The three-state Potts antiferromagnet on plane quadrangulations

Abstract

We study the antiferromagnetic 3-state Potts model on general (periodic) plane quadrangulations Γ. 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 Γ 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 Γ 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–Kotecký 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.