Antiferromagnetic Potts Models on the Square Lattice: A High-Precision Monte Carlo Study

Abstract

We study the antiferromagnetic q-state Potts model on the square lattice for q=3 and q=4, using the Wang-Swendsen-Kotecky (WSK) Monte Carlo algorithm and a powerful finite-size-scaling extrapolation method. For q=3 we obtain good control up to correlation length $ξ\sim 5000$; the data are consistent with $ξ(β) = A e^{2β} β^p (1 + a_1 e^{-β} + ...)$ as $β\to\infty$, with $p \approx 1$. The staggered susceptibility behaves as $χ_{stagg} \sim ξ^{5/3}$. For q=4 the model is disordered ($ξ\ltapprox 2$) even at zero temperature. In appendices we prove a correlation inequality for Potts antiferromagnets on a bipartite lattice, and we prove ergodicity of the WSK algorithm at zero temperature for Potts antiferromagnets on a bipartite lattice.