Potts-model critical manifolds revisited

TL;DR

This paper computes highly accurate critical manifolds for the q-state Potts model on all Archimedean lattices using advanced algorithms, improving known thresholds and revealing detailed phase features.

Contribution

It introduces a parallel algorithm implementation that computes exact critical polynomials for larger bases, enhancing precision in critical point determination.

Findings

Kagome-lattice threshold determined to eleven digits

(3,12^2) lattice threshold determined to thirteen digits

Detailed features of the antiferromagnetic region and Berker-Kadanoff phase

Abstract

We compute the critical polymials for the q-state Potts model on all Archimedean lattices, using a parallel implementation of the algorithm of (Jacobsen, J. Phys. A: Math. Theor. 47 135001) that gives us access to larger sizes than previously possible. The exact polynomials are computed for bases of size $6 \times 6$ unit cells, and the root in the temperature variable $v=e^K-1$ is determined numerically at $q=1$ for bases of size $8 \times 8$. This leads to improved results for bond percolation thresholds, and for the Potts-model critical manifolds in the real $(q,v)$ plane. In the two most favourable cases, we find now the kagome-lattice threshold to eleven digits and that of the $(3,12^2)$ lattice to thirteen. Our critical manifolds reveal many interesting features in the antiferromagnetic region of the Potts model, and determine accurately the extent of the Berker-Kadanoff phase for the lattices studied.