Three-dimensionalizing the eight-vertex model

TL;DR

This paper introduces a new three-dimensional generalization of the eight-vertex model using a simple ansatz for R-matrices that satisfy the tetrahedron equation, expanding the mathematical framework of integrable models.

Contribution

It proposes a novel ansatz for two-color R-matrices satisfying the tetrahedron equation, extending the eight-vertex model to three dimensions and generalizing permutation-type operators.

Findings

The R-matrix depends on one parameter.

The tetrahedron equation holds on an algebraic set with five irreducible components.

The approach unifies and generalizes previous models.

Abstract

A simple ansatz is proposed for two-color R-matrix satisfying the tetrahedron equation. It generalizes, on one hand, a particular case of the eight-vertex model to three dimensions, and on another hand - Hietarinta's permutation-type operators to their linear combinations. Each separate R-matrix depends on one parameter, and the tetrahedron equation holds provided the quadruple of parameters belongs to an algebraic set containing five irreducible two-dimensional components.