Critical Temperatures from Domain-Wall Microstate Counting: A Topological Solution for the Potts Universality Class

TL;DR

This paper introduces a topological microstate counting method to derive universal critical temperature relations for the Potts model across various lattice geometries, unifying phase transition understanding.

Contribution

It presents a novel topological framework that accurately predicts critical temperatures for the Potts model on multiple lattice types, including 3D geometries.

Findings

Exact solutions for 2D lattices like square, triangular, honeycomb

Sub-3% accuracy for 3D lattices such as simple cubic, bcc, fcc, diamond

Unifies Potts phase transitions into a single geometric classification

Abstract

We derive a universal relation for the critical temperatures of the $q$-state Potts model based on the counting of domain-wall microstates. By balancing interface energy against configurational entropy, we show that the critical temperature is determined by the ratio of the coordination-dependent energy cost to the logarithm of a total multiplicity factor. This factor decomposes into a lattice-topological constant, representing a projection from an underlying orthogonal Euclidean space, and a term representing Markovian sampling in the $q$-dimensional state space. The framework recovers exact solutions for two-dimensional square, triangular, and honeycomb lattices and achieves sub-3\% accuracy for three-dimensional simple cubic, bcc, fcc, and diamond geometries. This approach unifies the Potts universality class into a single geometric classification, revealing that the phase transition is governed by the saturation of interface propagation through the lattice manifold and providing a predictive tool that characterizes the entire $q$-state family from a single topological calibration.