Gaussian Ising Spin Glass Research Articles

High-precision studies of domain-wall properties in the 2D Gaussian Ising spin glass

In two dimensions, short-range spin glasses order only at zero temperature, where efficient combinatorial optimization techniques can be used to study these systems with high precision. The use of large system sizes and high statistics in disorder averages allows for reliable finite-size extrapolations to the thermodynamic limit. Here, we use a recently introduced mapping of the Ising spin-glass ground-state problem to a minimum-weight perfect matching problem on a sparse auxiliary graph to study square-lattice samples of up to 10 000 × 10 000 spins. We propose a windowing technique that allows to extend this method, that is formally restricted to planar graphs, to the case of systems with fully periodic boundary conditions. These methods enable highly accurate estimates of the spin-stiffness exponent and domain-wall fractal dimension of the 2D Edwards-Anderson spin-glass with Gaussian couplings. Studying the compatibility of domain walls in this system with traces of stochastic Loewner evolution (SLE), we find a strong dependence on boundary conditions and compatibility with SLE only for one out of several setups.

Bimodal and Gaussian Ising spin glasses in dimension two.

An analysis is given of numerical simulation data to size L=128 on the archetype square lattice Ising spin glasses (ISGs) with bimodal (±J) and Gaussian interaction distributions. It is well established that the ordering temperature of both models is zero. The Gaussian model has a nondegenerate ground state and thus a critical exponent η≡0, and a continuous distribution of energy levels. For the bimodal model, above a size-dependent crossover temperature T(*)(L) there is a regime of effectively continuous energy levels; below T(*)(L) there is a distinct regime dominated by the highly degenerate ground state plus an energy gap to the excited states. T(*)(L) tends to zero at very large L, leaving only the effectively continuous regime in the thermodynamic limit. The simulation data on both models are analyzed with the conventional scaling variable t=T and with a scaling variable τ(b)=T(2)/(1+T(2)) suitable for zero-temperature transition ISGs, together with appropriate scaling expressions. The data for the temperature dependence of the reduced susceptibility χ(τ(b),L) and second moment correlation length ξ(τ(b),L) in the thermodynamic limit regime are extrapolated to the τ(b)=0 critical limit. The Gaussian critical exponent estimates from the simulations, η=0 and ν=3.55(5), are in full agreement with the well-established values in the literature. The bimodal critical exponents, estimated from the thermodynamic limit regime analyses using the same extrapolation protocols as for the Gaussian model, are η=0.20(2) and ν=4.8(3), distinctly different from the Gaussian critical exponents.

Proposal of a checking parameter in the simulated annealing method applied to the spin glass model

We propose a checking parameter utilizing the breaking of the Jarzynski equality in the simulated annealing method using the Monte Carlo method. This parameter is based on the Jarzynski equality. By using this parameter, to detect that the system is in global minima of the free energy under gradual temperature reduction is possible. Thus, by using this parameter, one is able to investigate the efficiency of annealing schedules. We apply this parameter to the ±J Ising spin glass model. The application to the Gaussian Ising spin glass model is also mentioned. We discuss that the breaking of the Jarzynski equality is induced by the system being trapped in local minima of the free energy. By performing Monte Carlo simulations of the ±J Ising spin glass model and a glassy spin model proposed by Newman and Moore, we show the efficiency of the use of this parameter.

The Ising Spin Glass in dimension four

The critical behaviors of the bimodal and Gaussian Ising spin glass (ISG) models in dimension four are studied through extensive numerical simulations, and from an analysis of high temperature series expansion (HTSE) data of Klein et al. (1991). The simulations include standard finite size scaling measurements, thermodynamic limit regime measurements, and analyses which provide estimates of critical exponents without any consideration of the critical temperature. The higher order HTSE series for the bimodal model provide accurate estimates of the critical temperature and critical exponents. These estimates are independent of and fully consistent with the simulation values. Comparisons between ISG models in dimension four show that the critical exponents and the critical constants for dimensionless observables depend on the form of the interaction distribution of the model.

