A New Method to Calculate the Spin-Glass Order Parameter -the Spin-Glass Phase Transition of the Two-Dimensional +/-J Ising Model-

Hidetsugu Kitatani, A. Sinada

Published Aug 26, 1999• 0 citations

Abstract

A new method to calculate the n−th moment of the spin overlap of the two-dimensional ±J Ising model is presented, where the n−th moment of the spin overlap can be calculated as a simple average of the n−th moment of the total spins with a modified bond probability distribution. By this method, the values of the Binder parameter etc. have been calculated extensively with the linear size, L, up to L = 23. The accuracy of the present calculations is similar to that calculated by the conventional transfer matrix method averaging over about 10 5 bond samples. Our results show that the estimation of Tc is strongly affected by the corrections to scaling. Thus, there still remains the possibility of Tc = 0, contrary to the recent results which insist the existence of a finite temperature spin-glass phase transition. 75.50.Lk,02.70.Lq,64.60.Cn,05.50.+q Typeset using REVTEX 1 In the last two decades, investigations of spin-glass problems have been performed

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A new method to calculate the spin-glass order parameter of the two-dimensional ±JIsing model

Hidetsugu Kitatani, Akira Sinada

arXiv:cond-mat/9908370Journal of Physics A Mathematical and GeneralMay 5, 20009 citations

Comment on 'A new method to calculate the spin-glass order parameter of the two-dimensional $\pm J$ Ising model'

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We examine the phase transition of Ising spin glass models in two dimensions (2D), calculating complementary two quantities, i.e., the interface free energy $\overline{\Delta F(T)}$ and the Binder parameter g L , on finite L × L ( L ≤24) lattices at very low temperatures T . We find that these quantities exhibit quite different features depending on the distribution of bonds. For the ± J distribution, $\overline{\Delta F(T)}$ at very low temperatures slightly increases with L and g L intersect at T ∼0.25 J . These results suggest a non-zero temperature phase transition, T c ≠0. On the other hand, for the Gaussian distribution, $\overline{\Delta F(T)}$ decreases as L increases even at T = 0 and g L for different L converges to unity at T = 0. These results confirm the assumption of the zero temperature phase transition, T c = 0. Finite-size scaling analyses support those results. Thus we suggest that, in 2D, the existence of a finite-temperature phase transition depends on the distribution of bonds and it exists when the bond distribution is ± J .

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A New Method to Calculate the Spin-Glass Order Parameter -the Spin-Glass Phase Transition of the Two-Dimensional +/-J Ising Model-

Hidetsugu Kitatani, A. Sinada

Published Aug 26, 1999• 0 citations

OpenAlex

Abstract

Related papers

A new method to calculate the spin-glass order parameter of the two-dimensional ±JIsing model

Hidetsugu Kitatani, Akira Sinada

A new method to numerically calculate the n th moment of the spin overlap of the two-dimensional ± J Ising model is developed using the identity derived by one of the authors (HK) several years ago. By using the method, the n th moment of the spin overlap can be calculated as a simple average of the n th moment of the total spins with a modified bond probability distribution. The values of the Binder parameter etc have been extensively calculated with the linear size, L , up to L = 23. The accuracy of the calculations in the present method is similar to that in the conventional transfer matrix method with about 10 5 bond samples. The simple scaling plots of the Binder parameter and the spin-glass susceptibility indicate the existence of a finite-temperature spin-glass phase transition. We find, however, that the estimation of T c is strongly affected by the corrections to scaling within the present data ( L ⩽ 23). Thus, there still remains the possibility that T c = 0, contrary to the recent results which suggest the existence of a finite-temperature spin-glass phase transition.A new method to numerically calculate the n th moment of the spin overlap of the two-dimensional ± J Ising model is developed using the identity derived by one of the authors (HK) several years ago. By using the method, the n th moment of the spin overlap can be calculated as a simple average of the n th moment of the total spins with a modified bond probability distribution. The values of the Binder parameter etc have been extensively calculated with the linear size, L , up to L = 23. The accuracy of the calculations in the present method is similar to that in the conventional transfer matrix method with about 10 5 bond samples. The simple scaling plots of the Binder parameter and the spin-glass susceptibility indicate the existence of a finite-temperature spin-glass phase transition. We find, however, that the estimation of T c is strongly affected by the corrections to scaling within the present data ( L ⩽ 23). Thus, there still remains the possibility that T c = 0, contrary to the recent results which suggest the existence of a finite-temperature spin-glass phase transition.

arXiv:cond-mat/9908370Journal of Physics A Mathematical and GeneralMay 5, 20009 citations

Comment on 'A new method to calculate the spin-glass order parameter of the two-dimensional $\pm J$ Ising model'Comment on 'A new method to calculate the spin-glass order parameter of the two-dimensional $\pm J$ Ising model'

