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Physical Review Letters

Direct Determination of the Probability Distribution for the Spin-Glass Order Parameter

A. P. Young

Published Sep 26, 1983• 99 citations
Publisher

Abstract

The recent interpretation of Parisi's order-parameter function $q(x)$ 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 $q(x)$ are reproduced, in particular $q(x)\ensuremath{\propto}x$ as $x\ensuremath{\rightarrow}0$ and ${q}^{\ensuremath{'}}(x)=0$ at $q={q}_{max}$, 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 $x$.

DOI
10.1103/physrevlett.51.1206

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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∝xq(x)\ensuremath{\propto}xq(x)\ensuremath∝x as x\ensuremath→0x\ensuremath{\rightarrow}0x\ensuremath→0 and q\ensuremath′(x)=0{q}^{\ensuremath{'}}(x)=0q\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{q}^{\ensuremath{'}}(x)=0q\ensuremath′(x)=0 for a range of xxx.
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.
We discuss the results of numerical simulations of two spin-glass models, the infinite range Sherrington–Kirkpatrick (SK) model, and a nearest–neighbor model in three dimensions. The probability distribution PN(q) for the spin glass order parameter q has been evaluated for the SK model below Tc with N, the number of spins, up to 192. It shows the same basic features as the Parisi function dx/dq as expected, but with sizeable finite size corrections. Results for the short-range model show evidence for a finite temperature transition but further work is needed to confirm this.
The infinite-range Ising spin-glass in the presence of a transverse field is studied by Monte Carlo simulations. The overlap probability distribution P ( q ) is defined and calculated for the present quantum case. The P ( q ) in the spin-glass phase has a long tail extended down to q = 0 qualitatively similar to the classical Sherrington-Kirkpatrick model. The phase boundary of the glass transition temperature vs. the transverse field is obtained. Our results for the phase diagram agree with our recent replica symmetry-breaking solution but disagree with the results of Yokata and of Ray et al.
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 have studied the Parisi overlap distribution for the three-dimensional Ising spin glass in the Migdal-Kadanoff approximation. For temperatures $T\ensuremath{\simeq}{0.7T}_{c}$ and system sizes up to L=32L\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}32L=32, we found a P(q)P(q)P(q) as expected for full Parisi replica-symmetry breaking, just as was also observed in recent Monte Carlo simulations on a cubic lattice. However, for lower temperatures our data agree with predictions from the droplet or scaling picture. The failure to see droplet model behavior in Monte Carlo simulations is due to the fact that all existing simulations are done at temperatures too close to the transition temperature so that system sizes larger than the correlation length have not been achieved.
We describe results of Monte Carlo simulation studies on the Ising spin glass in four dimensions on a hypercubic lattice with nearest neighbor bonds. Studies of the equilibrium static properties show that the system undergoes a genuine phase transition to an ordered spin glass phase. Critical dynamical behavior is analyzed to obtain the dynamic exponent. Finally, we describe results on the spin glass phase, in particular the finite size scaling of the order parameter distribution function, and compare it with existing models of the spin glass phase, namely the droplet model and the Parisi solution for the low temperature phase of the infinite range spin glass.