Three-Dimensional Vertex Model in Statistical Mechanics, from Baxter-Bazhanov Model

TL;DR

This paper explores a three-dimensional vertex model in statistical mechanics derived from the Baxter-Bazhanov model, establishing duality relations, symmetry properties, and parametrizations based on cube geometry.

Contribution

It explicitly derives the Wu-Kadanoff duality for the Baxter-Bazhanov model and constructs a 3D vertex model with symmetry-based parametrization.

Findings

Explicit duality between cube and vertex tetrahedron equations.

Parametrization of weight functions via dihedral angles.

Symmetry relations under the cube's symmetry group.

Abstract

We find that the Boltzmann weight of the three-dimensional Baxter-Bazhanov model is dependent on four spin variables which are the linear combinations of the spins on the corner sites of the cube and the Wu-Kadanoff duality between the cube and vertex type tetrahedron equations is obtained explicitly for the Baxter-Bazhanov model. Then a three-dimensional vertex model is obtained by considering the symmetry property of the weight function, which is corresponding to the three-dimensional Baxter-Bazhanov model. The vertex type weight function is parametrized as the dihedral angles between the rapidity planes connected with the cube. And we write down the symmetry relations of the weight functions under the actions of the symmetry group $G$ of the cube. The six angles with a constrained condition, appeared in the tetrahedron equation, can be regarded as the six spectrums connected with the six spaces in which the vertex type tetrahedron equation is defined.