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

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.