Integrability of the Six-Vertex model and the Yang-Baxter Groupoid

TL;DR

This paper investigates the solutions of the Yang-Baxter equation for the six-vertex model, introducing new parametrizations and algebraic structures, including a groupoid framework, to deepen understanding of integrability in statistical models.

Contribution

It provides novel parametrizations of the Yang-Baxter equation and introduces a groupoid structure for non-free-fermionic solutions, extending existing algebraic frameworks.

Findings

Identified maximal commutative families of solutions.

Developed a new parametrization via a groupoid of matrices.

Formulated a conjecture on the associativity of the solution composition law.

Abstract

We study the Yang-Baxter equation for the $R$-matrices of the six-vertex model. We analyze the solutions and give new parametrizations of the Yang-Baxter equation. In particular, we find the maximal commutative families of parametrized solutions which generalize the $R$-matrices from the affine quantum (super)-groups. Then we give a new parametrization of the Yang-Baxter equation by a groupoid of non-free-fermionic matrices. In the appendix, we study the general algebraic structure of the solutions of the Yang-Baxter and formulate a conjecture that extends the conjecture by Brubaker, Bump, and Friedberg that the composition law on the Yang-Baxter solutions is always associative.