The three-state Potts antiferromagnet on plane quadrangulations

TL;DR

This paper investigates the phase behavior of the 3-state Potts antiferromagnet on plane quadrangulations, proposing a duality-based criterion to predict critical points and confirming it through simulations and analytical methods.

Contribution

It introduces a new criterion based on duality properties to predict phase diagrams of the model on various quadrangulations and validates it with high-precision computational techniques.

Findings

Self-dual quadrangulations exhibit a zero-temperature critical point with central charge c=1.

Non-self-dual quadrangulations have three low-temperature ordered phases and a finite-temperature critical point with c=4/5.

The proposed criterion accurately predicts phase behavior across studied quadrangulations.

Abstract

We study the antiferromagnetic 3-state Potts model on general (periodic) plane quadrangulations $\Gamma$. Any quadrangulation can be built from a dual pair $(G,G^*)$. Based on the duality properties of $G$, we propose a new criterion to predict the phase diagram of this model. If $\Gamma$ is of self-dual type (i.e., if $G$ is isomorphic to its dual $G^*$), the model has a zero-temperature critical point with central charge $c=1$, and it is disordered at all positive temperatures. If $\Gamma$ is of non-self-dual type (i.e., if $G$ is not isomorphic to $G^*$), three ordered phases coexist at low temperature, and the model is disordered at high temperature. In addition, there is a finite-temperature critical point (separating these two phases) which belongs to the universality class of the ferromagnetic 3-state Potts model with central charge $c=4/5$. We have checked these conjectures by studying four (resp. seven) quadrangulations of self-dual (resp. non-self-dual) type, and using three complementary high-precision techniques: Monte-Carlo simulations, transfer matrices, and critical polynomials. In all cases, we find agreement with the conjecture. We have also found that the Wang-Swendsen-Kotecky Monte Carlo algorithm does not have (resp. does have) critical slowing down at the corresponding critical point on quadrangulations of self-dual (resp. non-self-dual) type.