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Journal of Physics C Solid State Physics

Non-equilibrium correlations in the infinite-range Ising spin glass

R. N. Bhatt, A. P. Young

Published Jan 30, 1988• 8 citations
Publisher

Abstract

The authors use Monte Carlo simulations to investigate the possibility of obtaining equilibrium values of quantities which relax slowly to equilibrium by studying the evolution of their non-equilibrium behaviour. As an example they study the order parameter distribution P(q) for the infinite-range Ising spin glass during the approach to equilibrium. This is evaluated both by computing the overlap between two initially uncorrelated copies of the system and from the correlation between the same set of spins at two different times. They find that the value of the most slowly relaxing part of the distribution, P(q=0), decreases monotonically with time from the first method, while a monotonically increasing result is obtained from the second technique. Consequently they expect that one could obtain the equilibrium value of P(q=0) for larger sizes by extrapolation of non-equilibrium data.

DOI
10.1088/0022-3719/21/3/004

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Physical review. B, Condensed matter•Jan 1, 1994•13 citations
The authors use Monte Carlo simulations to investigate the possibility of obtaining equilibrium values of quantities which relax slowly to equilibrium by studying the evolution of their non-equilibrium behaviour. As an example they study the order parameter distribution P(q) for the infinite-range Ising spin glass during the approach to equilibrium. This is evaluated both by computing the overlap between two initially uncorrelated copies of the system and from the correlation between the same set of spins at two different times. They find that the value of the most slowly relaxing part of the distribution, P(q=0), decreases monotonically with time from the first method, while a monotonically increasing result is obtained from the second technique. Consequently they expect that one could obtain the equilibrium value of P(q=0) for larger sizes by extrapolation of non-equilibrium data.
For pt.I see ibid., vol.16, no.7, p.1219 (1983). The response of a simple infinite-range model of a spin glass to the imposition of a small external field is studied. The field is turned on at t=0, and the evolution of the spin correlations is followed as processes involving flips of successively larger and larger clusters of spins occur, until equilibrium is achieved at t= infinity . The evolution from one state to the other proceeds through a quasi-continuum of intermediate metastable states labelled by the fraction y of spins that flip simultaneously on that time scale. The time scale for such processes grows very long as the size N of the system increases (possibly like exp((Ny)12/Tg/T)), so in the infinite system, only an infinitesimal fraction of this evolution can be observed. The generalised equation of state describing the evolution of correlations has the form derived by Sompolinsky, but the physical interpretation is different. In particular, the author does not postulate any violation of the fluctuation-dissipation theorem in equilibrium; the terms in the equations which correspond to those which violate it in his theory have the form they do because a non-equilibrium situation is being described. The physical interpretation of the variable y also allows an additional relation between initial- and final-state correlation functions which leads to the Parisi equation for his order parameter q(x), with x=1-y.
The recent interpretation of Parisi's order-parameter function q(x)q(x)q(x) in terms of a probability distribution for the overlap between magnetizations in different phases is investigated by Monte Carlo computer simulation for the infinite-range Ising spin-glass model. The main features of the solution for q(x)q(x)q(x) are reproduced, in particular $q(x)\ensuremath{\propto}x$ as $x\ensuremath{\rightarrow}0$ and ${q}^{\ensuremath{'}}(x)=0$ at q=qmaxq={q}_{max}q=qmax​, the largest value. Finite-size effects prevent one from establishing with certainty whether there is a "plateau," i.e., ${q}^{\ensuremath{'}}(x)=0$ for a range of xxx.
We investigate the order-parameter distribution P(q) of the Ising spin glass with nearest-neighbor interactions in four dimensions using Monte Carlo simulations on lattices of linear dimension up to L=6. We find that, below the transition temperature Tc{\mathit{T}}_{\mathit{c}}Tc​, the weight at small q seems to saturate to a nonzero value as the size increases, similar to the infinite-range Sherrington-Kirkpatrick model. We discuss our results in the light of recent theoretical predictions for the nature of the spin-glass phase.
We present a numerical study of the non-equilibrium dynamics of the Sherrington-Kirkpatrick model. We analize the overlap distribution between the configurations visited at the time t and in particular its scaling behaviour with the size of the system. We find two different non-equilibrium dynamical regimes. The first is a proper Out of Equilibrium Regime, that is the relevant regime for the dynamics of an infinite system. The second is an Intermediate Regime that separates the Out of Equilibrium Regime from equilibrium. After studying the crossover beetween the two regimes, we focus on the Out of Equilibrium Regime and we reveal some of the geometrical features of phase space. According to recent analitical work, we find that the asymptotic out of equilibrium energy density is the equilibrium one. The same happens to the staggered-magnetisation, suggesting there is a deep geometrical similarity between equilibrium and out of equilibrium configurations. A procedure, that we call clonation, shows that the dynamics follows sort of canyons, as opposed to a system of independent traps.
We present results of a Monte Carlo study of the equilibrium dynamics of the one dimensional long-range Ising spin glass model. By tuning a parameter , this model interpolates between the mean field Sherrington–Kirkpatrick model and a proxy of the finite dimensional Edward-Anderson model. Activated scaling fits for the behavior of the relaxation time as a function of the number of spins N (Namely ) give values of that are not stable against inclusion of subleading corrections. Critical scaling ( ) gives more stable fits, at least in the non mean field region. We also present results on the scaling of the time decay of the critical remanent magnetization of the Sherrington–Kirkpatrick model, a case where the simulation can be done with quite large systems and that shows the difficulties in obtaining precise values for dynamical exponents in spin glass models.
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.