The three-state Potts antiferromagnet on plane quadrangulations

Abstract

We study the antiferromagnetic 3-state Potts model on general (periodic) plane quadrangulations $\hat{I}“$. Any quadrangulation can be built from a dual pair $(G,G^{\ast })$. Based on the duality properties of $G$, we propose a new criterion to predict the phase diagram of this model. If $\hat{I}“$ is of self-dual type (i.e., if $G$ is isomorphic to its dual $G^{\ast }$), the model has a zero-temperature critical point with central charge $c=1$, and it is disordered at all positive temperatures. If $\hat{I}“$ is of non-self-dual type (i.e., if $G$ is not isomorphic to $G^{\ast }$), 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.

55 pages, pdflatex. Contains 33 pdf figures. Uses tikz package. Minor changes with respect to v1. Final version