Exchange Monte Carlo Method and Application to Spin Glass Simulations

TLDR

The authors propose an efficient Monte Carlo algorithm that simulates hard‑to‑relax systems by running many replicas at different temperatures and exchanging configurations between them. The algorithm is applied to the three‑dimensional ±J Ising spin glass, using replica exchange across temperatures to facilitate sampling. The exchange process dramatically reduces ergodicity time compared to the multi‑canonical method, yielding nearly exponential decay of correlation functions and rapid relaxation even at low temperatures.

Abstract

We propose an efficient Monte Carlo algorithm for simulating a ``hardly-relaxing" system, in which many replicas with different temperatures are simultaneously simulated and a virtual process exchanging configurations of these replica is introduced. This exchange process is expected to let the system at low temperatures escape from a local minimum. By using this algorithm the three-dimensional $\pm J$ Ising spin glass model is studied. The ergodicity time in this method is found much smaller than that of the multi-canonical method. In particular the time correlation function almost follows an exponential decay whose relaxation time is comparable to the ergodicity time at low temperatures. It suggests that the system relaxes very rapidly through the exchange process even in the low temperature phase.