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Physical review. B, Condensed matter

Comment on "Canonical-ensemble results for the Ising model with random bonds in two dimensions"

Kurt Binder, I. Morgenstern

Published May 1, 1983• 8 citations
Publisher

Abstract

In numerical calculations of a quantity $\ensuremath{\Lambda}(T)\ensuremath{\equiv}\ensuremath{-}\frac{\ensuremath{\partial}\mathrm{ln}{\ensuremath{\chi}}_{\mathrm{EA}}^{{}^{\ensuremath{'}}}}{\ensuremath{\partial}T}$, where ${\ensuremath{\chi}}_{\mathrm{EA}}^{\ensuremath{'}}\ensuremath{\equiv}\ensuremath{\Sigma}{({〈{\ensuremath{\sigma}}_{i}{\ensuremath{\sigma}}_{j}〉}_{T}^{2})}_{J}$ for finite $L\ifmmode\times\else\texttimes\fi{}L$ Edwards-Anderson (EA) models, Fern\'andez [Phys. Rev. B 25, 417 (1982)] finds that $\ensuremath{\Lambda}(T)$ has a peak at ${T}_{0}\ensuremath{\approx}0.6$ (Gaussian model) or ${T}_{0}\ensuremath{\approx}1.0$, for $L=4,6,8,10,or 11$, respectively. He finds that the peak height ${\ensuremath{\Lambda}}_{max}=\ensuremath{\Lambda}({T}_{0})$ increases in direct proportion to $L$, and interprets his results in terms of a phase transition at ${T}_{0}$, where the correlation length ${\ensuremath{\xi}}_{\mathrm{EA}}\ensuremath{\propto}{(T\ensuremath{-}{T}_{0})}^{\ensuremath{-}1}$. This conclusion is in disagreement with our previous findings that ${\ensuremath{\xi}}_{\mathrm{EA}}$ is finite in this temperature range, although a (dynamic) "freezing transition" of the spins occurs at ${T}_{f}\ensuremath{\approx}1.0$ (Gaussian model) or ${T}_{f}\ensuremath{\approx}1.3$ ($\ifmmode\pm\else\textpm\fi{}J$ model), which lead us to suggest that the two-dimensional EA model has a phase transition at zero temperature only.

DOI
10.1103/physrevb.27.5826

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Canonical-ensemble results for the Ising model with random bonds in two dimensions

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The purpose of this paper is to show how some interesting numerically exact thermal averages can be computed for finite two-dimensional Ising models by a transfer-matrix method, and to apply these methods to the Ising version of the Edwards-Anderson (EA) model of a spin-glass to ascertain whether there is a phase transition. We do not study the spatial dependence of $〈{\ensuremath{\sigma}}_{0}{\ensuremath{\sigma}}_{l}〉$ for, as we show with an example, its behavior for finite systems can be misleading. It is first shown how to obtain ${\ensuremath{\chi}}_{\mathrm{EA}}^{\ensuremath{'}}$, defined by ${\ensuremath{\chi}}_{\mathrm{EA}}^{\ensuremath{'}}={N}^{\ensuremath{-}1}\ensuremath{\Sigma}{〈{〈{\ensuremath{\sigma}}_{i}{\ensuremath{\sigma}}_{j}〉}_{T}^{2}〉}_{J}$. We compute ${\ensuremath{\chi}}_{\mathrm{EA}}^{\ensuremath{'}}$ and study the quantity $\ensuremath{\Lambda}=\frac{\ensuremath{-}\ensuremath{\partial}\mathrm{ln}({\ensuremath{\chi}}_{\mathrm{EA}}^{\ensuremath{'}})}{\ensuremath{\partial}T}$, both as a function of temperature ($T$) and of the number of spins ($N$) in the system. The results obtained for square systems of up to 100 spins in the case where $J=\ifmmode\pm\else\textpm\fi{}1$ with equal probability and for square systems of up to 121 spins in the case where each $J$ is normally distributed about $J=0$ are in accord with the existence of a critical point at ${T}_{0}\ensuremath{\simeq}1.0$ and at ${T}_{0}\ensuremath{\simeq}0.6$, respectively. In addition the value $\ensuremath{\nu}\ensuremath{\approx}1$ is obtained. The value $q=0$ for $T>0$ is consistent with the results obtained. The low-temperature entropy per spin ($S$) is computed for long strips of different widths. Extrapolation to an infinite width yields $\frac{S}{k}\ensuremath{\simeq}0.07$. It is also shown how to calculate the probability, $P(\ensuremath{\eta})$, that the quantity, $\ensuremath{\eta}={N}^{\ensuremath{-}1}\ensuremath{\Sigma}{\ensuremath{\tau}}_{i}{\ensuremath{\sigma}}_{i}$, where each ${\ensuremath{\tau}}_{i}=0,\ifmmode\pm\else\textpm\fi{}1$ take any value in the range $1<~\ensuremath{\eta}<~1$. The probability, $P(\ensuremath{\eta})$, obtained for the EA model at low temperatures often has several maxima separated by regions of improbable values of $\ensuremath{\eta}$, as is to be expected of a system with metastable states.

