the derivations of our main results.  We start with\nverifying the most general expression ~\\eqref{eq:calMz} for the geometric\norbital magnetization. Then we derive the formula for the topological and\nnon-topological contributions in Eq.~\\eqref{eq:calm2}.\n\n\\subsection{Berry phases in adiabatic process}\\label{subsec:general}\nSuppose we are interested in the expectation value of the quantity $\\hat{X}$,\ngiven by\n\\begin{equation}\n\\hat{X}=\\partial_\\epsilon \\hat{H}_{\\epsilon}|_{\\epsilon=0}\\label{eq:other}\n\\end{equation}\nfor some parameter $\\epsilon$ in the Hamiltonian. For example, in the case of\nthe averaged current operator, $\\epsilon$ can be identified with the solenoidal\nflux $\\bm{\\phi}$ [see Eq.~\\eqref{eq:currentphi}]. Likewise, for the orbital\nmagnetization we use the external magnetic field $B_z$.\n\nNow suppose that the Hamiltonian $\\hat{H}_{t}$ has a periodic adiabatic\ndependence on $t$, and let $\\vert\\Phi_{t}\\rangle$ be the instantaneous\nground state with the energy eigenvalue $E_{t}$.  We assume an excitation\ngap $\\Delta_{t}$ and the time-dependence of the Hamiltonian must be slow\nenough so that $\\Delta_{t}T\\gg1$.  Using the density matrix\n$\\hat\\rho_t=\\vert\\Psi_t\\rangle\\langle\\Psi_t\\vert$ obeying the time-dependent\nSchr{\\\"o}dinger equation $\\partial_t\\hat\\rho_t=-i[\\hat H_t,\\hat\\rho_t ]$, we\nexpress the time-average of the expectation value of $\\hat{X}_t$ as\n\\begin{equation}\n\tX\\equiv\\int_0^T\\frac{dt}{T}{\\rm tr}[\\hat\\rho_t\\hat{X}_t].\\label{eq:Mz}\n\\end{equation}\nIn the absence of the time-evolution, the density matrix is identical to\n$\\vert\\Phi_{t}\\rangle\\langle\\Phi_{t}\\vert$.  It acquires contributions from\nexcited states $\\hat{H}_{t}|\\Phi_{t}^M\\rangle=E_{t}^M|\\Phi_{t}^M\\rangle$ due to\nthe time evolution. To the lowest-order perturbation theory with respect to\n$(\\Delta_{t}T)^{-1}$, the relevant matrix elements are given\nby~\\cite{thouless1983,watanabe2018}\n\\begin{equation}\n\\langle\\Phi_{t}^M\\vert\\hat\\rho_t\\vert\\Phi_{t}\\rangle=\\langle\\Phi_{t}\\vert\\hat\\rho_t\\vert\\Phi_{t}^M\\rangle^*=\\frac{i\\langle\\Phi_{t}^M\\vert\\partial_t\\vert\\Phi_{t}\\rangle}{E_{t}^M-E_{t}}.\n\\label{eq:rho_adiabatic}\n\\end{equation}\nNow we plug\n$\\hat{X}_t\\equiv\\partial_{\\epsilon}\\hat{H}_{t\\epsilon}|_{\\epsilon=0}$ and make\nuse of the Sternheimer identity\n\\begin{equation}\n(E_{t}-E_{t}^M)\\langle\\Phi_{t}^M\\vert\\partial_{\\epsilon}\\vert\\Phi_{t\\epsilon}\\rangle|_{\\epsilon=0}=\\langle\\Phi_{t}^M\\vert\\partial_{\\epsilon}\\hat{H}_{t\\epsilon}|_{\\epsilon=0}\\vert\\Phi_{t}\\rangle,\\label{SI}\n\\end{equation}\nwhich follows by differentiating\n$\\hat{H}_{t\\epsilon}\\vert\\Phi_{t\\epsilon}\\rangle=E_{t\\epsilon}\\vert\\Phi_{t\\epsilon}\\rangle$\nwith respect to $\\epsilon$ at $\\epsilon=0$. In our notation,\n$\\hat{H}_{t\\epsilon}|_{\\epsilon=0}=\\hat{H}_t$ and\n$\\vert\\Phi_{t\\epsilon}\\rangle|_{\\epsilon=0}=\\vert\\Phi_{t}\\rangle$. Combining\nthese equations, we find\n\\begin{align}\n\tX&=\\int_0^T\\frac{dt}{T}(\\partial_{\\epsilon} E_{t\\epsilon}+{\\cal F}_{t\\epsilon})\\rvert_{\\epsilon=0},\\label{eq:BerryF}\n\\end{align}\nwhere \n\\begin{align}\n\t{\\cal F}_{t\\epsilon}&\\equiv i\\partial_t\\langle\\Phi_{t\\epsilon}\\vert \\partial_{\\epsilon}\\vert\\Phi_{t\\epsilon}\\rangle-i\\partial_\\epsilon\\langle\\Phi_{t\\epsilon}\\vert \\partial_t\\vert\\Phi_{t\\epsilon}\\rangle,\n\\end{align}\nis the Berry curvature in $(t,\\epsilon)$ space. Further assuming the\nperiodicity in time\n\\begin{align}\n\t\\vert\\Phi_{T\\epsilon}\\rangle=\\vert\\Phi_{0\\epsilon}\\rangle,\\label{eq:GSp}\n\\end{align}\nwe arrive at our general expression\n\\begin{align}\n\tX&=X_{\\text{inst}}+X_{\\text{geom}},\\\\\n\tX_{\\text{inst}}&\\equiv\\int_0^T\\frac{dt}{T}\\langle\\Phi_t\\vert\\hat{X}_t\\vert\\Phi_t\\rangle=\\int_0^T\\frac{dt}{T}\\partial_{\\epsilon} E_{t\\epsilon}\\rvert_{\\epsilon=0},\\label{eq:Xinst}\\\\\n\tX_{\\text{geom}}&\\equiv -\\frac{1}{T}\\partial_\\epsilon\\varphi_\\epsilon\\rvert_{\\epsilon=0},\\quad\\varphi_\\epsilon\\equiv \\int_0^Tdt\\langle\\Phi_{t\\epsilon}\\vert i\\partial_t\\vert\\Phi_{t\\epsilon}\\rangle,\\label{eq:Xgeom}\n\\end{align}\nwhere $X_{\\text{inst}}$ is the time average of the expectation value using the\ninstantaneous ground state and $X_{\\text{geom}}$ is the geometric contribution\noriginating from the adiabatic time dependence.  This is the generalization of\nEq.~\\eqref{eq:jTh} for the electric current to physical observables written as\nthe derivative of Hamiltonian as in Eq.~\\eqref{eq:other}. The following\nbasis-independent expressions may also be useful \n\\begin{align}\n\tX_{\\text{geom}}&=\\int_0^T\\frac{dt}{T} i{\\rm tr}\\hat P_{t\\epsilon}[\\partial_t\\hat P_{t\\epsilon},\\partial_\\epsilon\\hat P_{t\\epsilon} ]|_{\\epsilon=0}\\label{eq:MzSPint}\\\\\n\t&={\\rm Re}\\int_0^T\\frac{dt}{T}\\oint\\frac{dz}{\\pi}{\\rm tr}[(\\partial_t\\hat{H}_{t})\\hat{G}_{t}^2\\partial_{\\epsilon}\\hat{H}_{t\\epsilon}|_{\\epsilon=0}\\hat{G}_{t}],\n\\end{align}\nwhere $\\hat P_{t\\epsilon}=|\\Phi_{t\\epsilon}\\rangle\\langle\\Phi_{t\\epsilon}\\vert$\nis the projector onto the many-body ground state,\n$\\hat{G}_{t}=(z-\\hat{H}_{t})^{-1}$ is the many-body Green function, and the\nintegration contour encloses only the ground state at $z=E_{t}$.\n\nLet us now specialize to the case $\\epsilon=B_z$. Then $X_{\\text{geom}}$ in\nEq.~\\eqref{eq:Xgeom} gives the geometric orbital magnetization $\\mathbcal{m}$\nin Eq.~\\eqref{eq:calMz}, while  $X_{\\text{inst}}$ is the persistent\ncurrent contribution. Note the additional minus sign because of\n$\\hat{M}_z=-\\partial_{B_z} \\hat{H}_{B_z}$. Previously,\nRefs.~\\onlinecite{ceresoli2002,stengel2018} considered the Berry curvature in\n$(t,B_z)$ space to describe orbital magnetization induced by rotation of\nmolecules. See also examples in Sec.~\\ref{subsec:nontgeoM}\nand~\\ref{subsec:rotom} below.\n\nWhen applying these formulae, one has to be careful about boundary conditions.\nIf open boundary conditions in at least one direction are imposed, the\nresult~(\\ref{eq:MzSPint}) is directly applicable. However, a process that is\nperiodic in time under periodic boundary condition may loose its\nperiodicity in time under open boundary conditions. For example, the system in\nFig.~\\ref{fig:magnetization}c is not periodic in time if the open boundary\ncondition in $y$ direction is imposed. Similarly, the periodicity in time\nrequires periodic boundary conditions in both directions for the $C_4$-symmetric\nsystem in Fig.~\\ref{fig:C4e}. Keeping (original) periodic boundary conditions\nin both directions in the presence of magnetic field, implies that the net flux\nthrough the system has to vanish. If the system is homogeneous, local\ncontributions to $\\mathbcal{m}$ cancel out and we cannot obtain a useful\ninformation about the system. (For single-particle problems there is a\nresolution as we discuss below.) Finally, one can impose \\textit{magnetic}\nperiodic boundary conditions assuming that the total magnetic flux applied to\nthe system $B_zL_xL_y$ is an integer multiple of\n$2\\pi$.~\\cite{brown1964,zak1964} However, each eigenstate of $\\hat{H}_{B_z}$\nmay not be analytic as a function of $B_z$ despite the fact that the magnetic\nfield $B_z=2\\pi\/L_xL_y$ itself can be made small for a large systems. The\nexpression~\\eqref{eq:MzSPint} is still applicable if the projector onto the\ninstantaneous ground state is analytic function of $B_z$, which is the case for\nband insulators with vanishing Chern\nnumber.~\\cite{essin2010,Malashevich2010,gonze2011,Chen2011} However, to our\nknowledge there is no general proof for gapped interacting systems. \n\n\\subsection{Noninteracting systems}\n\\label{nis}\nLet us apply this general expression to noninteracting\nelectrons described by the quadratic Hamiltonian\n\\begin{equation}\n\\hat{H}_{t\\epsilon}=\\sum_{n}\\varepsilon_{t\\epsilon n}\\hat{\\gamma}_{t\\epsilon n}^\\dagger \\hat{\\gamma}_{t\\epsilon n}.