Classical tau-function for quantum spin chains

TL;DR

This paper introduces a universal master T-operator for quantum integrable spin chains, linking quantum transfer matrices to classical soliton equations through Hirota bilinear relations, applicable across various R-matrix types.

Contribution

It establishes a model-independent framework connecting quantum transfer matrices with classical integrable hierarchies via the master T-operator.

Findings

Master T-operator generates commuting transfer matrices.

Functional relations are equivalent to Hirota bilinear equations.

Results apply to rational, trigonometric, elliptic, and supersymmetric spin chains.

Abstract

For an arbitrary generalized quantum integrable spin chain we introduce a "master T -operator" which represents a generating function for commuting quantum transfer matrices constructed by means of the fusion procedure in the auxiliary space. We show that the functional relations for the transfer matrices are equivalent to an infinite set of model-independent bilinear equations of the Hirota form for the master T -operator, which allows one to identify it with {\tau}-function of an integrable hierarchy of classical soliton equations. In this paper we consider spin chains with rational GL(N)-invariant R-matrices but the result is independent of a particular functional form of the transfer matrices and directly applies to quantum integrable models with more general (trigonometric and elliptic) R-matrices and to supersymmetric spin chains.