The three-state Potts antiferromagnet on plane quadrangulations

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IOP Publishing Ltd.
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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.
Duality, Potts antiferromagnet, Plane quadrangulation, Transfer matrix, Monte Carlo simulation, Critical polynomial, Wang-Swendsen-Koteky algorithm
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Journal of Physics A, Mathematical and Theoretical, 51(36), 3650012, July 2018, 44 pp.