\n\\end{equation}\nWe label single-particle states in such a way that $\\varepsilon_{t\\epsilon\nn+1}\\geq\\varepsilon_{t\\epsilon n}$ for all $n=1,2,\\cdots$. We also assume a\nfinite gap $\\Delta=\\varepsilon_{t\\epsilon N+1}-\\varepsilon_{t\\epsilon N}$\nbetween $N$-th and $(N+1)$-th levels. We write the single particle state\n$|\\gamma_{t\\epsilon n}\\rangle\\equiv \\hat{\\gamma}_{t\\epsilon\nn}^\\dagger|0\\rangle$.  Then the $N$-particle ground state can be written as\n\\begin{equation}\n|\\hat{\\Phi}_{t\\epsilon}\\rangle=\\prod_{n=1}^N\\hat{\\gamma}_{t\\epsilon n}^\\dagger |0\\rangle.\n\\end{equation}\nFor a later purpose, we allow for a unitary transformation \\textit{among the\noccupied levels}\n\\begin{equation}\n\\hat{\\psi}_{t\\epsilon \\ell}^\\dagger=\\sum_{n=1}^N\\hat{\\gamma}_{t\\epsilon n}^\\dagger U_{n\\ell},\\quad \\ell=1,2,\\cdots,N.\n\\end{equation}\nAlthough such a basis change may sound unnecessary, in the actual application\nof this framework it is sometimes important to work in the proper basis by\nchoosing $U_{n\\ell}$ appropriately. (See Sec.~\\ref{subsec:GOMBI} for an\nexample).  After all we find that the many-body Berry phase\n$\\varphi_{\\epsilon}$ is given by the sum of single-particle Berry phases\n$\\varphi_{\\epsilon\\ell}$ of occupied levels\n\\begin{align}\n\\varphi_{\\epsilon}= \\sum_{\\ell=1}^N\\varphi_{\\epsilon\\ell},\\label{eq:manysingle}\\quad\\varphi_{\\epsilon\\ell}\\equiv\\int_0^Tdt\\langle\\psi_{t\\epsilon\\ell}\\vert i\\partial_t\\psi_{t\\epsilon\\ell}\\rangle.\n\\end{align}\nTherefore, we get the following expressions for single-particle problems. The\nlatter two expressions are basis-independent\n\\begin{align}\nX_{\\text{geom}}\n&=\\int_0^T\\frac{dt}{T}\\,i\\sum_{\\ell=1}^N\\langle \\partial_t\\psi_{t\\epsilon\\ell}\\vert \\partial_\\epsilon\\psi_{t\\epsilon\\ell}\\rangle|_{\\epsilon=0}\\\\\n&=\\int_0^T\\frac{dt}{T}\\,i{\\rm tr}P_{t\\epsilon}[\\partial_t P_{t\\epsilon},\\partial_\\epsilon P_{t\\epsilon}]|_{\\epsilon=0}\\\\\n&={\\rm Re}\\oint\\frac{dz}{\\pi}\\int_0^T\\frac{dt}{T}\\tr[(\\partial_th_{t})g_{t}^2\\partial_{\\epsilon}h_{t\\epsilon}\\rvert_{\\epsilon=0}g_{t}],\n\t\\label{eq:MzSP}\n\\end{align}\nwhere\n$P_{t\\epsilon}=\\sum_{\\ell=1}^N\\vert\\psi_{t\\epsilon\\ell}\\rangle\\langle\\psi_{t\\epsilon\\ell}\\vert=\\sum_{\\ell=1}^N\\vert\\gamma_{t\\epsilon\nn}\\rangle\\langle\\gamma_{t\\epsilon n}\\vert$ is the projector onto occupied\nsingle-particle states, $h_{t\\epsilon}=\\sum_n\\varepsilon_{t\nn\\epsilon}\\vert\\gamma_{t \\epsilon n}\\rangle\\langle\\gamma_{t \\epsilon n}\\vert$\nis the single-particle Hamiltonian, $g_{t}=(z-h_{t})^{-1}$ is single-particle\nGreen function, and the integration contour encloses all the occupied states at\n$z=\\varepsilon_{t n}$ ($n=1,2,\\cdots,N$).\n\nFor the orbital magnetization, we again set $\\epsilon=B_z$. The\nsame remarks as in the previous section apply here. In the case of band\ninsulators, one may want to impose periodic boundary conditions to preserve the\ntranslation symmetry. As discussed in the previous section there are two\npossibilities to achieve this. One can change the the boundary condition to\nmagnetic periodic boundary conditions.~\\cite{brown1964,zak1964} For\nsingle-particle systems, assuming symmetric gauge $\\bm A^{\\rm ex}(\\bm x)=\\bm\nB\\times\\bm x\/2$, the magnetic periodic boundary conditions can be taken into\naccount explicitly by restricting the form of the projector $\\hat{P}_{tB_z}$\nto~\\cite{essin2010,gonze2011} $\\langle\\bm x_1\\vert\n\\hat{P}_{tB_z}\\vert\\bm x_2\\rangle=\\hat{P}^\\prime_{tB_z}(\\bm x_2,\\bm x_1)\ne^{ieB_z\\bm x_1\\times\\bm x_2\\cdot\\hat{\\bm z}\/2}$, where $P^\\prime_{tB_z}(\\bm\nx_1,\\bm x_2)$ is an arbitrary $N\\times N$ matrix function (not necessarily\nprojector) that satisfies $P^\\prime_{tB_z}(\\bm x_1+\\bm R,\\bm x_2+\\bm\nR)=P^\\prime_{tB_z}(\\bm x_1,\\bm x_2)$, where $\\bm R$ is an element of Bravais\nlattice.  The expression for $\\hat{P}^\\prime_{tB_z}(\\bm x)$ can be found\nperturbatively in $B_z$,~\\cite{essin2010,gonze2011} which, after substituting\nback to Eq.~(\\ref{eq:manysingle}), yields an expression for the Berry phase and\n$\\mathbcal{m}$. The second option is to apply a spatially modulating magnetic\nfield as we discuss below. \n\n\\subsection{Geometric orbital magnetization for band insulators}\n\\label{subsec:GOMBI}\nBelow we consider band insulators and show that geometric orbital magnetization\nhas two contributions as in Eq.~\\eqref{eq:calm2}. Since we assume the periodic\nboundary condition both in $x$ and $y$, we apply a slowly modulating magnetic\nfield~\\cite{shi2007,essin2010} in order to avoid changing of the boundary\ncondition as discussed in the previous subsection. We use the vector potential\n\\begin{align}\n&\\bm{A}^{\\text{ex}}(\\bm{x})=\\frac{\\epsilon}{2q}(-\\sin qy,\\sin qx,0)^{\\rm T},\\\\\n&\\bm{B}(\\bm{x})=\\bm{\\nabla}\\times\\bm{A}^{\\text{ex}}(\\bm{x})=\\bm{e}_z\\epsilon f(\\bm{x})\n\\end{align}\nwith $\\bm{e}_z\\equiv(0,0,1)^{\\rm T}$, $q\\equiv2\\pi\/L$, and $f(\\bm{x})=(\\cos\nqx+\\cos qy)\/2$. (To simplify the notation, we assume $L=L_x=L_y$ in this\nsubsection). Such a magnetic field induces the change of the Bloch function\n\\begin{align}\n&|\\partial_\\epsilon\\psi_{t\\epsilon n\\bm{k}}\\rangle|_{\\epsilon=0}\\notag\\\\\n&=-\\sum_{n'\\bm{k}'}|\\psi_{tn'\\bm{k}'}\\rangle\\frac{\\langle\\psi_{tn'\\bm{k}'}|\\partial_\\epsilon h_{t\\epsilon}|_{\\epsilon=0}|\\psi_{tn\\bm{k}}\\rangle}{\\varepsilon_{tn'\\bm{k}'}-\\varepsilon_{tn\\bm{k}}}\n\\end{align}\nand $|\\partial_\\epsilon w_{t\\epsilon n\\bm{R}}\\rangle|_{\\epsilon=0}$ is given\nvia Eq.~\\eqref{eq:Wannier}.\n\nWe compute the Berry phase using the formula~\\eqref{eq:manysingle} derived\nabove. It is important to work in the Wannier basis for which the magnetic\nfield effectively becomes uniform in the limit $q\\rightarrow0$. The\nsingle-particle Berry phase in this basis, summed over occupied bands, takes\nthe following form\n\\begin{align}\n\t\\partial_{\\epsilon}\\varphi_{\\epsilon \\bm{R}}|_{\\epsilon=0}&=-\\int_0^Tdt\\sum_{n\\in\\text{occ}}\\langle i\\partial_tw_{tn \\bm{R}}\\vert \\partial_{\\epsilon}w_{t\\epsilon n \\bm{R}}\\rangle|_{\\epsilon=0}+\\text{c.c.}\\notag\\\\\n\t&=Ta^2\\mathcal{m}_zf(\\bm{R}).\n\\end{align}\nIf we further sum over $\\bm{R}$, or equivalently if we work in the Bloch basis\n$|\\psi_{tn\\bm{k}}\\rangle$, we get $0$ reflecting the fact that for bulk systems\nFourier component $\\mathcal{m}_z(\\bm q)$ vanishes for $\\bm q\\neq0$. Thus, care\nmust be taken to correctly read off local contribution to $\\mathcal{m}_z$--- an\nunintentional integration over $\\bm R$ of a term proportional to $f(\\bm R)$\nmakes it impossible to find the correct value of $\\mathcal{m}_z$. The rest\ncalculation follows the appendix in Ref.~\\onlinecite{essin2010}. Upon taking\nthe limit $q\\rightarrow0$, we find\n\\begin{widetext}\n\\begin{align}\n\\mathcal{m}_z&=\\lim_{q\\rightarrow0}\\frac{1}{T}\\int_0^Tdt\\int\\frac{d^2k}{(2\\pi)^2}\\sum_{n\\in\\text{occ}}\\sum_{n'\\bm{k}'}\\,\\langle i\\partial_t\\psi_{tn\\bm{k}}|\\psi_{tn'\\bm{k}}\\rangle \\frac{\\langle\\psi_{tn'\\bm{k}}|\\partial_\\epsilon h_{t\\epsilon}|_{\\epsilon=0}|\\psi_{n\\bm{k}'}\\rangle}{\\varepsilon_{tn'\\bm{k}}-\\varepsilon_{tn\\bm{k}'}}+\\text{c.c}.\\notag\\\\\n&=-\\frac{e}{2T}\\int_0^Tdt\\int\\frac{d^2k}{(2\\pi)^2}\\sum_{n\\in\\text{occ}}\\sum_{n'}\\langle\\partial_t u_{n\\bm k}\\vert u_{n^\\prime\\bm k}\\rangle\n\\frac{\\langle u_{tn'\\bm k}\\vert\\bm\\nabla_{\\bm k}(h_{t\\bm k}+\\varepsilon_{tn\\bm k})\\times\\vert\\bm\\nabla_{\\bm k}u_{tn\\bm k}\\rangle}{\\varepsilon_{tn'\\bm k}-\\varepsilon_{tn\\bm k}}+\\text{c.c}.\t\n\\end{align}\nThis last expression can precisely be expressed as the sum of two terms,\n$\\mathbcal{m}^{\\text{top}}+\\mathbcal{m}^{\\text{non-top}}$. The topological\npiece $\\mathbcal{m}^{\\text{top}}$ reads\n\\begin{align}\n&\\mathbcal{m}^{\\text{top}}= \\bm{e}_zP_3\/T,\\label{eq:topc}\\\\\n&P_3\\equiv -\\frac{e}{2}\\int_0^Tdt\\int\\frac{d^2k}{(2\\pi)^2}\\tr[\\bm A_{\\bm{K}}\\cdot\\bm\\nabla_{\\bm{K}}\\times\\bm A_{\\bm{K}}+\\tfrac{2i}{3}\\bm A_{\\bm{K}}\\cdot\\bm A_{\\bm{K}}\\times\\bm A_{\\bm{K}}].