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Reply to "Comment on `Canonical-ensemble results for the Ising model with random bonds in two dimensions' "

Julio F. Fernández

In the previous Comment, Binder and Morgenstern use their previous result, ${〈{S}_{i}{S}_{j}〉}^{2}=c\mathrm{exp}(\ensuremath{-}\frac{{r}_{\mathrm{ij}}}{\ensuremath{\xi}})$ for small systems, plus their hypothesis that $\ensuremath{\xi}$ and $c$ depend weakly on $L$, to produce results strongly similar to what I have obtained previously. They thus suggest that my results, contrary to my own conclusion, do not indicate the existence of a critical point in the Edwards-Anderson model in two dimensions. I show here that their claim is not well founded.

Physical review. B, Condensed matter•May 1, 1983•1 citation

Reply to "Comment on `Canonical-ensemble results for the Ising model with random bonds in two dimensions"'

Julio F. Fernández

In the previous Comment, Binder and Morgenstern use their previous result, ${〈{S}_{i}{S}_{j}〉}^{2}=c \mathrm{exp}(\frac{{\ensuremath{-}r}_{\mathrm{ij}}}{\ensuremath{\xi}})$ for small systems, plus their hypothesis that $\ensuremath{\xi}$ and $c$ depend weakly on $L$, to produce results strongly similar to what I have obtained previously. They thus suggest that my results, contrary to my own conclusion, do not indicate the existence of a critical point in the Edwards-Anderson model in two dimensions. I show here that their claim is not well founded.

Physical review. B, Condensed matter•Apr 1, 1983•0 citations

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’’Exact’’ calculations for finite Ising spin glass lattices (invited)

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By a recursive method numerically exact free energies and correlation functions are calculated for finite two- and three-dimensional Ising lattices with bonds chosen randomly either ±J or from a gaussian distribution. While (dynamic) Monte Carlo simulations of precisely the same systems show the onset of Edwards-Anderson ordering at rather sharply defined nonzero freezing temperature Tf, the exact static calculations imply that there is no static order parameter qEA = [<Si≳2T]av at nonzero temperature. The correlation length, which would become infinite when qEA would start to become nonzero, is found at Tf to be of the order of a few lattice spacings only. As a result, the freezing-in of these models is identified as a dynamic phenomenon, and not a phase transition associated with static Edwards-Anderson order. The specific heat of the gaussian spin glass is found to vary nearly linear with temperature. On the other hand, in two-dimensional Ising systems with random fields a nonzero (Mattis-like) spin glass order is found.

