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arXiv:cond-mat/9604050 · 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, Giovanni Rinaldi

Published Jul 21, 1996• 107 citations
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Abstract

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 .

DOI
10.1088/0305-4470/29/14/018

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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 .
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 .
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).
With the help of exact ground states obtained by a polynomial algorithm we compute the domain wall energy Δ at zero temperature for the bond-random and the site-random Ising spin glass model in two dimensions. We find that in both models the stability of the ferromagnetic and the spin glass order ceases to exist at a unique concentration p c for the ferromagnetic bonds. In the vicinity of this critical point, the size and concentration dependence of the first and second moment of the domain wall energy are, for both models, described by a common finite-size scaling form. Moreover, below this concentration the stiffness exponent turns out to be slightly negative, θ S = − 0.056(6), indicating the absence of any intermediate spin glass phase at non-zero temperature.
Monte Carlo data of the two-dimensional Ising spin glass with bimodal interactions are presented with the aim of understanding the low-temperature physics of the model. An analysis of the specific heat, spin-glass susceptibility, finite-size correlation length, and the Binder ratio is performed to try to verify a recent proposal in which for large system sizes and finite but low temperatures the effective critical exponents are identical to the critical exponents of the two-dimensional Ising spin glass with Gaussian interactions. Our results show that with present system sizes the recently proposed scenario, in which the two-dimensional Ising spin glass with bimodally distributed interactions is in the same universality class as the model with Gaussian-distributed disorder at low but finite temperatures, cannot be reliably proven.
We present a finite-size scaling analysis of Monte Carlo simulation results on a four-dimensional Ising spin-glass. We study chaos with both coupling and temperature perturbations, and find the same chaos exponent in each case. Chaos is investigated both at the critical temperature and below where it seems to be more efficient (larger exponent). Dimension 4 seems to be above the critical dimension where chaos with temperature is no longer present in the critical region. Our results are consistent with the Gaussian and bimodal coupling distributions being in the same universality class.
A long-standing issue in a three-dimensional Ising spin glass (3DISG) is to settle the critical exponents. Numerical simulations and a conventional finite-size-scaling analysis have given controversial estimates, in particular, for the value of thermal exponent ν. Here a recently proposed extended scaling scheme is reviewed and illustrated for the canonical two-dimensional Ising ferromagnet. By applying the extended scaling to our extensive Monte Carlo data for 3DISG, consistent estimates of critical parameters are obtained in contrast to previous analyses.