\\label{eq:3Dtopo2}\n\\end{align}\nThe Berry connection $(\\bm A_{\\bm{K}})_{n,m}\\equiv-i\\langle\nu_{\\bm{K}n}|\\bm\\nabla_{\\bm{K}}u_{\\bm{K}m}\\rangle$ is defined using occupied\nBloch states as a function of $\\bm{K}\\equiv(t,\\bm{k})$. The smoothness and the\nperiodicity of $\\bm A_{\\bm{K}}$ are assumed in the integral in\nEq.~\\eqref{eq:3Dtopo2}. Such a choice is possible only  when both the pumped\ncharge through the bulk $\\bm Q$ in Eq.~\\eqref{eq:Jb} and the 2D Chern\nnumber for $(k_x,k_y)$ vanish.  \n\nThe non-topological contribution depends also on instantaneous eigenenergies of the Bloch Hamiltonian\n\\begin{align}\n\t\\mathbcal{m}^{\\text{non-top}}&=\\sum_{n\\in\\text{occ}}\\sum_{n'\\in\\text{unocc}}\\frac{e}{2T}\\int_0^Tdt\\int\\frac{d^2k}{(2\\pi)^2}\\frac{\n\t\\langle u_{tn\\bm k}|\\partial_tP_{t\\bm k}|u_{tn'\\bm k}\\rangle\\langle u_{tn'\\bm k}|\\{\\bm\\nabla_{\\bm k}h_{t\\bm k}\\times\\bm\\nabla_{\\bm k}P_{t\\bm k}\\}|u_{tn\\bm k}\\rangle\n\t}{\\varepsilon_{tn\\bm k}-\\varepsilon_{tn'\\bm k}}+\\text{c.c.}\\notag\\\\\n\t&=\\frac{e}{2T}\\int_0^Tdt\\int\\frac{d^2k}{(2\\pi)^2}\\oint\\frac{dz}{2\\pi i}{\\rm tr}\\left[ \\partial_tP_{t\\bm k}g_{t\\bm k}\\{\\bm\\nabla_{\\bm k}h_{t\\bm k}\\times\\bm\\nabla_{\\bm k}P_{t\\bm k}\\}g_{t\\bm k} \\right]+\\text{c.c.}\n\t\\label{eq:nontopcalM}\n\\end{align}\n\\end{widetext}\nHere, $h_{t\\bm k}$ is Bloch Hamiltonian, $P_{t\\bm k}=\\sum_{n\\in\\text{occ}}\\vert u_{tn\\bm{k}}\\rangle\\langle u_{tn\\bm{k}}\\vert$ is the projector\nonto occupied bands at $\\bm k$, $g_{t\\bm k}=(z-h_{t\\bm k})^{-1}$ is Bloch's\nGreen function, the curly brackets denote symmetrization $\\{\\bm A\\times\\bm\nB\\}=\\bm A\\times\\bm B+\\bm B\\times\\bm A$, and the integration contour encloses\nall the filled Bloch states at $z=\\varepsilon_{tn\\bm{k}}$. See the appendix of Ref.~\\onlinecite{essin2010}\nfor the details. Note that both $\\mathbcal{m}^{\\text{top}}$ and $\\mathbcal{m}^{\\text{non-top}}$ are not affected by the shift of the origin in Eq.~\\eqref{origin}.\n\n\\subsection{Topological contribution from response theory}\\label{subsec:topom}\nHere we give an alternative, easier derivation of $\\mathbcal{m}^{\\text{top}}$\nin Eq.~\\eqref{eq:topc} from the topological response theory.  To this end let\nus further reduce one spatial dimension in Eq.~\\eqref{eq:3Dtopo} to achieve the\ntopological quadratic response in $(2+1)$d~\\cite{qi2008}:\n\\begin{align}\n&\\mathcal{j}^\\mu(t,\\bm x)=-\\frac{1}{2\\pi}\\sum_{\\nu,\\lambda,\\rho}\\varepsilon^{\\mu\\nu\\lambda}G_2(\\theta,\\phi)\\partial_\\nu\\theta \\partial_\\lambda \\phi,\\label{eq:2Dtopo}\\\\\n&\\frac{1}{2\\pi}G_2(\\theta,\\phi)\\equiv-e\\int\\frac{d^2k}{32\\pi^2} \\,\\varepsilon^{\\mu\\nu\\rho\\sigma}\\tr F_{\\mu\\nu} F_{\\rho\\sigma},\\label{eq:2Dtopo2}\n\\end{align}\nwhere $F_{\\mu\\nu}\\equiv\\partial_\\mu A_{\\nu}-\\partial_\\nu\nA_{\\mu}+i[A_{\\mu},A_{\\nu}]$ is the Berry curvature in the\n$(k_x,k_y,\\theta,\\phi)$ space and $\\theta$ and $\\phi$ are two slowly varying\nfields:  $\\theta(t)$ denotes an adiabatic and periodic time dependence and\n$\\phi(\\bm{x})$ describes a smooth interface of domains\n(Fig.~\\ref{fig:Qboundary}). In this setting, we find\n\\begin{align}\n\\mathbcal{j}(t,\\bm{x})=\\frac{1}{2\\pi}G_2(\\theta,\\phi)\\partial_t\\theta(t)\\bm{\\nabla}\\phi(\\bm{x})\\times\\bm{e}_z\n\\end{align}\nso that\n\\begin{align}\n\\label{eq:j4}\n\t{\\mathbcal{j}}(\\bm{x})&\\equiv\\int_{0}^{T}\\frac{dt}{T}\\mathbcal{j}(t,\\bm{x})=\\partial_\\phi P_3(\\phi)\\bm{\\nabla}\\phi(\\bm{x})\\times\\bm{e}_z\/T\\notag\\\\\n\t&=\\bm{\\nabla}\\times[\\bm{e}_zP_3(\\phi(\\bm{x}))\/T]=\\bm{\\nabla}\\times\\mathbcal{m}^{\\text{top}}(\\bm{x}).\n\\end{align}\nThis reproduces Eq.~\\eqref{eq:topc}.  In the derivation we used the relation $\\int_0^{2\\pi}\\frac{d\\theta}{2\\pi}\nG_2(\\theta,\\phi)=\\partial_\\phi P_3(\\phi)$.  It is important to note that\n$\\mathbcal{j}(t,\\bm x)$ itself cannot be written as a curl of a vector field ---\nEq.~\\eqref{eq:j4} holds only after the time convolution (or equivalently the\ntime average). \n\nThe above derivation relies on the connection~(\\ref{eq:jmeso}) between\n$\\mathbcal{m}^{\\text{top}}$ and topological edge current in adiabatically\ndriven two-dimensional systems. To see this more concretely, let us consider\nthe boundary of two regions with $\\phi_0\\equiv\\phi(\\bm x_0)$ and\n$\\phi_1\\equiv\\phi(\\bm x_1)$ (see Fig.~\\ref{fig:Qboundary}). Just like in the\ncase of polarization, only the fractional part of the edge current is the bulk\ncontribution that depends only on $\\phi_0$ and $\\phi_1$. This can be understood\nby noticing that decorating the boundary with a 1D chain leads to an integer\ncharge transfer through the Thouless pump.~\\cite{thouless1983} To capture the\nfractional bulk contribution to the edge current, one can separately compute\n$\\mathcal{m}_z(\\bm x_0)$ and $\\mathcal{m}_z(\\bm x_1)$ without paying attention\nto their continuity. The geometric contribution to the charge transfer along\n$i$ direction, $i=x,y$ between two bulk systems with $\\mathcal{m}_z(\\bm{x}_0)$\nand $\\mathcal{m}_z(\\bm{x}_1)=\\mathcal{m}_z(\\bm{x}_1')$\n(Fig.~\\ref{fig:Qboundary}) is given by\n\\begin{align}\nI_i^{\\text{edge}}&\\equiv\\int_{\\bm  x_0}^{\\bm x_1} dx\\mathcal{j}_i(\\bm{x})=\\mathcal{m}_z(\\bm x_0)-\\mathcal{m}_z(\\bm x_1')\\mod e.\n\\end{align}\n\nNotice that $I^{\\text{edge}}$ of two adjacent edges may differ by an integer. To\nsee this formally, let us consider a charge flow $\\Delta Q^{\\text{corner}}$ into a corner\nsurrounded by a closed curve $\\bm x_\\alpha$ with $\\bm x_1=\\bm x_0$ (see\nFig.~\\ref{fig:Qboundary}). The net charge flow in the process is given by the\nsecond Chern number\n\\begin{equation}\n\t\\label{eq:Qc}\n\\Delta Q^{\\text{corner}}\\equiv T\\oint d\\bm{x}_\\alpha\\times\\mathbcal{j}(\\bm{x}_\\alpha)\\cdot\\bm{e}_z=\\int \\frac{d\\theta d\\phi}{2\\pi}G_2(\\theta,\\phi).\n\\end{equation}\nFor example, when the corner is formed by two edges along $x$ and $y$\ndirections, we have \n\\begin{equation}\n\\Delta Q^{\\text{corner}}=T(I_x^{{\\text{edge}}}-I_y^{\\text{edge}}),\n\\label{n1n2}\n\\end{equation}\nmeaning that the charge transfer along two intersecting edges can only differ\nby an integer multiple of $e$. Clearly, $\\Delta Q^{\\text{corner}}$ is \\textit{not} a bulk\ntopological invariant in general, since its value can be changed by closing the\nboundary gap, i.e., attaching 1D Thouless pump at certain boundaries\n(Figs.~\\ref{fig:Qboundary} and~\\ref{fig:C4}b).  \n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=0.7\\columnwidth]{fig2.pdf}\t\t\n\t\t\\caption{\\label{fig:Qboundary}The boundary current along the\n\t\tinterface of two adiabatic processes $h_{\\phi(\\bm x_i)\\theta(t)\\bm{k}}$ with $i=0$ and $1$. A 1D decoration with Thouless\n\t\tpump changes the edge charge transfer by an integer and leads\n\t\tto integer corner charge accumulation. Hatched parts denote the boundary area between the two systems.}\n\t\\end{center}\n\\end{figure}\n\n\\subsection{Symmetry constraints and corner charge}\n\\label{sec:symmetries}\nHere we consider adiabatic process of two-dimensional systems constrained by\ncertain symmetries that quantize $T\\mathcal{m}_z^\\text{top}$. We show that, if\nthe symmetry allows one to define the bulk contribution to quadrupole moment,\nthe quantized quadrupole moment is equal to $T\\mathcal{m}_z^\\text{top}$. Such adiabatic\nprocesses were recently discussed by van Miert and Ortix, who found the\nconnection between the quantized corner charge and higher-order topological\ninvariant.~\\cite{vanmiert2018}\n\nFor concreteness, let us consider the four-fold rotation $C_4$ mapping\n$\\bm{x}=(x,y,0)$ to $C_4\\bm{x}=(-y,x,0)$. It is easy to see that boundary\ndecorations by polarized one-dimensional chains do not affect the fractional\npart of the corner charge $\\Delta Q^{\\text{corner}}$, see Fig.