Virtual Bond Percolation for Ising Cluster Dynamics

TL;DR

This paper introduces a new class of cluster representations for the Ising model using virtual bond percolation, enabling more efficient Monte Carlo algorithms with reduced autocorrelation times.

Contribution

It develops a family of exact cluster representations based on renormalized percolation rules, extending beyond the traditional Fortuin-Kasteleyn mapping, and demonstrates improved simulation algorithms.

Findings

New cluster algorithms outperform Swendsen-Wang in 2D and 3D.

Cluster algorithms reduce autocorrelation times.

Numerical evidence supports effectiveness of virtual bond percolation.

Abstract

The Fortuin-Kasteleyn mapping between the Ising model and the site-bond correlated percolation model is shown to be only one of an infinite class of exact mappings. These new cluster representations are a result of "renormalized" percolation rules correlated to entire blocks of spins. For example these rules allow for percolation on "virtual" bonds between spins not present in the underlying Hamiltonian. As a consequence we can define new random cluster theories each with its own Monte Carlo cluster dynamics that exactly reproduce the Ising model. By tuning parameters on the critical percolation surface, it is demonstrated numerically that cluster algorithms can be formulated for the 2-d and 3-d Ising model that have smaller autocorrelations than the original Swendsen-Wang algorithm.