Free-fermionic ice, the Yang-Baxter equation and skein relations

TL;DR

This paper employs skein relations to prove Yang-Baxter equations for free-fermionic models, extending Kuperberg's algebraic approach to new classes of ice models like Gamma-Gamma ice.

Contribution

It introduces skein relation techniques to prove Yang-Baxter equations for free-fermionic ice models, expanding the algebraic framework beyond field-free conditions.

Findings

Proved Yang-Baxter equations for Gamma-Gamma ice.

Extended skein relation methods to free-fermionic models.

Connected algebraic proofs with integrable vertex models.

Abstract

In his famous ASM paper, Kuperberg uses a skein relation to give an algebraic proof of a Yang-Baxter equation where the Boltzmann weights satisfy the field-free condition. In this paper, we use Kuperberg's techniques to give proofs of a few Yang-Baxter equations where the Boltzmann weights satisfy the free-fermionic condition. In particular, we use skein relations to prove the Yang-Baxter equation for Gamma-Gamma ice which is a free-fermionic six-vertex model introduced by Brubaker, Bump and Friedberg.