On the universal behavior of two-dimensional Ising spin glasses

Ising spin glasses are studied, at zero temperature, on a hierarchical lattice as an approach to the square lattice. The stiffness exponent y, which governs the behavior of the interactions under changes of scale, is computed for several distinct continuous symmetric probability distributions for the couplings. All distributions considered lead to the same estimates, i.e., the exponent y is universal. Our results are compared with other estimates available for the two-dimensional Gaussian Ising spin glass.     

Aspect-ratio scaling and the stiffness exponent theta for Ising spin glasses

We introduce the technique of aspect-ratio scaling to study the scale dependence of interfacial energies in Ising spin glasses, and we show how one can use it to determine the stiffness exponent theta in a clean way, with results that are independent of the domain-wall-forcing boundary conditions imposed on the system. In space dimension d = 2 we obtain theta = -0.282(3) for a Gaussian distribution of exchange interactions.     

Conformal invariance and stochastic Loewner evolution processes in two-dimensional Ising spin glasses

We present numerical evidence that the techniques of conformal field theory might be applicable to two-dimensional Ising spin glasses with Gaussian bond distributions. It is shown that certain domain wall distributions in one geometry can be related to that in a second geometry by a conformal transformation. We also present direct evidence that the domain walls are stochastic Loewner (SLE) processes with kappa approximately 2.1. An argument is given that their fractal dimension d(f) is related to their interface energy exponent theta by d(f) - 1 = 3/[4(3 + theta)], which is consistent with the commonly quoted values d(f) approximately 1.27 and theta approximately -0.28.     

Corrections to scaling are large for droplets in two-dimensional spin glasses

The energy of a droplet of linear extent l in the droplet theory of spin glasses goes as l(theta) for large l. It is shown by numerical studies of large droplets in two-dimensional systems that this formula needs to be modified by the addition of a scaling correction l(-omega) in order to accurately describe droplet energies at the length scales currently probed in numerical simulations. Using this simple modification, it is now possible to explain many results which have been found in simulations of three-dimensional Ising spin glasses with the droplet model.     

Complexity of Ising spin glasses

We compute the complexity [logarithm of the number of Thouless-Anderson-Palmer (TAP) states] associated with minima and index-one saddle points of the TAP free energy. Higher-index saddles have smaller complexities. The two leading complexities are equal, consistent with the Morse theorem on the total number of turning points, and have the value given by Bray and Moore [J. Phys. C, ()]]. In the thermodynamic limit, TAP states of all free energies become marginally stable.     

Free energy fluctuations in Ising spin glasses

The sample-to-sample fluctuations of the free energy in finite-dimensional Ising spin glasses are calculated, using the replica method, from higher order terms in the replica number n. It is shown that the Parisi symmetry breaking scheme does not give the correct answers for these higher order terms. A modified symmetry breaking scheme with the same stability is shown to resolve the problem.     

Nonequilibrium dynamics in two-dimensional Ising spin glass

This paper investigates the nonequilibrium dynamics of two-dimensional Ising spin glass by dynamical Monte Carlo simulations. A new method is developed to quantitatively measure the age of domain growth. Using this method it investigates how temperature shift affects the effective age of domain growth. It finds that the T -shift dependence of the effective age follows the prediction of the droplet model quite well. It also investigates the overlap length between the spin glass states as well as the correlated flips of spins,which are not consistent with the theoretical predictions. The possible reasons are discussed.     

Ground-state landscape of J Ising spin glasses

Large numbers of ground states of two-dimensional Ising spin glasses with periodic boundary conditions in both directions are calculated for sizes up to 40². A combination of a genetic algorithm and Cluster-Exact Approximation is used. For each quenched realization of the bonds up to 40 independent ground states are obtained. For the infinite system a ground-state energy of e =-1.4015(3) is extrapolated. The ground-state landscape is investigated using a finite-size scaling analysis of the distribution of overlaps. The mean-field picture assuming a complex landscape describes the situation better than the droplet-scaling model, where for the infinite system mainly two ground states exist. Strong evidence is found that the ground states are not organized in an ultrametric fashion in contrast to previous results for three-dimensional spin glasses. Received 12 October 1998     

Hidden valley structure of Ising spin glasses

The hidden valley structure of the phase space of short-range spin glasses is investigated. In the spirit of a similar investigation for liquids by Stillinger and Weber, valleys are defined as the set of states which by steepest descent lead to the same metastable state. The lifetime of valleys and the size of spin clusters involved in intervalley transitions on thermal Monte Carlo trajectories are calculated as a function of temperature.     

On the damage spreading in Ising spin glasses

The spreading of the Hamming distance or damage has been investigated for ±J Ising spin glasses under heat bath dynamics. Dimensions d = 2, 3, 4, 6 and mean field were studied. For finite dimensions, the damage goes to zero at long times above a temperature T_D(d). Accurate values of this critical temperature were obtained, together with certain critical exponents. The spin glass ordering temperatures T_g were also estimated from the damage spreading data. The results are compared with other work and discussed from a phase space approach.     

Temperature and disorder chaos in three-dimensional Ising spin glasses

We study the effects of small temperature as well as disorder perturbations on the equilibrium state of three-dimensional Ising spin glasses via an alternate scaling ansatz. By using Monte Carlo simulations, we show that temperature and disorder perturbations yield chaotic changes in the equilibrium state and that temperature chaos is considerably harder to observe than disorder chaos.     

Cluster analysis and finite-size scaling for Ising spin systems

Based on the connection between the Ising model and a correlated percolation model, we calculate the distribution function for the fraction (c) of lattice sites in percolating clusters in subgraphs with n percolating clusters, f(n)(c), and the distribution function for magnetization (m) in subgraphs with n percolating clusters, p(n)(m). We find that f(n)(c) and p(n)(m) have very good finite-size scaling behavior and that they have universal finite-size scaling functions for the model on square, plane triangular, and honeycomb lattices when aspect ratios of these lattices have the proportions 1:square root[3]/2:square root[3]. The complex structure of the magnetization distribution function p(m) for the system with large aspect ratio could be understood from the independent orientations of two or more percolation clusters in such a system.     

Behavior of Ising spin glasses in a magnetic field

We study the existence of a spin-glass phase in a field using Monte Carlo simulations performed along a nontrivial path in the field-temperature plane that must cross any putative de Almeida-Thouless instability line. The method is first tested on the Ising spin glass on a Bethe lattice where the instability line separating the spin glass from the paramagnetic state is also computed analytically. While the instability line is reproduced by our simulations on the mean-field Bethe lattice, no such instability line can be found numerically for the short-range three-dimensional model.