~\\ref{fig:C4}a.  We\nconsider an \\textit{arbitrary} interpolation between the system of interest\n$h_{0\\bm k}$ and the reference system $h_{T\/2\\,\\bm k}$ that has no\ncorner charge. The second half of the cycle is performed in a $C_4$-symmetric\nmanner\n\\begin{equation}\n\tU_{C_4}h_{t\\bm k}U_{C_4}^\\dagger=h_{T-t\\,C_4\\bm k}.\n\t\\label{eq:C4cycle}\n\\end{equation}\nThe $C_4$ symmetry defined above behaves as the roto-inversion $IC_4$ in\n$(t,k_x,k_y)$-space, resulting in the following transformation law for\n$\\mathcal{m}_z^\\text{top}$:\n\\begin{equation}\n\tC_4:\\quad \\mathcal{m}_z^\\text{top}\\rightarrow-\\mathcal{m}_z^\\text{top}.\n\t\\label{eq:P3_C4}\n\\end{equation}\nThis does not mean that $T\\mathcal{m}_z^\\text{top}$ vanishes since it is defined only ${\\rm\nmod}\\,1$.  Thus in the presence of $C_4$ symmetry $T\\mathcal{m}_z$ is quantized\neither $0$ or $e\/2\\mod e$. When $T\\mathcal{m}_z=e\/2\\mod e$, the circulating\nedge current as in Fig.~\\ref{fig:Qboundary} violates $C_4$\nsymmetry constraint~(\\ref{eq:C4cycle})---the only allowed edge current\ndistribution is shown by black arrows in Fig.~\\ref{fig:C4}b.  Note that the\ninversion symmetry, for example, also quantizes $T\\mathcal{m}_z^\\text{top}$ but\nthe total corner charge accumulation during inversion-symmetric cycles need to\nvanish since the charge distribution of quadrupole moment is invariant under\nthe inversion (see Fig.~\\ref{fig:C4}c).\n\nNow we show that the parity of the corner charge accumulation $\\Delta\nQ^{\\text{corner}}$ is actually a bulk topological invariant for symmetric\nadiabatic processes satisfying constraint~\\eqref{eq:C4cycle}, see also\nFig.~\\ref{fig:C4}b. To this end, consider two perpendicular edges along $x$\nand $y$ direction, related to each other by $C_4$ symmetry. The\nrelations~\\eqref{n1n2} and \\eqref{eq:P3_C4} suggest that\n$I^{\\text{edge}}_y=-I^{\\text{edge}}_x$ and that\n\\begin{equation}\n\t\\Delta Q^{\\text{corner}}=2TI^{\\text{edge}}_x=2T\\mathcal{m}_z^\\text{top}(\\bm x_0)\\mod 2e.\n\t\\label{eq:qcorner}\n\\end{equation}\nFurthermore, Fig.~\\ref{fig:C4}c tells us that the corner charge accumulation\nduring the symmetric process is $\\Delta Q^{\\text{corner}}=2 q^{\\text{corner}}$. Therefore,\n\\begin{equation}\n\tq^{\\text{corner}}=T\\mathcal{m}_z^\\text{top}(\\bm x_0) =P_3(\\phi_0) \\mod e.\\label{corner}\n\\end{equation}\nWe will discuss an example  of quadrupole insulators with $P_3=e\/2$ in\nSec.~\\ref{subsec:tgeoM} using this result. On the other hand, a $C_4$-symmetric\nphase that hosts a corner charge of $q^{\\text{ corner}}=e\/4$ were recently\nreported.~\\cite{benalcazar2018} The fact that $\\Delta Q^{\\text{\ncorner}}\\in\\mathbb{Z}$ forces us to conclude that $C_4$-symmetric\nadiabatic process cannot be constructed for such a phase.\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=\\columnwidth]{C4.pdf}\n\t\t\\caption{\\label{fig:C4}  a): The four-fold rotation symmetry $C_4$ of $h_{0\\bm{k}}$ imposes constraint on\n\t\tthe time-independent boundary decorations and they cannot alter the corner\n\t\tcharge. b): Decorating the boundary with one-dimensional\n\t\tThouless pumps while respecting $C_4$ symmetry (combined with the time flip) of the adiabatic process can change $\\Delta\n\t\tQ^\\text{corner}$ by an \\textit{even} integer. c): Comparison of\n\t\taction of $C_4$ and the inversion $\\mathcal{I}$ on the corner charge distribution\n\t\tafter one period of adiabatic process.}\n\t\\end{center}\n\\end{figure}\n\nAlternatively, as discussed in detail in Ref.~\\onlinecite{vanmiert2018}, the\n$C_4$-symmetric adiabatic process $h_{t\\bm k}$ considered above, can be\nviewed as a 3D topological insulator protected by the roto-inversion symmetry\n$IC_4$ upon identification $k_z=2\\pi t\/T$. In fact, the 3D topological insulator\nwith $P_3=e\/2$ obtained this way is a second-order topological insulator. If we\nconsider a geometry with the open boundary conditions in $xy$-plane and the\nperiodic boundary conditions in $z$-direction, such a second-order phase can be\ntranslationally invariant in $z$-direction both in the bulk and on the boundary.\nThe boundary hosts an odd number of chiral modes running along each of four\nhinges in $IC_4$-symmetric manner. Going back to the picture of an adiabatic\nprocess, it becomes clear that the corner charge accumulation is an odd integer\nas $t$ is varied from $0$ to $T$, which is consistent with the above\nresult~(\\ref{eq:qcorner}).\n\n\\section{Examples: noninteracting systems}\\label{sec:examples}\nIn this section, we discuss a simple model of noninteracting spinless electrons\nin a periodic potential, which highlights the distinction of two contributions\nto the bulk orbital magnetization, $\\bm{m}_{\\text{pers}}$ and $\\mathbcal{m}$.\nAdditionally, we want to consider examples where there is only topological\ngeometric magnetization, Sec.~\\ref{subsec:tgeoM}, only non-topological\ngeometric magnetization, Sec.~\\ref{subsec:nontgeoM}, and both topological and\nnon-topological contributions, Sec.~\\ref{subsec:rotoM}. To keep the discussions\nsimple while capturing the relevant physics, we focus on isolated orbitals\nwithout any overlap between them.\n\n\\subsection{Bloch functions in the localized limit}\nLet us consider a time-dependent deep potential $v_t^0(\\bm{x})$ centering at\n$\\bm{x}=\\bm{r}(t)$ that accommodates at least one bound state.  Let $\\phi_t^0(\\bm x)$ be the wavefunction of the instantaneous lowest-energy bound state, satisfying\n$h_t^0\\phi_t^0(\\bm{x})=\\varepsilon_t^0\\phi_t^0(\\bm{x})$ with \n\\begin{equation}\nh_{t}^0=\\frac{1}{2m}[\\tfrac{1}{i}\\bm{\\nabla}-e\\bm{A}_t^{\\text{ex}}(\\bm{x})]^2+v_t^0(\\bm{x}).\\label{simplemodel}\n\\end{equation}\nHere $\\bm{A}_t^{\\text{ex}}(\\bm{x})$ describes an external field.  In these expressions, the superscript $0$ implies the quantities for an isolated orbit.  \nWhen the\npotential $v_t^0(\\bm{x})$ is deep enough, $\\phi_t^0(\\bm{x})$ should be\nwell-localized around $\\bm{x}=\\bm{r}(t)$ with the localization length $\\xi\\ll a$.\nHence, we assume that \n\\begin{equation}\n\\int d^2x |\\phi_t^0(\\bm{x})|^2=1,\\quad \\int d^2x \\bm{x}|\\phi_t^0(\\bm{x})|^2=\\bm{r}(t)\\label{exp}\n\\end{equation}\nand that both $\\phi_t^0(\\bm{x})$ and $v_t^0(\\bm{x})$ decays fast enough, i.e.,\n$|v_t^0(\\bm{x})|,|\\phi_t^0(\\bm{x})|\\rightarrow0$ as $|\\bm{x}-\\bm{r}(t)|\\gg\\xi$.  \n\nWith these building blocks, we construct a periodic potential and the\ncell-periodic Bloch state.\n\\begin{align}\n&v_t(\\bm{x})\\equiv\\sum_{\\bm{R}}v_t^0(\\bm{x}-\\bm{R}),\\label{pot}\\\\\n&u_{t\\bm k}(\\bm{x})\\equiv\\frac{a}{\\sqrt{V}}\\sum_{\\bm{R}}e^{i\\bm{k}\\cdot(\\bm{R}-\\bm{x})}\\phi_t^0(\\bm{x}-\\bm{R}),\\label{eq:Bloch}\n\\end{align}\nWe assume that $\\bm{A}^{\\text{ex}}(\\bm{x})$ respects the\nperiodicity, i.e.,\n$\\bm{A}^{\\text{ex}}(\\bm{x}-\\bm{R})=\\bm{A}^{\\text{ex}}(\\bm{x})$.  Then, as far\nas $\\phi_t^0(\\bm{x}-\\bm{R})^*\\phi_t^0(\\bm{x})$ and\n$v_t^0(\\bm{x}-\\bm{R})\\phi_t^0(\\bm{x})$ ($\\bm{R}\\neq\\bm{0}$) are entirely neglected,\n$u_{t\\bm k}(\\bm{x})$ is an eigenstate of the periodic Hamiltonian\n\\begin{equation}\nh_{t\\bm{k}}=\\frac{1}{2m}[\\tfrac{1}{i}\\bm{\\nabla}-e\\bm{A}_t^{\\text{ex}}(\\bm{x})+\\bm{k}]^2+v_t(\\bm{x})\\label{egH}\n\\end{equation}\nwith a completely flat band dispersion $\\varepsilon_{t\\bm{k}}=\\varepsilon_t^0$. \n\n\\subsection{Polarization and instantaneous magnetization}\nLet us first demonstrate the modern theory formula for the polarization and the\norbital magnetization by deriving $\\bm{p}$ and $\\bm{m}$ in two different ways.\n\nFirst, we present direct calculation of the polarization and the orbital\nmagnetization from the microscopic charge distribution and the persistent\ncurrent densities in this insulator. The instantaneous contribution to the\nlocal charge and current distribution from a single orbit $\\phi_t^0(\\bm{x})$\ncan be written as\n\\begin{align}\n\t\\label{eq:nt0}\n&n_t^0(\\bm{x})\\equiv e|\\phi_t^0(\\bm{x})|^2,\\\\\n&\\bm{j}_t^0(\\bm{x})\\equiv\\frac{e}{m i}\\phi_t^0(\\bm{x})^*(\\bm{\\nabla}-ie\\bm{A}^{\\text{ex}}(\\bm{x}))\\phi_t^0(\\bm{x}).