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In numerical calculations of a quantity \ensuremathΛ(T)\ensuremath≡\ensuremath−\ensuremath∂ln\ensuremathχEA\ensuremath′\ensuremath∂T\ensuremath{\Lambda}(T)\ensuremath{\equiv}\ensuremath{-}\frac{\ensuremath{\partial}\mathrm{ln}{\ensuremath{\chi}}_{\mathrm{EA}}^{{}^{\ensuremath{'}}}}{\ensuremath{\partial}T}\ensuremathΛ(T)\ensuremath≡\ensuremath−\ensuremath∂T\ensuremath∂ln\ensuremathχEA\ensuremath′​​, where \ensuremathχEA\ensuremath′\ensuremath≡\ensuremathΣ(〈\ensuremathσi\ensuremathσj〉T2)J{\ensuremath{\chi}}_{\mathrm{EA}}^{\ensuremath{'}}\ensuremath{\equiv}\ensuremath{\Sigma}{({〈{\ensuremath{\sigma}}_{i}{\ensuremath{\sigma}}_{j}〉}_{T}^{2})}_{J}\ensuremathχEA\ensuremath′​\ensuremath≡\ensuremathΣ(〈\ensuremathσi​\ensuremathσj​〉T2​)J​ for finite L\ifmmode×\else\texttimes\fiLL\ifmmode\times\else\texttimes\fi{}LL\ifmmode×\else\texttimes\fiL Edwards-Anderson (EA) models, Fern'andez [Phys. Rev. B 25, 417 (1982)] finds that \ensuremathΛ(T)\ensuremath{\Lambda}(T)\ensuremathΛ(T) has a peak at T0\ensuremath≈0.6{T}_{0}\ensuremath{\approx}0.6T0​\ensuremath≈0.6 (Gaussian model) or T0\ensuremath≈1.0{T}_{0}\ensuremath{\approx}1.0T0​\ensuremath≈1.0, for L=4,6,8,10,or11L=4,6,8,10,or 11L=4,6,8,10,or11, respectively. He finds that the peak height \ensuremathΛmax=\ensuremathΛ(T0){\ensuremath{\Lambda}}_{max}=\ensuremath{\Lambda}({T}_{0})\ensuremathΛmax​=\ensuremathΛ(T0​) increases in direct proportion to LLL, and interprets his results in terms of a phase transition at T0{T}_{0}T0​, where the correlation length \ensuremathξEA\ensuremath∝(T\ensuremath−T0)\ensuremath−1{\ensuremath{\xi}}_{\mathrm{EA}}\ensuremath{\propto}{(T\ensuremath{-}{T}_{0})}^{\ensuremath{-}1}\ensuremathξEA​\ensuremath∝(T\ensuremath−T0​)\ensuremath−1. This conclusion is in disagreement with our previous findings that \ensuremathξEA{\ensuremath{\xi}}_{\mathrm{EA}}\ensuremathξEA​ is finite in this temperature range, although a (dynamic) "freezing transition" of the spins occurs at Tf\ensuremath≈1.0{T}_{f}\ensuremath{\approx}1.0Tf​\ensuremath≈1.0 (Gaussian model) or Tf\ensuremath≈1.3{T}_{f}\ensuremath{\approx}1.3Tf​\ensuremath≈1.3 (\ifmmode±\else\textpm\fiJ\ifmmode\pm\else\textpm\fi{}J\ifmmode±\else\textpm\fiJ model), which lead us to suggest that the two-dimensional EA model has a phase transition at zero temperature only.
The purpose of this paper is to show how some interesting numerically exact thermal averages can be computed for finite two-dimensional Ising models by a transfer-matrix method, and to apply these methods to the Ising version of the Edwards-Anderson (EA) model of a spin-glass to ascertain whether there is a phase transition. We do not study the spatial dependence of $〈{\ensuremath{\sigma}}_{0}{\ensuremath{\sigma}}_{l}〉$ for, as we show with an example, its behavior for finite systems can be misleading. It is first shown how to obtain ${\ensuremath{\chi}}_{\mathrm{EA}}^{\ensuremath{'}}$, defined by ${\ensuremath{\chi}}_{\mathrm{EA}}^{\ensuremath{'}}={N}^{\ensuremath{-}1}\ensuremath{\Sigma}{〈{〈{\ensuremath{\sigma}}_{i}{\ensuremath{\sigma}}_{j}〉}_{T}^{2}〉}_{J}$. We compute ${\ensuremath{\chi}}_{\mathrm{EA}}^{\ensuremath{'}}$ and study the quantity $\ensuremath{\Lambda}=\frac{\ensuremath{-}\ensuremath{\partial}\mathrm{ln}({\ensuremath{\chi}}_{\mathrm{EA}}^{\ensuremath{'}})}{\ensuremath{\partial}T}$, both as a function of temperature (TTT) and of the number of spins (NNN) in the system. The results obtained for square systems of up to 100 spins in the case where $J=\ifmmode\pm\else\textpm\fi{}1$ with equal probability and for square systems of up to 121 spins in the case where each JJJ is normally distributed about J=0J=0J=0 are in accord with the existence of a critical point at ${T}_{0}\ensuremath{\simeq}1.0$ and at ${T}_{0}\ensuremath{\simeq}0.6$, respectively. In addition the value $\ensuremath{\nu}\ensuremath{\approx}1$ is obtained. The value q=0q=0q=0 for T>0T>0T>0 is consistent with the results obtained. The low-temperature entropy per spin (SSS) is computed for long strips of different widths. Extrapolation