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Physical review. B, Condensed matter

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

Julio F. Fernández

Published May 1, 1983• 1 citation
Publisher

Abstract

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.

DOI
10.1103/physrevb.27.5829

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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, ${〈{S}_{i}{S}_{j}〉}^{2}=c \mathrm{exp}(\frac{{\ensuremath{-}r}_{\mathrm{ij}}}{\ensuremath{\xi}})$ 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

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

Kurt Binder, I. Morgenstern

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.

Physical review. B, Condensed matter•May 1, 1983•8 citations

Reply to the Comment ‘‘Critical temperature of the two coupled Ising planes’’

P. Białas, Z. Burda, P. Sawicki +1 more

We agree with Pelizzola's comment and present independent results which support the same conclusion. Our calculations for multilayer models also support this result and illustrate transition to the three-dimensional Ising model. The role of the critical equalities is emphasized.

Physical review. B, Condensed matter•May 1, 1995•2 citations

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

Julio F. Fernández

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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Comment on “Finite-size scaling of the 5D Ising model"

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In a recent letter [1], Mon addresses the disagreement between a renormalization prediction [2] and Monte Carlo (MC) analyses of the critical value of the Binder cumulant in the five-dimensional Ising model. Whereas there is firm evidence supporting his overall conclusion, namely that the discrepancy can be explained by strong finite-size corrections [3], there is no such evidence for Mon's identification of the nature of this correction. This is highly relevant for the discussion of this problem, because neglecting the dominant correction to scaling can evidently explain the incorrect conclusions drawn from previous simulations.

Europhysics Letters (EPL)•Mar 1, 1997•9 citations

Reply to “Comment on ‘Accurate ground-state phase diagram of the one-dimensional extended Hubbard model at half filling’ ”

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We respond to Jeckelman's Comment on our paper [G. P. Zhang, Phys. Rev. B 68, 153101 (2003)], which discusses his previous work [E. Jeckelmann, Phys. Rev. Lett. 89, 236401 (2002)]. We feel that it is important to point out that his previous results show a trend contradicting the weak-coupling limit, a conclusion that is independent of our calculation. Therefore, our comments on his previous results remain valid and our conclusion is unchanged.

Physical Review B•May 4, 2005•2 citations
In the previous Comment, Binder and Morgenstern use their previous result, 〈SiSj〉2=cexp(\ensuremath−rij\ensuremathξ){〈{S}_{i}{S}_{j}〉}^{2}=c\mathrm{exp}(\ensuremath{-}\frac{{r}_{\mathrm{ij}}}{\ensuremath{\xi}})〈Si​Sj​〉2=cexp(\ensuremath−\ensuremathξrij​​) for small systems, plus their hypothesis that \ensuremathξ\ensuremath{\xi}\ensuremathξ 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.
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,or11L=4,6,8,10,or 11L=4,6,8,10,or11, respectively. He finds that the peak height ${\ensuremath{\Lambda}}_{max}=\ensuremath{\Lambda}({T}_{0})$ 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 ${\ensuremath{\xi}}_{\mathrm{EA}}\ensuremath{\propto}{(T\ensuremath{-}{T}_{0})}^{\ensuremath{-}1}$. 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.
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 a recent letter [1], Mon addresses the disagreement between a renormalization prediction [2] and Monte Carlo (MC) analyses of the critical value of the Binder cumulant in the five-dimensional Ising model. Whereas there is firm evidence supporting his overall conclusion, namely that the discrepancy can be explained by strong finite-size corrections [3], there is no such evidence for Mon's identification of the nature of this correction. This is highly relevant for the discussion of this problem, because neglecting the dominant correction to scaling can evidently explain the incorrect conclusions drawn from previous simulations.