Overcoming the critical slowing down of flat-histogram Monte Carlo simulations: Cluster updates and optimized broad-histogram ensembles

TL;DR

This paper investigates the limitations of flat-histogram Monte Carlo methods in simulating energy landscapes of Ising/Potts models and introduces cluster updates and ensemble optimization techniques to significantly improve sampling efficiency.

Contribution

It identifies the sub-optimal performance of single-spin flip flat-histogram methods and proposes cluster updates and ensemble optimization to overcome critical slowing down.

Findings

Mean first-passage time scales as N^2L^z with z decreasing with dimension

Cluster updates and ensemble optimization reduce z to near zero in 1D and 2D models

Proposed methods improve energy space sampling efficiency in Monte Carlo simulations

Abstract

We study the performance of Monte Carlo simulations that sample a broad histogram in energy by determining the mean first-passage time to span the entire energy space of d-dimensional ferromagnetic Ising/Potts models. We first show that flat-histogram Monte Carlo methods with single-spin flip updates such as the Wang-Landau algorithm or the multicanonical method perform sub-optimally in comparison to an unbiased Markovian random walk in energy space. For the d=1,2,3 Ising model, the mean first-passage time \tau scales with the number of spins N=L^d as \tau \propto N^2L^z. The critical exponent z is found to decrease as the dimensionality d is increased. In the mean-field limit of infinite dimensions we find that z vanishes up to logarithmic corrections. We then demonstrate how the slowdown characterized by z>0 for finite d can be overcome by two complementary approaches - cluster dynamics in connection with Wang-Landau sampling and the recently developed ensemble optimization technique. Both approaches are found to improve the random walk in energy space so that \tau \propto N^2 up to logarithmic corrections for the d=1 and d=2 Ising model.