Critical Temperatures from Domain-Wall Microstate Counting: A Topological Solution for the Potts Universality Class
Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Vaknin, David
Institution: Iowa State University
Subfield: Condensed Matter Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7127414041
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