On the solutions of the Yang-Baxter equations with general inhomogeneous eight-vertex $R$-matrix: Relations with Zamolodchikov's tetrahedral algebra

TL;DR

This paper classifies the most general one-parameter solutions to the Yang-Baxter equations for six- and eight-vertex models with particle number conservation mod(2), and explores their relation to Zamolodchikov's tetrahedral algebra.

Contribution

It provides a complete classification of inhomogeneous solutions to the Yang-Baxter equations for specific vertex models and links them to tetrahedral algebra structures.

Findings

Derived the most general solutions with particle number conservation mod(2).

Identified particular solutions depending on two spectral parameters.

Discussed applications to Zamolodchikov's tetrahedral algebra.

Abstract

We present most general one-parametric solutions of the Yang-Baxter equations (YBE) for one spectral parameter dependent $R_{ij}(u)$-matrices of the six- and eight-vertex models, where the only constraint is the particle number conservation by mod(2). A complete classification of the solutions is performed. We have obtained also two spectral parameter dependent particular solutions $R_{ij}(u,v)$ of YBE. The application of the non-homogeneous solutions to construction of Zamolodchikov's tetrahedral algebra is discussed.