Takayuki Shirakura, Fumitaka Matsubara, Michinori Shiomi

Contrary to the suggestion of Kitatani and Sinada (2000 J. Phys. A: Math. Gen. {\bf 33} 3545-3553), the scaling analysis of the order parameter distributions suggests the existence of a finite-temperature spin-glass phase transition $T_c\ne 0$ , even if the corrections to scaling are taken into account.Contrary to the suggestion of Kitatani and Sinada (2000 J. Phys. A: Math. Gen. {\bf 33} 3545-3553), the scaling analysis of the order parameter distributions suggests the existence of a finite-temperature spin-glass phase transition $T_c\ne 0$, even if the corrections to scaling are taken into account.

arXiv:cond-mat/0011235Nov 14, 20000 citations

Phase Transition of Ising Spin Glass Models in Two Dimensions

Michinori Shiomi, Fumitaka Matsubara, Takayuki Shirakura

We examine the phase transition of Ising spin glass models in two dimensions (2D), calculating complementary two quantities, i.e., the interface free energy $\overline{\Delta F(T)}$ and the Binder parameter g L , on finite L × L ( L ≤24) lattices at very low temperatures T . We find that these quantities exhibit quite different features depending on the distribution of bonds. For the ± J distribution, $\overline{\Delta F(T)}$ at very low temperatures slightly increases with L and g L intersect at T ∼0.25 J . These results suggest a non-zero temperature phase transition, T c ≠0. On the other hand, for the Gaussian distribution, $\overline{\Delta F(T)}$ decreases as L increases even at T = 0 and g L for different L converges to unity at T = 0. These results confirm the assumption of the zero temperature phase transition, T c = 0. Finite-size scaling analyses support those results. Thus we suggest that, in 2D, the existence of a finite-temperature phase transition depends on the distribution of bonds and it exists when the bond distribution is ± J .We examine the phase transition of Ising spin glass models in two dimensions (2D), calculating complementary two quantities, i.e., the interface free energy $\overline{\Delta F(T)}$ and the Binder parameter g L , on finite L × L ( L ≤24) lattices at very low temperatures T . We find that these quantities exhibit quite different features depending on the distribution of bonds. For the ± J distribution, $\overline{\Delta F(T)}$ at very low temperatures slightly increases with L and g L intersect at T ∼0.25 J . These results suggest a non-zero temperature phase transition, T c ≠0. On the other hand, for the Gaussian distribution, $\overline{\Delta F(T)}$ decreases as L increases even at T = 0 and g L for different L converges to unity at T = 0. These results confirm the assumption of the zero temperature phase transition, T c = 0. Finite-size scaling analyses support those results. Thus we suggest that, in 2D, the existence of a finite-temperature phase transition depends on the distribution of bonds and it exists when the bond distribution is ± J .

Journal of the Physical Society of JapanSep 1, 200010 citations

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We have studied the three-dimensional Ising spin glass with a ± J distribution by Monte Carlo simulations. Using larger sizes and much better statistics than in earlier work, a finite size scaling analysis shows quite strong evidence for a finite transition temperature, Tc, with ordering below Tc. Our estimate of the transition temperature is rather lower than in earlier work, and the value of the correlation length exponent, ν, is somewhat higher. Because there may be (unknown) corrections to finite size scaling, we do not completely rule out the possibility that Tc = 0 or that Tc is finite but with no order below Tc. However, from our data, these possibilities seem less likely.We have studied the three-dimensional Ising spin glass with a ± J distribution by Monte Carlo simulations. Using larger sizes and much better statistics than in earlier work, a finite size scaling analysis shows quite strong evidence for a finite transition temperature, Tc, with ordering below Tc. Our estimate of the transition temperature is rather lower than in earlier work, and the value of the correlation length exponent, ν, is somewhat higher. Because there may be (unknown) corrections to finite size scaling, we do not completely rule out the possibility that Tc = 0 or that Tc is finite but with no order below Tc. However, from our data, these possibilities seem less likely.

International Journal of Modern Physics CJun 1, 19963 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.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.

Physical review. B, Condensed matterJan 1, 198222 citations

Comment on 'A new method to calculate the spin-glass order parameter of the two-dimensional

\pm J

Ising model'

T_c\ne 0

, even if the corrections to scaling are taken into account.

We examine the phase transition of Ising spin glass models in two dimensions (2D), calculating complementary two quantities, i.e., the interface free energy

\overline{\Delta F(T)}

and the Binder parameter g L , on finite L × L ( L ≤24) lattices at very low temperatures T . We find that these quantities exhibit quite different features depending on the distribution of bonds. For the ± J distribution,

\overline{\Delta F(T)}

at very low temperatures slightly increases with L and g L intersect at T ∼0.25 J . These results suggest a non-zero temperature phase transition, T c ≠0. On the other hand, for the Gaussian distribution,

\overline{\Delta F(T)}

decreases as L increases even at T = 0 and g L for different L converges to unity at T = 0. These results confirm the assumption of the zero temperature phase transition, T c = 0. Finite-size scaling analyses support those results. Thus we suggest that, in 2D, the existence of a finite-temperature phase transition depends on the distribution of bonds and it exists when the bond distribution is ± J .

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.