\n\\label{eq:jt0}\n\\end{align}\nWe introduce vector fields $\\bm{p}_t^0(\\bm{x})$ and $\\bm{m}_t^0(\\bm{x})$ such\nthat\n\\begin{equation}\nn_t^0(\\bm{x})=\\bar{n}^0-\\bm{\\nabla}\\cdot\\bm{p}_t^0(\\bm{x}),\\quad\n\\bm{j}_t^0(\\bm{x})=\\bm{\\nabla}\\times\\bm{m}_t^0(\\bm{x}).\\label{defm0}\n\\end{equation}\nThe existence of such $\\bm{m}_t^0(\\bm{x})$ is guaranteed by the divergence-free\nnature of the instantaneous current density $\\bm{j}_t^0(\\bm{x})$. The current\ndensity induced by the adiabatic motion of $\\bm{r}(t)$ is captured by\n$\\mathbcal{j}_t^0(\\bm{x})$ in Eq.~\\eqref{idcurrent} whose divergence may not\nvanish. We assume both $\\bm{p}_t^0(\\bm{x})$ and $\\bm{m}_t^0(\\bm{x})$ decay\nrapidly for $|\\bm{x}-\\bm{r}(t)|>\\xi$, which specifies the boundary\ncondition for differential equations~(\\ref{defm0}).\n\nPhysical quantities of the insulator composed of periodically arranged\nlocalized orbits can be written as the sum of the contributions from each\norbit. For example microscopic current is given by\n\\begin{equation}\n\\bm{j}^{\\text{micro}}(t,\\bm{x})\\equiv\\sum_{\\bm{R}}\\bm{j}_t^0(\\bm{x}-\\bm{R})\n\\end{equation}\nand analogously for $n$, $\\bm{p}$, and $\\bm{m}$. These \\textit{microscopic}\nexpressions have a strong spatial dependence, periodically oscillating at the\nscale of $a$.  To derive to a smooth mesoscopic description, we need to perform\na convolution in space (Sec. 6.6 of Ref.~\\onlinecite{jackson1999}). Here we\nchoose the Gaussian $g(\\bm{x})=(\\pi R^2)^{-1}e^{-|\\bm{x}|^2\/R^2}$ ($R\\gg a$)\n\\begin{equation}\n\\bm{j}(t,\\bm{x})\\equiv\\int d^2x' g(\\bm{x}-\\bm{x}')\\bm{j}^{\\text{micro}}(t,\\bm{x}').\n\\end{equation}\nWe do the same for other quantities. Relations such as\n$\\bm{j}(t,\\bm{x})=\\bm{\\nabla}\\times\\bm{m}(t,\\bm{x})$ are preserved by the\nconvolution. Because the convolution is identical to the average for the\nperiodic distribution, we find $n(t,\\bm{x})=\\bar{n}=\\frac{e}{a^2}$,\n$\\bm{j}(t,\\bm{x})=\\bm{0}$,\n\\begin{align}\n&\\bm{p}(t,\\bm{x})=\\frac{1}{a^2}\\int d^2x'\\bm{p}_t^0(\\bm{x}'),\\label{modd2}\\\\\n&\\bm{m}_{\\text{pers}}(t,\\bm{x})=\\frac{1}{a^2}\\int d^2x'\\bm{m}_t^0(\\bm{x}').\\label{mod2}\n\\end{align}\n This is the part of the orbital magnetization produced by the persistent current as illustrated in Fig.~\\ref{fig:magnetization}a.\n\nLet us check that we get the same results using the general formulae of the\nmodern theory. Because of the non-overlapping assumption of $\\phi_t^0(\\bm{x})$,\nit can be readily shown that the formula in Eqs.~\\eqref{eq:modthp},\n\\eqref{eq:modthm} for the Bloch function~\\eqref{eq:Bloch} can be simplified to \n\\begin{align}\n&\\bm{p}(t)=\\frac{1}{a^2}\\int d^2x\\bm{x}n_t^0(\\bm{x})\\,\\,\\left(=\\frac{e}{a^2}\\bm{r}(t)\\right),\\label{modd1}\\\\\n&\\bm{m}_{\\text{pers}}(t)=\\frac{1}{2a^2}\\int d^2x\\bm{x}\\times\\bm{j}_t^0(\\bm{x}).\\label{mod1}\n\\end{align}\nwhere we used Eqs.~\\eqref{eq:nt0} and \\eqref{eq:jt0}. These are well-known\nexpressions in classical electrodynamics for the charge and current\ndistributions in a confined region (see Sec.~4.1 and 5.6 of\nRef.~\\onlinecite{jackson1999}). The equivalence of Eqs.~\\eqref{modd2},\n\\eqref{mod2}  and \\eqref{modd1}, \\eqref{mod1} can be easily checked by using\nthe definition of $\\bm{p}_t^0$ and $\\bm{m}_t^0$ in Eq.~(\\ref{defm0}) and\nintegrating by parts. The second equality of Eq.~\\eqref{modd1} follows from\nEqs.~\\eqref{exp}, \\eqref{eq:nt0}, and \\eqref{modd1}.\n\n\\subsection{Topological geometric magnetization}\\label{subsec:tgeoM}\nNext, we discuss the topological geometric contribution\n$\\mathbcal{m}^{\\text{top}}$ for this model. To this end, suppose that the\nposition of the potential minimum $\\bm{r}(t)$ adiabatically moves as a function\nof  $t\\in[0,T]$ and forms a closed curve as illustrated in\nFig.~\\ref{fig:magnetization}b. We assume the form of the potential, and thus\nthe localization length, remains unchanged during the adiabatic process.\n\nWe first apply our general expression for $\\mathbcal{m}^{\\text{top}}$ in\nEq.~\\eqref{eq:topc} to the Bloch function~\\eqref{eq:Bloch}. Thanks to the\nnon-overlapping assumption, the vector potential $\\bm{A}_{(t,\\bm{k})}$ is\n$\\bm{k}$-independent:\n\\begin{equation}\n\t\\bm{A}_{\\bm{K}}=(A_t,-\\bm{r}(t))^{\\rm T}\\label{eq:AK}\n\\end{equation}\nwith $A_t\\equiv -i\\int d^2x\\phi_t^0(\\bm{x})^*\\partial_t\\phi_t^0(\\bm{x})$.\nPlugging this into Eq.~\\eqref{eq:3Dtopo2}, we find\n\\begin{equation}\nP_3=\\frac{e}{2a^2}\\int_0^{T} dt\\,\\bm{r}(t)\\times \\partial_t\\bm{r}(t)\\cdot\\bm{e}_z=\\frac{eS_{\\bm{r}}}{a^2},\n\\end{equation}\nwhere $S_{\\bm{r}}$ represents the area enclosed by the orbit of $\\bm{r}(t)$ in\none cycle. Therefore,\n\\begin{equation}\n\t\\mathbcal{m}^{\\text{top}}=\\bm{e}_z\\frac{eS_{\\bm{r}}}{Ta^2}=\\frac{e}{2Ta^2}\\oint \\bm{r}(t)\\times d\\bm{r}(t).\\label{MArea1}\n\\end{equation}\nObserve the analogy to $\\bm{m}$ in Eq.~\\eqref{mod1}. This expression does not\nhave integer ambiguity because it is given by \\textit{abelian} third\nChern-Simons form.\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=0.5\\columnwidth]{C4e.pdf}\t\t\n\t\t\\caption{\\label{fig:C4e}  An adiabatic process with four-fold\n\t\trotational symmetry $C_4$.  Each unit cell contains two\n\t\toccupied Wannier orbitals, whose trajectories during adiabatic\n\t\tprocess are shown with dashed red and blue lines. The hatched\n\t\tarea is Arahonov-Bohm flux per unit cell acquired by such\n\t\tadiabatic process under applied magnetic field. Letters $a$\n\t\tand $b$ denote Wyckoff positions.}\n\t\\end{center}\n\\end{figure}\n\nLet us verify this result from a direct calculation. The adiabatic motion of\nthe single orbit following the potential minimum at $\\bm{x}=\\bm{r}(t)$ induces\na local current distribution\n\\begin{equation}\n\\mathbcal{j}_t^0(\\bm{x})=\\partial_t\\bm{r}(t)n_t^0(\\bm{x}).\\label{idcurrent}\n\\end{equation}\nIt becomes divergence-free if averaged over one period\n\\begin{align}\n&\\mathbcal{j}^0(\\bm{x})=\\int_0^T\\frac{dt}{T}\\mathbcal{j}_t^0(\\bm{x})=\\frac{1}{T}\\oint  d\\bm{r}(t)n_t^0(\\bm{x}),\\\\\n&\\bm{\\nabla}\\cdot\\mathbcal{j}^0(\\bm{x})=-\\frac{e}{T}\\oint  d\\bm{r}\\cdot\\bm{\\nabla}_{\\bm{r}}n_t^0(\\bm{x})=0.\n\\end{align}\nAs we have seen above, the sum of such microscopic currents from each unit cell\nproduces the bulk magnetization\n\\begin{align}\n\\mathbcal{m}^{\\text{top}}&=\\frac{1}{2a^2}\\int d^2x\\,\\bm{x}\\times\\mathbcal{j}^0(\\bm{x})\\notag\\\\\n&=\\frac{e}{2Ta^2}\\oint \\left(\\int d^2x \\bm{x}|\\phi_0(\\bm{x}-\\bm{r}(t))|^2\\right)\\times d\\bm{r}(t).\n\\end{align}\nThis agrees with Eq.~\\eqref{MArea1} because the integral in the parenthesis is\nprecisely $\\bm{r}(t)$ due to Eq.~\\eqref{exp}.\n\nThe result in Eq.~\\eqref{MArea1} can be readily generalized to the case with\nmulti-orbitals, such as examples in Fig.~\\ref{fig:magnetization}c and e. Let\nus introduce potential minima $\\bm{x}=\\bm{r}_n(t)$ ($n=1,2,\\cdots$) in each\nunit cell, which are adiabatically varied as a function of $t\\in[0,T]$. This\ntime, each orbit is allowed to form an \\textit{open} curve, as far as the total\npolarization $\\bm{p}(t)=(e\/a^2)\\sum_{n}\\bm{r}_n(t)$ satisfies\n$\\bm{p}(T)=\\bm{p}(0)$.  Under such an assumption, we find that\n\\begin{align}\n\t\\label{eq:P3ukN}\n\t\\mathbcal{m}^{\\text{top}}=&\\sum_{n}\\frac{e}{Ta^2}\\bigg(\\bm{S}_{\\bm{r}_n}+\\frac{1}{2}\\bm{r}_n(0)\\times\\bm{r}_n(T)\\bigg),\\\\\n\t\\bm S_{\\bm{r}_n}&\\equiv\\frac{1}{2}\\int_{\\bm{r}_n(0)}^{\\bm{r}_n(T)}\\bm{r}_n(t)\\times d\\bm{r}_n(t).\\label{Sn}\n\\end{align}\nWe present the proof in the Appendix~\\ref{app:MzKP}.  