Synchronous Monte Carlo method and its application to a mean-field spin glass model

Abstract

Standard Markov chain Monte Carlo methods update variables one at a time to satisfy detailed balance, which prevents them from fully exploiting massively parallel hardware such as graphics processing units (GPUs). We study a Monte Carlo method for systems with pairwise interactions in which all variables are updated simultaneously, made possible by auxiliary Gaussian fields introduced through the Gaussian integral identity. The method satisfies detailed balance and is therefore guaranteed to converge to the canonical distribution. For mean-field spin glasses, its dynamics can be analyzed exactly by dynamical mean-field theory (DMFT). Applying the method and the DMFT to the random orthogonal model, which exhibits a random first-order transition, we find that the fluctuation-dissipation theorem (FDT) is clearly violated below the dynamical transition temperature, and that the relation between response and correlation takes the two-slope form of a generalized FDT, with simulations and theory in quantitative agreement.