Chaos and stiffness exponents for short-range Gaussian Ising spin glasses

Two important exponents in spin-glass theory, namely, the chaos (ζ) and stiffness (y) exponents, are studied for Ising spin glasses with nearest-neighbor Gaussian interactions on different approaches to Bravais lattices. We consider hierarchical lattices of the Migdal–Kadanoff type (both diamond and tress families), with varying fractal dimensions, as well as two lattices of the Wheatstone-bridge family, more specifically, those with fractal dimensions D ≈ 2.32 and D ≈ 3.58. Whenever it is possible to compare, our estimates agree with those obtained from extensive numerical simulations on Bravais lattices, suggesting the present results represent good approximations for these exponents.

Exact location of the multicritical point for finite-dimensional spin glasses: a conjecture

We present a conjecture on the exact location of the multicritical point in the phase diagram of spin glass models in finite dimensions. By generalizing our previous work, we combine duality and gauge symmetry for replicated random systems to derive formulas which make it possible to understand all the relevant available numerical results in a unified way. The method applies to non-self-dual lattices as well as to self dual cases, in the former case of which we derive a relation for a pair of values of multicritical points for mutually dual lattices. The examples include the +-J and Gaussian Ising spin glasses on the square, hexagonal and triangular lattices, the Potts and Z_q models with chiral randomness on these lattices, and the three-dimensional +-J Ising spin glass and the random plaquette gauge model.

Low-temperature behavior of two-dimensional Gaussian Ising spin glasses

We perform Monte Carlo simulations of large two-dimensional Gaussian Ising spin glasses down to very low temperatures $\beta=1/T=50$. 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 $T_c=0$. All results are compatible with an algebraic divergence of the correlation length with an exponent $\nu$. We find $-1/\nu=-0.295(30)$, which is compatible with the value for the domain-wall and droplet exponent $\theta\approx-0.29$ found previously in ground-state studies. Hence the thermodynamic behavior of this model seems to be governed by one single exponent.

Energetics of clusters in the two-dimensional Gaussian Ising spin glass

We study numerically the properties of local low-energy excitations in the two-dimensional Ising spin glass. Given the ground state, we determine the lowest-lying connected cluster of flipped spins containing one given spin, either with a fixed volume, or with a volume constrained to lie in a certain range. Our aim is to understand corrections to the scaling predicted by the droplet picture of spin glasses and to resolve contradictory results reported in the literature for the stiffness exponent. We find no clear trace of corrections to scaling, and the obtained stiffness exponent is in relatively good agreement with standard domain wall calculations.

Statistics of lowest droplets in two-dimensional Gaussian Ising spin glasses

An approach to determine the value of the zero-temperature thermal exponent $\ensuremath{\theta}$ in spin glasses is presented. It consists in describing the energy level spectrum in spin glasses only in terms of the properties of the lowest energy droplets and the lowest droplet exponents (LDE's) ${\ensuremath{\lambda}}_{l},{\ensuremath{\theta}}_{l}$ that describe the statistics of their sizes and gaps. We show how these LDE's yield the standard thermal exponent of droplet theory $\ensuremath{\theta}$ through the relation $\ensuremath{\theta}={\ensuremath{\theta}}_{l}+d{\ensuremath{\lambda}}_{l}.$ The present approach provides a new way to measure the thermal exponent $\ensuremath{\theta}$ without any assumption about the correct procedure to generate typical low-lying excitations as is commonly done in many perturbation methods including domain wall calculations. To illustrate the usefulness of the method we present a detailed investigation of the properties of the lowest energy droplets in two-dimensional Gaussian Ising spin glasses. By independent measurements of both LDE's and an aspect-ratio analysis, we find $\ensuremath{\theta}(2d)\ensuremath{\simeq}\ensuremath{-}0.46(1)<{\ensuremath{\theta}}_{\mathrm{DW}}(2d)\ensuremath{\simeq}\ensuremath{-}0.287$ where ${\ensuremath{\theta}}_{\mathrm{DW}}$ is the thermal exponent obtained in domain-wall theory. We also discuss the origin of finite-volume corrections in the behavior of the LDE ${\ensuremath{\theta}}_{l}$ and relate them to the finite-volume corrections in the statistics of extreme values. Finally, we analyze some geometrical properties of the lowest energy droplets, finding results in agreement with those recently reported by Kawashima and Aoki [J. Phys. Soc. Jpn. $69,$ 169 (2000)]. All in all, we show that typical large-scale droplets are not probed by most of the present perturbation methods, since they probably do not have a compact structure as has been recently suggested. We speculate that a multifractal scenario could be at the roots of the reported discrepancies on the value of the thermal exponent $\ensuremath{\theta}$ in the two-dimensional Gaussian Ising spin glass.

