Determinant of the Potts model transfer matrix and the critical point

TL;DR

This paper derives an exact formula for the determinant of the Potts model transfer matrix on a 3D lattice and uses it to conjecture an approximation for the critical temperature in higher dimensions.

Contribution

It introduces a novel decomposition method for the transfer matrix of the Potts model and proposes a new formula to estimate critical points in hypercubic lattices.

Findings

Exact determinant calculation for the 3D Potts model transfer matrix.

A conjectured formula for approximating the critical temperature in d-dimensional lattices.

Potentially improved understanding of phase transitions in higher-dimensional systems.

Abstract

By using a decomposition of the transfer matrix of the $q$-state Potts Model on a three dimensional $ m \times n \times n $ simple cubic lattice its determinant is calculated exactly. By using the calculated determinants a formula is conjectured which approximates the critical temperature for a d-dimensional hypercubic lattice.