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arXiv:cond-mat/0603526

Free-Energy Barriers in the SK Model

Elmar Bittner, Wolfhard Janke E

Published Mar 20, 2006• 0 citations
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Abstract

The Sherrington-Kirkpatrick spin-glass model is investigated by means of Monte Carlo simulations employing a combination of the multi-overlap algorithm with parallel tempering methods. We investigate the finite-size scaling behaviour of the free-energy barriers which are visible in the probability density of the Parisi overlap parameter. Assuming that the mean barrier height diverges with the number of spins N as N^alpha, our data show good agreement with the theoretical value alpha = 1/3.

DOI
10.48550/arxiv.cond-mat/0603526

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The Sherrington-Kirkpatrick spin-glass model is investigated by means of Monte Carlo simulations employing a combination of the multi-overlap algorithm with parallel tempering methods. We investigate the finite-size scaling behaviour of the free-energy barriers which are visible in the probability density of the Parisi overlap parameter. Assuming that the mean barrier height diverges with the number of spins N as N^alpha, our data show good agreement with the theoretical value alpha = 1/3.
For the Edwards-Anderson Ising spin-glass model in three and four dimensions (3d(3d(3d and 4d)4d)4d) we have performed high statistics Monte Carlo calculations of those free-energy barriers FBq{F}_{B}^{q}FBq​ which are visible in the probability density PJ(q){P}_{\mathcal{J}}(q)PJ​(q) of the Parisi overlap parameter q. The calculations rely on the recently introduced multioverlap algorithm. In both dimensions, within the limits of lattice sizes investigated, these barriers are found to be non-self-averaging and the same is true for the autocorrelation times of our algorithm. Further, we present evidence that barriers hidden in q dominate the canonical autocorrelation times.
We present results of recent high-statistics Monte Carlo simulations of the Edwards-Anderson Ising spin-glass model in three and four dimensions. The study is based on a non-Boltzmann sampling technique, the multi-overlap algorithm which is specifically tailored for sampling rare-event states. We thus concentrate on those properties which are difficult to obtain with standard canonical Boltzmann sampling such as the free-energy barriers F^q_B in the probability density P_J(q) of the Parisi overlap parameter q and the behaviour of the tails of the disorder averaged density P(q) = [P_J(q)]_av.
We report large-scale simulations of the three-dimensional Edwards-Anderson Ising spin glass system using the recently introduced multi-overlap Monte Carlo algorithm. In this approach the temperature is fixed and two replica are coupled through a weight factor such that a broad distribution of the Parisi overlap parameter q is achieved. Canonical expectation values for the entire q-range (multi-overlap) follow by reweighting. We present an analysis of the performance of the algorithm and in particular discuss results on spin glass free-energy barriers which are hard to obtain with conventional algorithms. In addition we discuss the non-trivial scaling behavior of the canonical q-distributions in the broken phase.
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.
Abstract We report large‐scale simulations of the three‐dimensional Edwards‐Anderson Ising spin glass system using the recently introduced multi‐overlap Monte Carlo algorithm. In this approach the temperature is fixed and two replica are coupled through a weight factor such that a broad distribution of the Parisi overlap parameter q is achieved. Canonical expectation values for the entire q ‐range (multi‐overlap) follow by reweighting. We present an analysis of the performance of the algorithm and in particular discuss results on spin glass free‐energy barriers which are hard to obtain with conventional algorithms. In addition we discuss the non‐trivial scaling behavior of the canonical q ‐distributions in the broken phase.