Although the above\nexpression appears to be the sum of single-band contributions, the ``would-be''\ncontribution from each band depends on the specific choice of the origin when\nit does not form a closed loop.  Only after performing the summation over\nall occupied bands, or in other words, only after fully taking into account the\n\\textit{non-abelian} nature of the third Chern-Simons form, the result restores\nthe independence from the origin choice.\n\nAs the application of the formula \\eqref{eq:P3ukN}, let us discuss the corner\ncharge of the $C_4$-symmetric quadrupole insulator introduced in\nRefs.~\\onlinecite{benalcazar2017,benalcazar2017}.  \nFor the wallpaper group\n$p4$, there exist three spacial Wyckoff positions: the unit cell origin at\n$\\bm{x}_a=(0,0)$, the center of the plaquette at $\\bm{x}_b=(a\/2,a\/2)$, and the\ncenter of bonds at $\\bm{x}_c=(a\/2,0)$, $(0,a\/2)$.~\\cite{ITC}  In the nontrivial\nphase, the two occupied Wannier orbitals locate at $\\bm{x}_b$, while in the\ntrivial phase they are at $\\bm{x}_a$. We consider a periodic adiabatic process\nillustrated in Fig.~\\ref{fig:C4e} starting with the nontrivial phase at $t=0$\nand passing the trivial phase at $t=T\/2$. The\ninstantaneous Hamiltonian $h_{t\\bm{k}}$ itself breaks the $C_4$-symmetry down\nto $C_2$ symmetry except when $t=0$ and $T\/2$, while the adiabatic process as a\nwhole implements the full $C_4$ in the sense of Eq.~\\eqref{eq:C4cycle}.  We can\nreadily compute $P_3$ of this process using Eq.~\\eqref{eq:P3ukN} which turns\nout to be $e\/2$. This is the corner charge of the quadrupole insulator as\npredicted by Eq.~\\eqref{corner}, which agrees with the original\nstudy.~\\cite{benalcazar2017,benalcazar2017} A variant of this adiabatic\nprocess was also discussed in Ref.~\\onlinecite{vanmiert2018}.\n\n\\subsection{Non-topological geometric magnetization}\\label{subsec:nontgeoM}\nIn this example we first consider a single electron in an anisotropic and\nrotating two-dimensional well,~\\cite{goldman2014} see\nFig.~\\ref{fig:magnetization}d. We assume a harmonic confining potential, i.e.,\nHamiltonian~(\\ref{simplemodel}) with \n\\begin{equation}\n\tv_t^0(\\bm x)=\\frac{1}{2}m(\\omega_x^2x_t^2+\\omega_y^2y_t^2),\n\t\\label{eq:assymv0}\n\\end{equation}\nwhere $\\bm x_t\\equiv(x\\cos\\Omega t+y\\sin\\Omega t,-x\\sin\\Omega t+y\\cos\\Omega t)$\nand $\\Omega=2\\pi\/T$. To obtain the geometric orbital magnetization for this\nmodel, we consider external magnetic field $\\bm B=B_z\\bm e_z$ described by the\nvector potential $\\bm A^{\\rm ex}(\\bm x)=\\bm B\\times\\bm x\/2$. (Strictly speaking,\nthis form is valid only around the origin as it lacks the required\nperiodicity.) The wave function of the instantaneous ground state of this model\ncan be obtained based on Ref.~\\onlinecite{rebane2012}:\n\\begin{align}\n\t\\phi_{t}^0(\\bm x)=\\mathcal{N}e^{\\frac{im\\omega_c(\\omega_y-\\omega_x)x_ty_t}{2(\\omega_x+\\omega_y)}-\\frac{m\\sqrt{(\\omega_x+\\omega_y)^2+\\omega_c^2}(\\omega_xx_t^2+\\omega_yy_t^2)}{2(\\omega_x+\\omega_y)}},\n\t\\label{eq:phi0assym}\n\\end{align}\nwhere $\\cal N$ is the normalization factor and $\\omega_c\\equiv eB_z\/m$ is the\ncyclotron frequency. The Berry phase $\\varphi_{B_z}$ during the adiabatic\nprocess $t\\in[0,T]$ is\n\\begin{align}\n\t\\varphi_{B_z}^0&=\\int_0^{T}dt\\int d^2 x\\phi_{t}^0(\\bm x)^*i\\partial_t\\phi_{t}^0(\\bm x)\\notag\\\\\n\t&=\\frac{\\pi \\omega_c(\\omega_y-\\omega_x)^2}{2\\omega_x\\omega_y\\sqrt{(\\omega_x+\\omega_y)^2+\\omega_c^2}}.\n\t\\label{eq:phiberry}\n\\end{align}\nFrom Eq.~(\\ref{eq:calMz}) it follows that the adiabatic\nprocess~(\\ref{eq:assymv0}) has non-zero geometric orbital magnetic moment\n$\\mathcal{m}_z^0$\n\\begin{align}\n\t\t\\mathcal{m}_z^0=\\frac{e(\\omega_y-\\omega_x)^2\\Omega}{4ma^2\\omega_x\\omega_y(\\omega_x+\\omega_y)}.\\label{cmz0}\n\\end{align}\n\n\nNow we construct the Bloch function \\eqref{eq:Bloch} using $\\phi_{t}^0(\\bm x)$\nas the building block and compute the geometric orbital magnetization\n$\\mathcal{m}_z$ based on Eq.~\\eqref{eq:nontopcalM} for the corresponding band\ninsulator.  To this end we need  instantaneous eigenstates and eigenenergies in\nabsence of external magnetic field including unoccupied bands.  The Hamiltonian\n$h_{t\\bm k}$ is given by Eq.~(\\ref{egH}) with $v_t^0(\\bm x)$ given by\nEq.~(\\ref{eq:assymv0}) and $\\bm A^{\\rm ex}=0$.  We assume that there is no\noverlap between wavefunctions belonging to different unit cells as before.\n(When $\\omega_{x}, \\omega_{y}$ are large enough, such an assumption is valid at\nleast for relevant low-energy states.) Bloch\nwavefunctions read \n\\begin{align}\nu_{t\\bm{n}\\bm k}(\\bm{x})&\\equiv\\frac{a}{\\sqrt{V}}\\sum_{\\bm{R}}e^{i\\bm{k}\\cdot(\\bm{R}-\\bm{x})}\\phi_{t\\bm{n}}^0(\\bm{x}-\\bm{R}),\n\\end{align}\nwhere $\\bm{n}\\equiv(n_x,n_y)$ labels energy levels of two-dimensional the\nanisotropic harmonic oscillator and $\\bm{n}=(0,0)$ corresponds to the ground\nstate in Eq.~\\eqref{eq:phi0assym} with $\\omega_c=0$. Substituting above\nexpressions to Eqs.~(\\ref{eq:topc}) and~(\\ref{eq:nontopcalM}), we find\n\\begin{align}\n\t&\\mathcal{m}_z^{\\text{top}}=0,\\\\\n\t&\\mathcal{m}_z^{\\text{non-top}}=\\sum_{\\bm{n}\\neq(0,0)}\\frac{\\vert\\langle\\phi_{t}^0\\vert\\bm x\\times\\bm\\nabla\\vert\\phi_{t\\bm{n}}^0\\rangle\\vert^2e\\Omega}{4ma^2(n_x\\omega_x+n_y\\omega_y)}\\notag\\\\\n\t&=\\frac{\\vert\\langle\\phi_{t}^0\\vert\\bm x\\times\\bm\\nabla\\vert\\phi_{t(1,1)}^0\\rangle\\vert^2e\\Omega}{4ma^2(\\omega_x+\\omega_y)}=\\mathcal{m}_z^0.\n\t\\label{eq:barnett}\n\\end{align}\n\n\n\\subsection{Geometric magnetization by rotation}\\label{subsec:rotoM}\nHere we calculate the contribution to the geometric orbital magnetization of a rotating\nuncharged body and compare it to the classical Barnett\neffect.~\\cite{barnett1915,barnett1935} The Barnett effect predicts magnetization\n$\\chi\/\\gamma\\Omega$, where $\\chi$ is the paramagnetic susceptibility, $\\gamma$\nis the electron gyromagnetic ratio, and $\\Omega$ is the rotation frequency. Since the rotation\naxis does not necessarily coincide with potential well minima we have $v_t^0(\\bm x-\\bm\nr(t))$ with $v_t^0$ from Eq.~(\\ref{eq:assymv0}) and $\\bm r(t)=(R\\cos\\Omega\nt,R\\sin\\Omega t,0)^{\\rm T}$, where $R$ is the distance of the potential well minima to the\nrotation axis. The lowest-energy instantaneous wavefunction $\\phi_t(\\bm x)$\ncan be obtained from Eq.~(\\ref{eq:phi0assym}) by performing gauge\ntransformation\n\\begin{align}\n\t\\label{eq:phi0}\n\t\\phi_{t}^0(\\bm x)=\\mathcal{N}e^{\\frac{im\\omega_c(\\omega_y-\\omega_x)(x_t-R)y_t}{2(\\omega_x+\\omega_y)}+\\frac{i}{2}m\\omega_c\\bm{e}_z\\cdot\\bm r(t)\\times\\bm x}\\\\\n\t\\times e^{-\\frac{m\\sqrt{(\\omega_x+\\omega_y)^2+\\omega_c^2}(\\omega_x(x_t-R)^2+\\omega_yy_t^2)}{2(\\omega_x+\\omega_y)}}.\\notag\n\\end{align}\nAs compared to Eq.~\\eqref{eq:phiberry}, the Berry phase $\\varphi_{B_z}$ during\nthe adiabatic process $t\\in[0,T]$ acquires an additional contribution $eB_z \\pi\nR^2$ from the  Aharonov-Bohm phase. Therefore electrons contribute to the\nfollowing geometric orbital magnetization \n\\begin{align}\n\t\\mathcal{m}_z&=\\frac{eR^2\\Omega}{2a^2}+\\mathcal{m}_z^0.\n\\end{align}\nThe first term can be identified with $\\mathcal{m}_z^{\\text{top}}$ in\nEq.~(\\ref{MArea1}) and the second term is the contribution in Eq.~\\eqref{cmz0}.\nSince the body is uncharged, the contribution from ions cancels the topological\ncontribution, while $\\mathcal{m}_z^{\\text{non-top}}=\\mathcal{m}_z^0$ remains\nsince ions are much more localized compared to electrons. Assuming anisotropy\n$\\omega_x\/\\omega_y=2$, and confinement of electrons on the scale of angstroms,\nwe find that contribution~(\\ref{eq:barnett}) is on the same order as Barnett\neffect for paramagnets with paramagnetic susceptibility $\\chi\\sim10^{-5}$. For\ncomparison, paramagnets have typically magnetic susceptibility\n$\\chi\\sim10^{-3}-10^{-5}$.~\\cite{wiki:paramagnetism}\n\n\\section{Examples: finite interacting systems}\\label{sec:examples2}\nIn this section we demonstrate the validity of Eq.