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Journal of Physics C Solid State Physics

Order parameters of the spin glass mean field theory and initial conditions

C. De Dominicis, A. P. Young

Published Jun 30, 1983• 10 citations
Publisher

Abstract

The time-dependent order parameter q(t 2 -t 1 ) is re-examined as a function of the time difference t 2 -t 1 and number of spins N, for the infinite-range spin glass. In the statistical mechanics limit, t 2 -t 1 to infinity before N to infinity , the authors find q(t 2 -t 1 )=q= integral q(x)dx, where q(x) is the order parameter function of Parisi (1979, 1980, 1983), provided the system has a canonical (Boltzmann) distribution at the earlier time t 1 . In the opposite limit, N to infinity before t 2 -t 1 to infinity , they obtain q(t 2 -t 1 )=q(x=1). If the initial distribution differs slightly from a canonical one they obtain q(t 2 , t 1 )=q(x=0) (i.e. the statistical mechanics limit) and the Sompolinsky solution (1981) appears to be correct under these (non-equilibrium) conditions.

DOI
10.1088/0022-3719/16/18/008

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Physical review. B, Condensed matter•Jan 1, 1982•85 citations
The time-dependent order parameter q(t 2 -t 1 ) is re-examined as a function of the time difference t 2 -t 1 and number of spins N, for the infinite-range spin glass. In the statistical mechanics limit, t 2 -t 1 to infinity before N to infinity , the authors find q(t 2 -t 1 )=q= integral q(x)dx, where q(x) is the order parameter function of Parisi (1979, 1980, 1983), provided the system has a canonical (Boltzmann) distribution at the earlier time t 1 . In the opposite limit, N to infinity before t 2 -t 1 to infinity , they obtain q(t 2 -t 1 )=q(x=1). If the initial distribution differs slightly from a canonical one they obtain q(t 2 , t 1 )=q(x=0) (i.e. the statistical mechanics limit) and the Sompolinsky solution (1981) appears to be correct under these (non-equilibrium) conditions.
We discuss the dynamics of the infinite-range Sherrington-Kirkpatrick spin-glass model for which relaxation times diverge when NNN, the number of spins, tends to infinity. Calculations on a large but finite system are very difficult, so we mimic a large finite system in equilibrium by working with $N=\ensuremath{\infty}$ and imposing, by hand, a canonical distribution at an initial time. For short times, where no barrier hopping has occurred, we find that the Edwards-Anderson order parameter, qEA{q}_{\mathrm{EA}}qEA​, is identical to that obtained from an analysis of the mean-field equations of Thouless, Anderson, and Palmer and, with further assumptions, gives q(x=1)q(x=1)q(x=1) in Parisi's theory, in agreement with earlier work. For times longer than the longest relaxation time (of the finite system), true equilibrium is reached and our theory agrees with previous statistical-mechanics calculations using the replica trick. There is no violation of the fluctuation-dissipation theorem.
The authors argue that standard treatments of dynamics are incorrect for non-ergodic systems such as the Sherrington-Kirkpatrick model of a spin glass. Since the system never loses the memory of its initial state it is necessary to average over initial states with a Boltzmann weight to obtain time-dependent correlation functions as conventionally defined. It is now necessary to introduce replicas. The 'dynamical' order parameter, q EA , is equal to q(x=1) in Parisi's scheme whereas the statistical mechanics order parameter, q, is given by integral 0 I q(x) dx. They also propose an interpretation of the function q(x).
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.
It is proposed that the spin-glass phase is described by two order parameters q(x)=[〈Si(0)Si(tx)〉]avq(x)={[〈{S}_{i}(0){S}_{i}({t}_{x})〉]}_{\mathrm{a}\mathrm{v}}q(x)=[〈Si​(0)Si​(tx​)〉]av​ and $\ensuremath{\chi}(x)=\ensuremath{\int}{0}^{{t}_{x}}\ensuremath{\chi}(t)\mathrm{dt}$, that measure the relaxations of the average autocorrelation and susceptibility along macroscopic time scales, tx{t}_{x}tx​, which are parametrized in a decreasing order by $x\ensuremath{\in}[0,1]$. The function qqq decays from a finite value at t1{t}_{1}t1​ to zero at t0{t}_{0}t0​, while $\ensuremath{\chi}$ increases from $\ensuremath{\chi}(1)$ to a value $\ensuremath{\chi}(0)$ which is independent of temperature in the mean-field limit. The equilibrium results agree with Parisi's replica solution.
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.
The low-temperature behavior of the infinite-range Ising spin-glass model with a Gaussian distribution of interactions is studied by evaluating the statistical mechanics exactly for small systems (up to N=20N=20N=20 spins). Our results can be well understood by a simple picture of elementary cluster excitations from the ground state which overturn, on average, of order N12{N}^{\frac{1}{2}}N21​ spins. The conventional definition of the spin-glass order parameter is shown to be unsatisfactory and a new definition is proposed. It is speculated that ensemble averages may give different results from time averages for $N\ensuremath{\rightarrow}\ensuremath{\infty}$ even in the infinite time limit.