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Journal of Physics C Solid State Physics

Dynamics of Ising spin glasses

Alan J. Bray, M A Moore, P Reed

Published Jun 28, 1979• 14 citations
Publisher

Abstract

A simple approximation procedure is introduced which quantitatively accounts for the low-temperature logarithmic decay in time of the autocorrelation function for the Edwards-Anderson model of an Ising spin glass. The relationship between the autocorrelation function and the susceptibility is used to derive the dependence on observation time of the susceptibility. The order parameter q(t) is calculated and is in good agreement with the Monte Carlo simulations on two- and three-dimensional systems with Gaussian bond distributions, which show that q(t) tends to zero for long times t.

DOI
10.1088/0022-3719/12/12/006

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•Jan 1, 1993•101 citations
A simple approximation procedure is introduced which quantitatively accounts for the low-temperature logarithmic decay in time of the autocorrelation function for the Edwards-Anderson model of an Ising spin glass. The relationship between the autocorrelation function and the susceptibility is used to derive the dependence on observation time of the susceptibility. The order parameter q(t) is calculated and is in good agreement with the Monte Carlo simulations on two- and three-dimensional systems with Gaussian bond distributions, which show that q(t) tends to zero for long times t.
The nature of the freezing process in two- and three-dimensional Ising spin glasses with bonds J ij =+or-J has been investigated. Monte Carlo simulations indicate that the characteristic cusp in the susceptibility chi is a non-equilibrium phenomenon resulting from the finite observation times available for the computation of chi (or its experimental measurement). This picture of the freezing process, which is illustrated by explicit calculation for the one-dimensional Ising spin glass, implies chi =1/k B T for all temperatures, or that the Edwards-Anderson order parameter q vanishes. Investigation of the ground states of certain periodic models in two and three dimensions show that every spin belongs to a finite cluster which may be turned over with no cost of energy, so that q is rigorously zero for such models. It is argued that this result holds also for the fully random models.
The low temperature dynamics of the three-dimensional Ising spin-glass in zero field with a discrete bond distribution is investigated via MC simulations. The thermoremanent magnetization is found to decay algebraically and the temperature dependent exponents agree very well with the experimentally determined values. The nonequilibrium autocorrelation function C(t, t w ) shows a crossover at the waiting (or aging) time t w from algebraic quasiequilibrium decay for times t<<t w to another, faster algebraic decay for t>>t w with an exponent similar to one for the remanent magnetization.
I present an analysis of the dynamic behavior of short-range Ising spin glasses observed in stochastic simulations. The time dependence of the order parameter q(t)=〈Sx{S}_{x}Sx​(0)Sx{S}_{x}Sx​(t)〉\ifmmode\bar\else\textasciimacron\fi{}---which is the same as that of the structure factor---and the time dependence of the related dynamic correlation functions have been recorded with good statistics and very long observation times. The spin-glass model with a symmetric distribution of discrete nearest-neighbor \ifmmode\pm\else\textpm\fi{}J interactions on a simple-cubic lattice was used. Simulations were performed with a special fast computer, allowing for the first-time investigation of the equilibrium dynamics for a wide range of temperatures (0.7\ensuremath{\le}kT/J\ensuremath{\le}5.0) and lattice sizes (83{8}^{3}83, 163{16}^{3}163, 323{32}^{3}323, and 643{64}^{3}643). I have found that the empirical formula q(t)=${\mathrm{ct}}^{\mathrm{\ensuremath{-}}x}$exp(-\ensuremath{\omega}${t}^{\ensuremath{\beta}}$) with temperature-dependent exponents x(T) and \ensuremath{\beta}(T) describes the decay very well at all temperatures above the spin-glass transition. In the spin-glass phase, only the algebraic decay q(t)=${\mathrm{ct}}^{\mathrm{\ensuremath{-}}x}$ could be observed, with different temperature dependences of the exponent x(T). The dynamic scaling hypothesis and finite-size scaling explain well the observed temperature and size dependence of the data, and the functional form of the correlation functions is com- patible with the scaling form if corrections to scaling are taken into account. The scaling behavior and the dynamic and static critical exponents found in my simulations are in reasonable agreement with recent experiments performed on insulating spin glasses, showing that despite its simplicity the discrete model of spin glasses analyzed in this work displays behavior similar to that seen in nature.
The authors examine the dynamical properties of the 2D+or-J Edwards-Anderson Ising spin glass in the anomalous phase which corresponds to the Griffiths phase in the diluted ferromagnetic system. By using Monte Carlo simulation they find that the autocorrelation function can be described as a function of a power of scaled time. This new scaling includes both the stretched exponential law for the intermediate stage and the logarithmic law for the early stage of relaxation. The relaxation time obtained from the scaling is found to diverge exponentially with exponent proportional to T -2 .
In order to determine if the effects of an overlap length are visible in an Ising Edwards-Anderson spin-glass model, we have performed Monte Carlo simulations that mimic experimental procedures. The spin system was quenched from a random initial condition to a temperature T1{\mathit{T}}_{1}T1​, equilibrated in zero field for a time tw1{\mathit{t}}_{\mathit{w}1}tw1​, whereupon the system was subjected to a perturbation in field \ensuremath{\Delta}H, temperature \ensuremath{\Delta}T, or bond \ensuremath{\Delta}J with a duration tw2{\mathit{t}}_{\mathit{w}2}tw2​. After the perturbation of the original field, the temperature or bonds were restored. After waiting for a time tw3{\mathit{t}}_{\mathit{w}3}tw3​, a weak magnetic field H was applied. The susceptibility \ensuremath{\chi}(t)=[M+{\mathit{M}}_{+}M+​(t)-${\mathit{M}}_{\mathrm{\ensuremath{-}}}$(t)]/2H, where ${\mathit{M}}_{\ifmmode\pm\else\textpm\fi{}}$(t) is the magnetization due to positive or negative field, and the spin autocorrelation function q(t) were studied as functions of time. For small perturbations, \ensuremath{\Delta}HH0{\mathit{H}}_{0}H0​, \ensuremath{\Delta}TT0{\mathit{T}}_{0}T0​, or \ensuremath{\Delta}JJ0{\mathit{J}}_{0}J0​, respectively, the system is undisturbed. For larger perturbations, the correlations built up during tw1{\mathit{t}}_{\mathit{w}1}tw1​ are gradually destroyed. Similar results are found in all three types of simulations and the qualitative features are in agreement with experimental results. These features are found for both two-dimensional and three-dimensional spin-glass systems. The results can be interpreted in the domain models for spin glasses. \ensuremath{\Delta}H0{\mathit{H}}_{0}H0​ is found to be strongly dependent on the duration tw2{\mathit{t}}_{\mathit{w}2}tw2​ of the perturbation. If tw1{\mathit{t}}_{\mathit{w}1}tw1​ is constant and tw2{\mathit{t}}_{\mathit{w}2}tw2​ increases, \ensuremath{\Delta}H0{\mathit{H}}_{0}H0​ decreases. If tw1{\mathit{t}}_{\mathit{w}1}tw1​ is set equal to tw2{\mathit{t}}_{\mathit{w}2}tw2​ and tw1{\mathit{t}}_{\mathit{w}1}tw1​=tw2{\mathit{t}}_{\mathit{w}2}tw2​ is increased, \ensuremath{\Delta}H0{\mathit{H}}_{0}H0​ decreases, in agreement with the expected behavior. \ensuremath{\Delta}H0{\mathit{H}}_{0}H0​ was found to increase with increasing temperature T, which is in agreement with experimental results, but contrary to the theoretical expectations.
The low temperature dynamics of the three dimensional Ising spin glass in zero field with a discrete bond distribution is investigated via MC simulations. The thermoremanent magnetization is found to decay algebraically and the temperature dependent exponents agree very well with the experimentally determined values. The nonequilibrium autocorrelation function C(t,tw) shows a crossover at the waiting (or aging) time tw from algebraic quasi-equilibrium decay for times t≪tw to another, faster algebraic decay for t≫tw with an exponent similar to one for the remanent magnetization.