Duality and the universality class of the three-state Potts antiferromagnet on plane quadrangulations

TL;DR

This paper introduces a graph duality-based criterion to determine the critical behavior and universality class of the 3-state Potts antiferromagnet on plane quadrangulations, validated by high-precision computations.

Contribution

It presents a novel criterion based on graph duality to predict critical points and universality classes for the 3-state Potts antiferromagnet on quadrangulations, supported by computational evidence.

Findings

Self-dual quadrangulations have zero-temperature critical points with central charge c=1.

Non-self-dual quadrangulations have finite-temperature critical points in the 3-state Potts ferromagnet universality class.

The Wang-Swendsen-Kotecký algorithm exhibits no critical slowing-down for self-dual cases, but does for non-self-dual cases.

Abstract

We provide a new criterion based on graph duality to predict whether the 3-state Potts antiferromagnet on a plane quadrangulation has a zero- or finite-temperature critical point, and its universality class. The former case occurs for quadrangulations of self-dual type, and the zero-temperature critical point has central charge $c=1$. The latter case occurs for quadrangulations of non-self-dual type, and the critical point belongs to the universality class of the 3-state Potts ferromagnet. We have tested this criterion against high-precision computations on four lattices of each type, with very good agreement. We have also found that the Wang-Swendsen-Koteck\'y algorithm has no critical slowing-down in the former case, and critical slowing-down in the latter.