to an infinite width yields $\frac{S}{k}\ensuremath{\simeq}0.07$. It is also shown how to calculate the probability, $P(\ensuremath{\eta})$, that the quantity, $\ensuremath{\eta}={N}^{\ensuremath{-}1}\ensuremath{\Sigma}{\ensuremath{\tau}}_{i}{\ensuremath{\sigma}}_{i}$, where each ${\ensuremath{\tau}}_{i}=0,\ifmmode\pm\else\textpm\fi{}1$ take any value in the range $1<~\ensuremath{\eta}<~1$. The probability, $P(\ensuremath{\eta})$, obtained for the EA model at low temperatures often has several maxima separated by regions of improbable values of $\ensuremath{\eta}$, as is to be expected of a system with metastable states.
In the previous Comment, Binder and Morgenstern use their previous result, ${〈{S}_{i}{S}_{j}〉}^{2}=c\mathrm{exp}(\ensuremath{-}\frac{{r}_{\mathrm{ij}}}{\ensuremath{\xi}})$ for small systems, plus their hypothesis that $\ensuremath{\xi}$ and ccc depend weakly on LLL, to produce results strongly similar to what I have obtained previously. They thus suggest that my results, contrary to my own conclusion, do not indicate the existence of a critical point in the Edwards-Anderson model in two dimensions. I show here that their claim is not well founded.
In the previous Comment, Binder and Morgenstern use their previous result, ${〈{S}_{i}{S}_{j}〉}^{2}=c \mathrm{exp}(\frac{{\ensuremath{-}r}_{\mathrm{ij}}}{\ensuremath{\xi}})$ for small systems, plus their hypothesis that $\ensuremath{\xi}$ and ccc depend weakly on LLL, to produce results strongly similar to what I have obtained previously. They thus suggest that my results, contrary to my own conclusion, do not indicate the existence of a critical point in the Edwards-Anderson model in two dimensions. I show here that their claim is not well founded.
This Comment is dedicated to the investigation of many-body localization in a quantum Ising model with long-range power-law interactions ${r}^{\ensuremath{-}\ensuremath{\alpha}}$, relevant for a variety of systems ranging from electrons in Anderson insulators to spin excitations in chains of cold atoms. It has earlier been argued [arXiv:cond-mat/0611387 (2005); Phys. Rev. B 91, 094202 (2015)] that this model obeys the dimensional constraint suggesting the delocalization of all finite-temperature states in the thermodynamic limit for $\ensuremath{\alpha}\ensuremath{\le}2d$ in a ddd-dimensional system. This expectation conflicts with the recent numerical studies of the specific interacting spin model of Li et al. [Phys. Rev. A 94, 063625 (2016)]. To resolve this controversy we reexamine the model of Li et al. [Phys. Rev. A 94, 063625 (2016)] and demonstrate that the infinite-temperature states there obey the dimensional constraint. The earlier developed scaling theory for the critical system size required for delocalization is extended to small exponents $0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}d$. The disagreements between the two works are explained by the nonstandard selection of investigated states in the ordered phase in the work of Li et al. [Phys. Rev. A 94, 063625 (2016)].
By a recursive method numerically exact free energies and correlation functions are calculated for finite two- and three-dimensional Ising lattices with bonds chosen randomly either ±J or from a gaussian distribution. While (dynamic) Monte Carlo simulations of precisely the same systems show the onset of Edwards-Anderson ordering at rather sharply defined nonzero freezing temperature Tf, the exact static calculations imply that there is no static order parameter qEA = [<Si≳2T]av at nonzero temperature. The correlation length, which would become infinite when qEA would start to become nonzero, is found at Tf to be of the order of a few lattice spacings only. As a result, the freezing-in of these models is identified as a dynamic phenomenon, and not a phase transition associated with static Edwards-Anderson order. The specific heat of the gaussian spin glass is found to vary nearly linear with temperature. On the other hand, in two-dimensional Ising systems with random fields a nonzero (Mattis-like) spin glass order is found.