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

A new method to numerically calculate the $n$th moment of the spin overlap of the two-dimensional $\pm J$ 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 $T_{\rm c}$ is strongly affected by the corrections to scaling within the present data ($L\leq 23$). Thus, there still remains the possibility that $T_{\rm c}=0$, contrary to the recent results which suggest the existence of a finite-temperature spin-glass phase transition.Comment: 10 pages,8 figures: final version to appear in J. Phys.

Finite Size Scaling Analysis of Exact Ground States for +/-J Spin Glass Models in Two Dimensions

With the help of EXACT ground states obtained by a polynomial algorithm we compute the domain wall energy 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 dependency 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 \theta_S = -0.056(6) indicating the absence of any intermediate spin glass phase at non-zero temperature.Comment: 7 pages Latex, 5 postscript-figures include

Numerical Study of Competing Spin-Glass and Ferromagnetic Order

Two and three dimensional random Ising models with a Gaussian distribution of couplings with variance $J$ and non-vanishing mean value $J_0$ are studied using the zero-temperature domain-wall renormalization group (DWRG). The DWRG trajectories in the ($J_0,J$) plane after rescaling can be collapsed on two curves: one for $J_0/J > r_c$ and other for $J_0/J < r_c$. In the first case the DWRG flows are toward the ferromagnetic fixed point both in two and three dimensions while in the second case flows are towards a paramagnetic fixed point and spin-glass fixed point in two and three dimensions respectively. No evidence for an extra phase is found.Comment: a bit more data is taken, 5 pages, 4 eps figures included, to appear in PR

Lower Critical Dimension of Ising Spin Glasses

Exact ground states of two-dimensional Ising spin glasses with Gaussian and bimodal (+- J) distributions of the disorder are calculated using a ``matching'' algorithm, which allows large system sizes of up to N=480^2 spins to be investigated. We study domain walls induced by two rather different types of boundary-condition changes, and, in each case, analyze the system-size dependence of an appropriately defined ``defect energy'', which we denote by DE. For Gaussian disorder, we find a power-law behavior DE ~ L^\theta, with \theta=-0.266(2) and \theta=-0.282(2) for the two types of boundary condition changes. These results are in reasonable agreement with each other, allowing for small systematic effects. They also agree well with earlier work on smaller sizes. The negative value indicates that two dimensions is below the lower critical dimension d_c. For the +-J model, we obtain a different result, namely the domain-wall energy saturates at a nonzero value for L\to \infty, so \theta = 0, indicating that the lower critical dimension for the +-J model exactly d_c=2.Comment: 4 pages, 4 figures, 1 table, revte

The critical exponents of the two-dimensional Ising spin glass revisited: Exact Ground State Calculations and Monte Carlo Simulations

The critical exponents for $T\to0$ 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 $y(=\theta)=-0.281\pm0.002$, for the magnetic exponent $\delta=1.48 \pm 0.01$ and for the chaos exponent $\zeta=1.05\pm0.05$. From Monte Carlo simulations we get the thermal exponent $\nu=3.6\pm0.2$. The scaling prediction $y=-1/\nu$ is fulfilled within the error bars, whereas there is a disagreement with the relation $y=1-\delta$.Comment: 8 pages RevTeX, 7 eps-figures include

Finite-Size Scaling in the Energy-Entropy Plane for the 2D +- J Ising Spin Glass

For $L \times L$ square lattices with $L \le 20$ the 2D Ising spin glass with +1 and -1 bonds is found to have a strong correlation between the energy and the entropy of its ground states. A fit to the data gives the result that each additional broken bond in the ground state of a particular sample of random bonds increases the ground state degeneracy by approximately a factor of 10/3. For $x = 0.5$ (where $x$ is the fraction of negative bonds), over this range of $L$, the characteristic entropy defined by the energy-entropy correlation scales with size as $L^{1.78(2)}$. Anomalous scaling is not found for the characteristic energy, which essentially scales as $L^2$. When $x= 0.25$, a crossover to $L^2$ scaling of the entropy is seen near $L = 12$. The results found here suggest a natural mechanism for the unusual behavior of the low temperature specific heat of this model, and illustrate the dangers of extrapolating from small $L$.Comment: 9 pages, two-column format; to appear in J. Statistical Physic

Ground states of two-dimensional ±\pmJ Edwards-Anderson spin glasses

We present an exact algorithm for finding all the ground states of the two-dimensional Edwards-Anderson $\pm J$ spin glass and characterize its performance. We investigate how the ground states change with increasing system size and and with increasing antiferromagnetic bond ratio $x$. We find that that some system properties have very large and strongly non-Gaussian variations between realizations.Comment: 15 pages, 21 figures, 2 tables, uses revtex4 macro