The master T-operator for vertex models with trigonometric $R$-matrices as classical tau-function

TL;DR

This paper demonstrates that the master T-operator for certain integrable vertex models with trigonometric R-matrices functions as a classical tau-function, linking quantum integrable systems with classical soliton equations.

Contribution

It applies the master T-operator construction to trigonometric R-matrix models, showing it acts as a tau-function satisfying Hirota equations and relates to classical particle systems.

Findings

Master T-operator generates commuting transfer matrices.

It satisfies classical Hirota bilinear equations.

Connection established with Ruijsenaars-Schneider particle system.

Abstract

The construction of the master T-operator recently suggested in Alexandrov et al. (arXiv:1112.3310) is applied to integrable vertex models and associated quantum spin chains with trigonometric R-matrices. The master T-operator is a generating function for commuting transfer matrices of integrable vertex models depending on infinitely many parameters. At the same time it turns out to be the tau-function of an integrable hierarchy of classical soliton equations in the sense that it satisfies the the same bilinear Hirota equations. The class of solutions of the Hirota equations that correspond to eigenvalues of the master T-operator is characterized and its relation to the classical Ruijsenaars-Schneider system of particles is discussed.