~\\eqref{eq:calMz} for\nfinite interacting systems. We consider two canonical ways of introducing the\ntime-dependence to the Hamiltonian: rotating~\\cite{ceresoli2002,stengel2018}\nand translating the whole system.~\\cite{goldman2014,juraschek2017,dong2018}\n\nWe consider many-body systems under the \\textit{open boundary condition} in two\nspatial dimensions.  We start with a \\textit{time-independent} Hamiltonian\n$\\hat{H}$ that can contain arbitrary interactions.  The total charge, current,\npolarization, and orbital magnetization operator for this Hamiltonian can be\nwritten as $\\hat{\\bm{N}}=\\int_Vd^2x \\hat{\\bm{n}}(\\bm{x})$, $\\hat{\\bm{J}}=\\int\nd^2x \\hat{\\bm{j}}(\\bm{x})$, $\\hat{\\bm{X}}=\\int_Vd^2x\n\\bm{x}\\hat{\\bm{n}}(\\bm{x})$, $\\hat{\\bm{M}}=(1\/2)\\int_Vd^2x\n\\bm{x}\\times\\hat{\\bm{j}}(\\bm{x})$.  We stress that these expressions are valid\nonly when the system is confined in a finite region; they need to be modified\nin extended systems under the periodic boundary conditions as done by the\nmodern theory.  We denote the many-body ground state of $\\hat{H}$ and its\nenergy by $|\\Phi\\rangle$ and $E$, respectively.\n\nTo compute the many-body Berry phase, let $\\hat{H}_{\\bm{B}}$ be the Hamiltonian\nwith the vector potential in the symmetric gauge\n$\\bm{A}^{\\text{ex}}(\\bm{x})=(1\/2)\\bm{B}\\times\\bm{x}$ with $\\bm{B}=B_z\\bm{e}_z$.\nExpanding to the linear order in $B_z$ and using\n$\\hat{\\bm{j}}(\\bm{x})=-\\partial_{\\bm{A}(\\bm{x})}\\hat{H}$, we get\n\\begin{equation}\n\\hat{H}_{B_z}=\\hat{H}-\\hat{M}_zB_z+O(B_z^2).\\label{perturbationmB}\n\\end{equation}\nTherefore, the ground state of $\\hat{H}_{B_z}$ to the leading order in $B_z$\ncan be expressed as\n\\begin{equation}\n|\\Phi_{B_z}\\rangle=|\\Phi\\rangle+\\hat{Q}\\frac{1}{\\hat{H}-E}\\hat{Q}\\hat{M}_zB_z|\\Phi\\rangle+O(B_z^2).\\label{firstB}\n\\end{equation}\nHere $\\hat{Q}\\equiv1-|\\Phi\\rangle\\langle\\Phi|$ is the projector onto excited\nstates.\n\n\\subsection{Rotation}\\label{subsec:rotom}\nHere we consider the time-dependence of the problem induced by the rotation of\nthe whole system\n\\begin{equation}\n\\hat{H}_t\\equiv e^{-i\\hat{L}_z\\Omega t}\\hat{H}e^{i\\hat{L}_z\\Omega  t},\n\\end{equation}\nwhere $\\Omega=\\bm{e}_z\\Omega$ is the rotation frequency and $\\hat{\\bm{L}}$ is\nthe angular momentum operator. For the time-dependent Hamiltonian $\\hat{H}_t$, \nthe orbital magnetization operator $\\hat{\\bm{M}}_t\\equiv (1\/2)\\int_Vd^2x\n\\bm{x}\\times\\hat{\\bm{j}}_t(\\bm{x})$\n\\begin{equation}\n\\hat{\\bm{M}}_t=e^{-i\\hat{L}_z\\Omega t}\\hat{\\bm{M}}e^{i\\hat{L}_z\\Omega t}.\\label{eq:rotm}\n\\end{equation}\n\nWe evaluate the instantaneous contribution $\\bm{m}_{\\text{pers}}$ and the\ngeometric contribution $\\mathbcal{m}$ to the orbital magnetization via the\nformulae in Eqs.~\\eqref{eq:Xinst} and \\eqref{eq:Xgeom}. The instantaneous\ncontribution is given by the instantaneous ground state\n$|\\Phi_t\\rangle=e^{-i\\hat{L}_z\\Omega  t}|\\Phi\\rangle$\n\\begin{equation}\n\\bm{m}_{\\text{pers}}\\equiv\\int_0^T\\frac{dt}{T}\\frac{\\langle\\Phi_t|\\hat{\\bm{M}}_t|\\Phi_t\\rangle}{V}=\\frac{\\langle\\Phi|\\hat{\\bm{M}}|\\Phi\\rangle}{V}.\\label{M1}\n\\end{equation}\nThe geometric contribution is given by the many-body Berry phase.  Since the\ninstantaneous ground state of the Hamiltonian $\\hat{H}_{tB_z}\\equiv\ne^{-i\\hat{L}_z\\Omega  t}\\hat{H}_{B_z}e^{i\\hat{L}_z\\Omega  t}$ is given by\n$|\\Phi_{tB_z}\\rangle=e^{-i\\hat{L}_z\\Omega  t}|\\Phi_{B_z}\\rangle$, we have \n\\begin{equation}\n\\varphi_{B_z}=\\int_0^Tdt\\langle\\Phi_{tB_z}|i\\partial_t|\\Phi_{tB_z}\\rangle=T\\langle\\Phi_{B_z}|\\hat{L}_z\\Omega|\\Phi_{B_z}\\rangle.\n\\end{equation}\nThis is the expectation value of $\\hat{L}_z\\Omega$ in the presence of the\nperturbation $-\\hat{m}_z B_z$ in Eq.~\\eqref{perturbationmB}. Using\nEq.~\\eqref{firstB}, we get\n\\begin{equation}\n\t\\mathbcal{m}=\\langle\\Phi|\\hat{L}_z\\Omega\\hat{Q}\\frac{1}{\\hat{H}-E}\\hat{Q}\\frac{\\hat{\\bm{M}}}{V}|\\Phi\\rangle+\\text{c.c.}\\label{M2}\n\\end{equation}\n\nWe verify these results by solving time-dependent problem.  The solution to the\n\\textit{time-dependent} Schr\\\"odinger equation\n$i\\partial_t|\\Psi_t\\rangle=\\hat{H}_t|\\Psi_t\\rangle$ can be readily constructed\nusing the ground state $|\\Phi_{\\Omega}\\rangle$ of the \\textit{time-independent}\nHamiltonian\n\\begin{equation}\n\\hat{H}_{\\Omega}\\equiv \\hat{H}-\\hat{L}_z\\Omega.\\label{perturbationOL}\n\\end{equation}\nThe solution that is smoothly connected to the ground state in the static limit $\\Omega\\rightarrow0$ reads\n\\begin{equation}\n|\\Psi_t\\rangle=e^{-i\\hat{L}_z\\Omega  t-iE_{\\Omega}t}|\\Phi_{\\Omega}\\rangle.\\label{eq:rotphi2}\n\\end{equation}\nThe time-average of the orbital magnetization is thus given by\n\\begin{equation}\n\\bm{m}=\\int_0^T\\frac{dt}{T}\\frac{\\langle\\Psi_t|\\hat{\\bm{M}}_t|\\Psi_t\\rangle}{V}=\\frac{\\langle\\Phi_{\\Omega}|\\hat{\\bm{M}}|\\Phi_{\\Omega}\\rangle}{V}.\n\\end{equation}\nThis is the expectation value of $\\hat{\\bm{M}}$ in the presence of the\nperturbation $-\\hat{L}_z\\Omega$ as in Eq.~\\eqref{perturbationOL}.  The\nfirst-order perturbation theory with respect to $\\Omega$ gives\n\\begin{equation}\n\t\\bm{m}=\\bm m_{\\text{pers}}+\\langle\\Phi|\\hat{L}_z\\Omega\\hat{Q}\\frac{1}{\\hat{H}-E}\\hat{Q}\\frac{\\hat{\\bm{M}}}{V}|\\Phi\\rangle+\\text{c.c.}\n\\end{equation}\nThis is precisely $\\bm{m}_{\\text{pers}}+\\mathbcal{m}$ predicted above in\nEqs.~\\eqref{M1} and ~\\eqref{M2}. As it is clear from the derivation, the\nagreement of the two independent approaches is guaranteed by the Maxwell\nrelation for the free energy\n$\\hat{F}\\equiv\\hat{H}-\\hat{L}_z\\Omega-\\hat{M}_zB_z$\n\n\\begin{equation}\n\\partial_{B_z}\\langle\\hat{L}_z\\rangle=-\\partial_{B_z}\\partial_{\\Omega}\\langle\\hat{F}\\rangle=-\\partial_{\\Omega}\\partial_{B_z}\\langle\\hat{F}\\rangle=\\partial_{\\Omega}\\langle\\hat{M}_z\\rangle.\n\\end{equation}\n\n\\subsection{Translation}\nNext let us introduce the time-dependence by the translation.  All discussions\nproceed in essentially the same way, while there are still a few differences.\nFirst we define the time-dependent Hamiltonian by\n\\begin{equation}\n\\hat{H}_t'\\equiv \\hat T_t\\hat{H}\\hat T_t^\\dagger,\n\\end{equation}\nwhere $\\hat T_t=e^{-i\\hat{\\bm P}\\cdot\\bm r(t)}$ is the translation by amount $\\bm\nr(t)$ and $\\hat{\\bm{P}}$ is the momentum operator. For $\\hat{H}_t$  the orbital\nmagnetization operator becomes\n\\begin{equation}\n\\hat{\\bm{M}}_t'= \\hat T_t\\left(\\hat{\\bm{M}}+\\frac{1}{2}\\bm{r}(t)\\times\\hat{\\bm{J}}\\right)\\hat T_t^\\dagger\\label{eq:transm}\n\\end{equation}\nwhere the second term in the parenthesis is due to the change of the origin. The\ninstantaneous ground state  $|\\Phi_t'\\rangle=\\hat T_t|\\Phi\\rangle$\ngives $\\bm{m}_{\\text{pers}}$ as in Eq.~(\\ref{M1}), where we used\n$\\langle\\Phi|\\hat{\\bm{J}}|\\Phi\\rangle=\\bm{0}$.\n\nNext, we compute the geometric contribution via the many-body Berry phase.\nIn the presence of magnetic field translation operator $\\hat\nT_{tB_z}\\equiv\\hat T_{B_z}(\\bm r(t))$ becomes translation followed by gauge\ntransformation~\\cite{brown1964,zak1964}\n\\begin{equation}\n\\partial_t\\hat{T}_{tB_z}\\equiv -i(\\hat{\\bm{P}}+\\tfrac{e}{2}\\bm{B}\\times\\hat{\\bm{X}})\\cdot\\partial_t\\bm{r}(t)\\hat{T}_{tB_z}.\n\\end{equation}\nThe instantaneous ground state of $\\hat{H}_{tB_z}\\equiv\n\\hat{T}_{tB_z}\\hat{H}_{B_z}\\hat{T}_{tB_z}^\\dagger$ is\n$|\\Phi_{tB_z}\\rangle=\\hat{T}_{tB_z}|\\Phi_{B_z}\\rangle$, thus the many-body\nBerry phase reads\n\\begin{equation}\n\\varphi_{B_z}=\\int_0^Tdt\\langle\\Phi_{B_z}|\\hat{T}_{tB_z}^\\dagger i\\partial_t\\hat{T}_{tB_z}|\\Phi_{B_z}\\rangle=eN\\bm{S}_{\\bm{r}}\\cdot \\bm{B},\n\\end{equation}\nHere, $\\bm{S}_{\\bm{r}}\\equiv\\frac{1}{2}\\oint \\hat{\\bm{r}}\\times d\\bm{r}$\nrepresents the area swept by $\\bm{r}(t)$ in one cycle. In the derivation, we\nused\n\\begin{align}\n&\\hat{T}_{tB_z}^\\dagger i\\partial_t\\hat{T}_{tB_z}\\notag\\\\\n&=(\\hat{\\bm{P}}+\\tfrac{e}{2}\\bm{B}\\times\\hat{\\bm{X}})\\cdot\\dot{\\bm{r}}(t)+\\tfrac{eN}{2}\\bm{r}(t)\\times\\partial_t\\bm{r}(t)\\cdot \\bm{B}.