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 $\xi \sim 5000$; the data are consistent with $\xi (\beta )=A{e}^{2\beta }{\beta }^p(1+a_1{e}^{-\beta }+...)$ as $\beta \to \infty$, with $p\approx 1$. The staggered susceptibility behaves as ${\chi }_{stagg}\sim {\xi }^{5/3}$. For q=4 the model is disordered ($\xi \text{ltapprox}2$) 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.

Comments: LaTeX, 67 pages including 19 figures. Discussion of alternative scenario added to Sections 4.2.1 and 7.1