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.

Real-space renormalization-group approaches for two-dimensional Gaussian Ising spin glass

Two-dimensional Ising spin glass with Gaussian couplings is studied through different real-space renormalization-group procedures. The stiffness exponent y, associated with the evolution of the coupling distribution at zero temperature, is computed. Our results are discussed and compared with other estimates available.

Violation of the fluctuation-dissipation theorem in finite-dimensional spin glasses

We study the violation of the fluctuation-dissipation theorem in the three and four dimensional Gaussian Ising spin glasses using on and off equilibrium simulations. We have characterized numerically the function X(C) that determine the violation and we have studied its scaling properties. Moreover we have computed the function x(C) which characterize the breaking of the replica symmetry directly from equilibrium simulations. The two functions are numerically equal and in this way we have established that the conjectured connection between the violation of fluctuation dissipation theorem in the off-equilibrium dynamics and the replica symmetry breaking at equilibrium holds for finite dimensional spin glasses. These results point to a spin glass phase with spontaneously broken replica symmetry in finite dimensional spin glasses.

Local phase-space structure and low-temperature dynamics of short-range Ising spin glasses.

We study the phase-space geometry and the low-temperature relaxation of short-range Gaussian Ising spin-glasses, by finding all the configurations and their interconnection within a ``pocket'' of states surrounding a low-lying energy minimum. Thermalization within the pocket is modeled by a master equation, which is solved numerically, yielding the time-dependent propagator. The findings show that the relaxation proceeds by a series of local equilibrations in nested ``valleys,'' and support the dynamical predictions of hierarchical relaxation models.

Monte Carlo study on dynamical aspects of the 2D Gaussian Ising spin glass

Dynamical aspects of the 2D Gaussian Ising spin glass are studied by a Monte Carlo simulation. The relaxation times tau i of individual spins in the system are determined by analysing their autocorrelation functions. The distribution of In tau i turns out to be broad, and rather homogeneous, and with the decreasing temperature its upper limit increases more rapidly than expected from the simple Arrhenius law. These tau i are shown to reproduce excellently most of the qualitative features of the total spin relaxation function and the complex AC susceptibility observed experimentally in various spin-glass systems. The authors argue then that the change of the tau i distribution by temperatures originates from the cooperative aspect of spin clusters, which themselves are by no means rigid even in the limited temperature range investigated.

Monte-Carlo Simulation of a Gaussian Isign Spin Glass–Bond Patterns and Clustering Feature–

For the purpose to study clustering feature in one sample of spin glasses by means of Monte-Carlo simulations, we devise a `bond pattern' of the system, which is constructed by drawing stable right bonds (\( \langle \sigma _{i}\sigma _{j}\text{sgn}(J_{ij}) \rangle \cong 1\)) and stable wrong bonds (\( \langle \sigma _{i}\sigma _{j}\text{sgn}(J_{ij}) \rangle \cong{-}1\)) in a figure. We apply this idea to a Gaussian Ising spin glass, and investigate temperature- and time-dependences of clusters in this system, which are sets of spins connected by such stable bonds.