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arXiv:cond-mat/9908370 · Journal of Physics A Mathematical and General

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

Hidetsugu Kitatani, Akira Sinada

Published May 5, 2000• 9 citations
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Abstract

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.

DOI
10.1088/0305-4470/33/18/302

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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

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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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.

arXiv:cond-mat/0011235•Nov 14, 2000•0 citations

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We have calculated finite-size correlation lengths for strips of the square-lattice Ising spin glass using the transfer-matrix method. In order to minimize sampling errors we study strip lengths up to 5\ifmmode\times\else\texttimes\fi{}${10}^{6}$ lattice spacings. A phenomenological renormalization-group analysis indicates that there are strong corrections to simple power-law scaling near the zero-temperature critical point, as is to be expected near a lower critical dimension. We examine models with Gaussian, exp(-${J}^{2}$/2), and exponential, exp(-\ensuremath{\Vert}J\ensuremath{\Vert}), distributions of couplings; the Gaussian distribution shows stronger finite-size corrections. The correlation-length exponent is estimated to be \ensuremath{\nu}=4.2\ifmmode\pm\else\textpm\fi{}0.5, although we do not want to rule out the possibility that \ensuremath{\nu} is significantly larger than this.

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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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Due to an incorrect generation of the bond configuration we obtained incorrect Monte Carlo results. New simulation runs with a corrected and enlarged set of samples has led to the results shown in figures 1 and 2 which replace figures 3 and 4 of the original paper. The best data collapse of XL we obtained (see figure 1) for and of the Binder ratio gL (see figure 2) for Figure 1. Scaling plot of XL for different system sizes and temperatures. The value of the correlation length exponent is v = 1.5. Thus, the average value of the correlation length exponent now reads It should be noted that all other results of the paper where independently generated samples were used are not affected by these corrections. Figure 2. Scaling plot of the Binder ration gL . We obtained the best data collapse for v = 1.9. We note that the new estimate is much closer to the results [1] and [2] and is not compatible with the scaling prediction y = -1/ v .

Journal of Physics A Mathematical and General•Dec 21, 1997•24 citations
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 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
Comment on 'A new method to calculate the spin-glass order parameter of the two-dimensional ±J\pm J±J Ising model'
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 Tc≠0T_c\ne 0Tc​=0, even if the corrections to scaling are taken into account.
We have calculated finite-size correlation lengths for strips of the square-lattice Ising spin glass using the transfer-matrix method. In order to minimize sampling errors we study strip lengths up to 5\ifmmode\times\else\texttimes\fi{}106{10}^{6}106 lattice spacings. A phenomenological renormalization-group analysis indicates that there are strong corrections to simple power-law scaling near the zero-temperature critical point, as is to be expected near a lower critical dimension. We examine models with Gaussian, exp(-J2{J}^{2}J2/2), and exponential, exp(-\ensuremath{\Vert}J\ensuremath{\Vert}), distributions of couplings; the Gaussian distribution shows stronger finite-size corrections. The correlation-length exponent is estimated to be \ensuremath{\nu}=4.2\ifmmode\pm\else\textpm\fi{}0.5, although we do not want to rule out the possibility that \ensuremath{\nu} is significantly larger than this.
We examine the phase transition of Ising spin glass models in two dimensions (2D), calculating complementary two quantities, i.e., the interface free energy ΔF(T)‾\overline{\Delta F(T)}ΔF(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, ΔF(T)‾\overline{\Delta F(T)}ΔF(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, ΔF(T)‾\overline{\Delta F(T)}ΔF(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 .
Due to an incorrect generation of the bond configuration we obtained incorrect Monte Carlo results. New simulation runs with a corrected and enlarged set of samples has led to the results shown in figures 1 and 2 which replace figures 3 and 4 of the original paper. The best data collapse of XL we obtained (see figure 1) for and of the Binder ratio gL (see figure 2) for Figure 1. Scaling plot of XL for different system sizes and temperatures. The value of the correlation length exponent is v = 1.5. Thus, the average value of the correlation length exponent now reads It should be noted that all other results of the paper where independently generated samples were used are not affected by these corrections. Figure 2. Scaling plot of the Binder ration gL . We obtained the best data collapse for v = 1.9. We note that the new estimate is much closer to the results [1] and [2] and is not compatible with the scaling prediction y = -1/ v .