A master solution of the quantum Yang-Baxter equation and classical discrete integrable equations

TL;DR

This paper introduces a new master solution to the quantum Yang-Baxter equation, unifying known solutions and linking statistical mechanics models with classical discrete integrable equations, especially near zero temperature.

Contribution

It presents a novel master solution to the star-triangle relation that encompasses all known continuous and discrete spin solutions, connecting quantum integrability with classical discrete equations.

Findings

New master solution of the star-triangle relation with positive Boltzmann weights.

Connection between zero-temperature limit and classical discrete integrable equations.

Reduction to known models like Kashiwara-Miwa and chiral Potts in special cases.

Abstract

We obtain a new solution of the star-triangle relation with positive Boltzmann weights which contains as special cases all continuous and discrete spin solutions of this relation, that were previously known. This new master solution defines an exactly solvable 2D lattice model of statistical mechanics, which involves continuous spin variables, living on a circle, and contains two temperature-like parameters. If one of the these parameters approaches a root of unity (corresponds to zero temperature), the spin variables freezes into discrete positions, equidistantly spaced on the circle. An absolute orientation of these positions on the circle slowly changes between lattice sites by overall rotations. Allowed configurations of these rotations are described by classical discrete integrable equations, closely related to the famous $Q_4$-equations by Adler Bobenko and Suris. Fluctuations between degenerate ground states in the vicinity of zero temperature are described by a rather general integrable lattice model with discrete spin variables. In some simple special cases the latter reduces to the Kashiwara-Miwa and chiral Potts models.