Critical Temperatures from Domain-Wall Microstate Counting: A Topological Solution for the Potts Universality Class
Abstract
We derive a universal relation for the critical temperatures of the -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 -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 -state family from a single topological calibration.
Cite
@article{arxiv.2602.00962,
title = {Critical Temperatures from Domain-Wall Microstate Counting: A Topological Solution for the Potts Universality Class},
author = {David Vaknin},
journal= {arXiv preprint arXiv:2602.00962},
year = {2026}
}