Partition function of two- and three-dimensional Potts ferromagnets for arbitrary values of q>0

TL;DR

This paper introduces a high-precision numerical algorithm to compute the partition function of Potts models for any real q>0, enabling detailed analysis of phase transition behaviors in 2D and 3D systems.

Contribution

A novel algorithm for calculating the Potts model partition function for arbitrary q>0 at any temperature with high accuracy.

Findings

Confirmed the critical q_c=4 in 2D Potts models.

Determined the critical q_c≈2.35 in 3D Potts models.

Analyzed large system sizes up to 1000x1000 and 100x100x100.

Abstract

A new algorithm is presented, which allows to calculate numerically the partition function Z_q of the d-dimensional q-state Potts models for arbitrary real values q>0 at any given temperature T with high precision. The basic idea is to measure the distribution of the number of connected components in the corresponding Fortuin-Kasteleyn representation and to compare with the distribution of the case q=1 (graph percolation), where the exact result Z_1=1 is known. As application, d=2 and d=3-dimensional ferromagnetic Potts models are studied, and the critical values q_c, where the transition changes from second to first order, are determined. Large systems of sizes N=1000^2 respectively N=100^3 are treated. The critical value q_c(d=2)=4 is confirmed and q_c(d=3)=2.35(5) is found.