On the non-ergodicity of the Swendsen-Wang-Kotecky algorithm on the kagome lattice

TL;DR

This paper proves that the Wang-Swendsen-Kotecky cluster algorithm is non-ergodic when applied to the 3-state kagome-lattice Potts antiferromagnet at zero temperature with periodic boundary conditions, indicating limitations in its sampling ability.

Contribution

It provides a rigorous proof of non-ergodicity for the algorithm on certain kagome lattice configurations, highlighting a fundamental limitation in its use for this model.

Findings

The algorithm is non-ergodic on symmetric kagome lattice subsets.

Single-site dynamics also fail to be ergodic under the same conditions.

Not all configurations are accessible from a given initial state.

Abstract

We study the properties of the Wang-Swendsen-Kotecky cluster Monte Carlo algorithm for simulating the 3-state kagome-lattice Potts antiferromagnet at zero temperature. We prove that this algorithm is not ergodic for symmetric subsets of the kagome lattice with fully periodic boundary conditions: given an initial configuration, not all configurations are accessible via Monte Carlo steps. The same conclusion holds for single-site dynamics.