Lattice models, differential forms, and the Yang-Baxter equation

TL;DR

This paper introduces novel mathematical frameworks for analyzing lattice models in statistical mechanics, providing new proofs, enumeration methods, and conditions related to the Yang-Baxter equation for six- and eight-vertex models.

Contribution

It presents a differential forms approach for the six-vertex model, a vector space interpretation for the eight-vertex model, and necessary conditions for the Yang-Baxter equation in the latter.

Findings

Proof of correspondence between states and 3-colorings for six-vertex model

Enumeration of admissible states for eight-vertex model

Necessary conditions for Yang-Baxter equation in eight-vertex model

Abstract

We introduce new methods to describe admissible states of the six-vertex and the eight-vertex lattice models of statistical mechanics. For the six-vertex model, we view the admissible states as differential forms on a grid graph. This yields a new proof of the correspondence between admissible states and 3-colorings of a rectangular grid. For the eight-vertex model, we interpret the set of admissible states as an $\mathbb{F}_2$-vector space. This viewpoint lets us enumerate the set of admissible states. Finally, we find necessary conditions for a Yang-Baxter equation to hold for the general eight-vertex model.