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Journal of Physics A Mathematical and General

The critical exponents of the two-dimensional Ising spin glass revisited: Exact ground-state calculations and Monte Carlo simulations

Heiko Rieger, Ludger Santen, Ulrich Blasum, M. Diehl, Michael Jünger

Published Dec 21, 1997• 24 citations
Publisher

Abstract

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 .

DOI
10.1088/0305-4470/30/24/038

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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 .
The critical exponents for of the two-dimensional Ising spin-glass model with Gaussian couplings are determined with the help of exact ground states for system sizes up to L = 50 and by a Monte Carlo study of a pseudo-ferromagnetic order parameter. We obtain: for the stiffness exponent , for the magnetic exponent and for the chaos exponent . From Monte Carlo simulations we get the thermal exponent . The scaling prediction is fulfilled within the error bars, whereas there is a disagreement with the relation .
The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction ~1/L^0.25 of a very small magnitude also has been detected, as it can be expected from our recently developed GFD (grouping of Feynman diagrams) theory. Monte Carlo simulations of the squared magnetization of 3D Ising model have been performed by Wolff's algorithm in the range of the reduced temperatures t =< 0.000086 and system sizes L =< 410. The effective critical exponent beta_eff(t) tends to increase above the currently accepted numerical values. The critical coupling K_c=0.22165386(51) has been extracted from the Binder cumulant data within 96 =< L =< 384. The critical exponent 1/nu, estimated from the finite-size scaling of the derivatives of the Binder cumulant, tends to decrease slightly below the RG value 1.587 for the largest system sizes. The finite-size scaling of accurately simulated maximal values of the specific heat C_v in 3D Ising model confirms a logarithmic rather than power-like critical singularity of C_v.
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.
We perform high-statistics Monte Carlo simulations of three-dimensional Ising spin glass models on cubic lattices of size LLL: the $\ifmmode\pm\else\textpm\fi{}J$ (Edwards-Anderson) Ising model for two values of the disorder parameter ppp, p=0.5p=0.5p=0.5 and p=0.7p=0.7p=0.7 (up to L=28L=28L=28 and L=20L=20L=20, respectively), and the bond-diluted bimodal model for bond-occupation probability pb=0.45{p}_{b}=0.45pb​=0.45 (up to L=16L=16L=16). The finite-size behavior of the quartic cumulants at the critical point allows us to check very accurately that these models belong to the same universality class. Moreover, it allows us to estimate the scaling-correction exponent $\ensuremath{\omega}$ related to the leading irrelevant operator: $\ensuremath{\omega}=1.0(1)$. Shorter Monte Carlo simulations of the bond-diluted bimodal models at pb=0.7{p}_{b}=0.7pb​=0.7 and pb=0.35{p}_{b}=0.35pb​=0.35 (up to L=10L=10L=10) and of the Ising spin glass model with Gaussian bond distribution (up to L=8L=8L=8) also support the existence of a unique Ising spin glass universality class. A careful finite-size analysis of the Monte Carlo data which takes into account the analytic and the nonanalytic corrections to scaling allows us to obtain precise and reliable estimates of the critical exponents. We obtain $\ensuremath{\nu}=2.45(15)$ and $\ensuremath{\eta}=\ensuremath{-}0.375(10)$.
We perform Monte Carlo simulations of large two-dimensional Gaussian Ising spin glasses down to very low temperatures $\ensuremath{\beta}=1∕T=50$. Equilibration is ensured by using a cluster algorithm including Monte Carlo moves consisting of flipping fundamental excitations. We study the thermodynamic behavior using the Binder cumulant, the spin-glass susceptibility, the distribution of overlaps, the overlap with the ground state, and the specific heat. We confirm that Tc=0{T}_{c}=0Tc​=0. All results are compatible with an algebraic divergence of the correlation length with an exponent $\ensuremath{\nu}$. We find $\ensuremath{-}1∕\ensuremath{\nu}=\ensuremath{-}0.295(40)$, which is compatible with the value for the domain-wall and droplet exponent $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.29$ found previously in ground-state studies. Hence the thermodynamic behavior of this model seems to be governed by one single exponent.
We perform high-statistics Monte Carlo simulations of three three-dimensional Ising spin glass models: the ± J Ising model for two values of the disorder parameter p , p = 1/2 and 0.7, and the bond-diluted ± J model for bond-occupation probability p b = 0.45. A finite-size scaling analysis of the quartic cumulants at the critical point shows conclusively that these models belong to the same universality class and allows us to estimate the scaling-correction exponent ω related to the leading irrelevant operator, ω = 1.0(1). We also determine the critical exponents ν and η. Taking into account the scaling corrections, we obtain ν = 2.53(8) and η = −0.384(9).