\n\\end{align}\nTherefore, when the whole system is translated, the geometric contribution to\nthe orbital magnetization captures the Aharonov-Bohm phase\n\\begin{equation}\n\\mathbcal{m}=\\frac{eN}{TV}\\bm{S}_{\\bm{r}}.\\label{M4}\n\\end{equation}\n\nTo verify the above results, we consider the time-dependent Schr\\\"odinger\nequation $i\\partial_t|\\Psi_t'\\rangle=\\hat{H}_t'|\\Psi_t'\\rangle$. An\n approximate solution is given by $|\\Psi_t'\\rangle=\\hat\nT_t|\\Phi_{t\\bm{r}}\\rangle$, where $|\\Phi_{t\\bm{r}}\\rangle$ is the instantaneous\nground state of the Hamiltonian $\\hat{H}_{t\\bm{r}}\\equiv\n\\hat{H}-\\hat{\\bm{P}}\\cdot\\partial_t\\bm{r}(t)$. Therefore, the time-average of\nthe orbital magnetization is\n\\begin{align}\n&\\bm{m}=\\int_0^T\\frac{dt}{T}\\frac{\\langle\\Psi_t'|\\hat{\\bm{M}}_t'|\\Psi_t'\\rangle}{V}\\\\\n&=\\int_0^T\\frac{dt}{T}\\frac{\\langle\\Phi_{t\\bm{r}}|\\hat{\\bm{M}}|\\Phi_{t\\bm{r}}\\rangle}{V}+\\frac{eN}{2TV}\\int_0^Tdt\\,\\bm{r}(t)\\times \\partial_t\\bm{r}(t).\\notag\n\\end{align}\nIn the adiabatic limit, this reproduces $\\bm{m}_{\\text{pers}}+\\mathbcal{m}$ in\nEqs.~\\eqref{M1} and \\eqref{M4}. In the derivation we used\n$\\langle\\Phi_{t\\bm{r}}|\\hat{\\bm{J}}|\\Phi_{t\\bm{r}}\\rangle=eN\\partial_t\\bm{r}(t)$\nfor the ground state of $\\hat H_{t\\bm r}$.\n\n\\section{Conclusion}\\label{sec:conclusions}\nIn order to obtain current and charge distribution in a medium, one needs to\nsolve Maxwell's equation together with two constitutive relations [see\nEq.~(\\ref{eq:jmeso})] that fully characterize the medium at the mesoscopic\nscale. The modern theories, developed in the last 30 years, provide handy\nformulae to calculate electric\npolarization~\\cite{king-smith1993,vanderbilt1993,resta1994,resta2007} and\norbital magnetization~\\cite{xiao2005,thonhauser2005,ceresoli2006,shi2007} for\nrealistic materials.\n\nThe focus of this work is on spinless short-range entangled systems under\nperiodic adiabatic evolution. Our main result is to identify an additional\ncontribution to the orbital magnetization that we name geometric orbital\nmagnetization $\\mathbcal{m}$. This new contribution is defined only after\nperforming the time-average over the period of the adiabatic process, which\nmakes the current density divergence-free. We find that the geometric orbital\nmagnetization can be expressed compactly as derivative of the many-body Berry\nphase with respect to an externally applied magnetic field. For band\ninsulators, we obtain handy formulae for the bulk geometric orbital\nmagnetization $\\mathbcal{m}$ in terms of instantaneous Bloch states and\nenergies. Interestingly, we find\nthat for band insulators\n$\\mathbcal{m}=\\mathbcal{m}^{\\text{top}}+\\mathbcal{m}^{\\text{non-top}}$ consists\nof two pieces, where topological piece $\\mathbcal{m}^{\\text{top}}$  depends\nonly on the Bloch states of occupied bands. For spinless systems only electric\npolarization and orbital magnetization enter constitutive relations, since the\ncontributions from higher moments are typically negligible.~\\cite{jackson1999}\nIn this sense, our results together with ``the modern theories'' provide a\ncomplete mesoscopic description of a medium under periodic adiabatic time\nevolution. In this work we have not considered adiabatic processes with ground\nstate degeneracy,~\\cite{meidan2011} it would be interesting to see to which\nextent our findings can be generalized to such systems.\n\nIn the present work, the adiabaticity assumption is crucial for validity of the\nobtained results. In practice, for band insulators with band gaps on the order\nof electronvolt, this conditions requires that the period $T$ is larger than\ncouple of femtoseconds. Nevertheless, shorter period $T$ results in a larger\ngeometric orbital magnetization. It would be therefore interesting to extend\nour results to the case of strong drive that excites unoccupied bands. In the\ncase of Thouless pumps such extension was very fruitful and resulted in recent\ndiscovery of shift currents.~\\cite{balz1981,morimoto2016}\n\nAlthough higher (than dipole) electric and magnetic multiple moments typically\ndo not enter constitutive relations, the knowledge of these quantities may be\nuseful for certain systems.~\\cite{benalcazar2017,benalcazar2018} In fact, it is\na topic of current research whether higher moments can be established as bulk\nquantities in general.~\\cite{gao2018b,shitade2018,kang2018,metthew2018,ono2019}\nIn the presence of certain crystalline symmetries, both electric\npolarization and topological geometric orbital magnetization can be quantized,\nin which case they can serve as a topological invariants. In this context, we\nshowed that the quantized quadrupole moment is related to\n$\\mathcal{m}^{\\text{top}}_z$ in systems with proper symmetries that allow bulk\ndefinition of the quadrupole moment.~\\cite{ono2019}\n\nIn this work we succeeded in separating $\\mathbcal{m}$ into the topological and\nthe non-topological piece only for band insulators. There, we found that the\ntopological contribution is expressed as the third Chern-Simons form ($P_3$), in\n$(t,k_x,k_y)$ space. For interacting systems, based on examples considered in\nSec.~\\ref{sec:examples2}, we conclude that it is possible to separate\nAharonov-Bohm contribution originating from the center of mass motion. In fact,\nthis contribution can be captured by calculating $P_3$ formally defined for the\nmany-body ground state as a function of time and two solenoidal fluxes.\nClearly, the many-body $P_3$ defined in such manner is abelian and does not\ncapture all possible topological contributions. For example, it vanishes for\nthe model in Fig.~\\ref{fig:C4}. As a future direction, it would be interesting\nto see if separation achieved for band insulators is possible for general\nsingle-particle, or even many-body systems. The affirmative answer to this\nquestion would provide a way to define $P_3$ in two dimensional systems with\nadiabatic time-dependence lacking the translational invariance or the\nsingle-particle description. The formula for $P_3$ in many-body\nthree-dimensional systems already exist in the\nliterature,~\\cite{wang2014,shiozaki2018b} where it was argued that $P_3$ is\nrelated to the magnetoelectric polarizability. The magnetoelectric\npolarizability of three-dimensional materials contains, at\nleast for the case of band insulators, not only topological but also\nnon-topological contribution,~\\cite{essin2010} thus the analogous ``separation\nquestion'' arises also in that context. Additionally, defining quantized\nquadrupole moment for interacting systems is one of the open\nquestions.~\\cite{kang2018,metthew2018,ono2019} Since for band insulators we\nfind connection between $\\mathcal{m}_z^{\\text{top}}$ and quantized quadrupole\nmoment, separating $\\mathcal{m}_z^{\\text{top}}$ contribution in interacting\nsystems might provide useful many-body definition of quantized quadrupole\nmoment.\n\nWe hope that our work will also have practical implication as it contributes to\nemerging field of ``dynamical material design'' by providing a way to calculate\nadditional orbital magnetization contribution that appears in these\nsystems.~\\cite{ceresoli2002,juraschek2017,juraschek2018,dong2018,stengel2018}\n\n\\begin{acknowledgements}\nWe would like to thank David Vanderbilt for drawing\nRefs.~\\onlinecite{ceresoli2002,juraschek2017,juraschek2018,dong2018,stengel2018}\nto our attention.  The work of S.O. is supported by Materials Education program\nfor the future leaders in Research, Industry, and Technology (MERIT).  The work\nof H.W. is supported by JSPS KAKENHI Grant No.~JP17K17678 and by JST PRESTO\nGrant No.~JPMJPR18LA. \n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}