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

TL;DR

This study uses high-precision Monte Carlo simulations to analyze the antiferromagnetic Potts model on a square lattice for q=3 and q=4, revealing detailed correlation behaviors and disordered states at zero temperature.

Contribution

It provides the first high-precision numerical analysis of the antiferromagnetic q-state Potts model on the square lattice for q=3 and q=4, including new correlation inequalities and algorithm ergodicity proofs.

Findings

For q=3, correlation length up to ~5000 with exponential growth in inverse temperature.

For q=4, the model remains disordered with very short correlation length.

Proved correlation inequality and ergodicity of the Monte Carlo algorithm on bipartite lattices.

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 ($\xi \ltapprox 2$) 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.