Quantum Integrable Systems and Elliptic Solutions of Classical Discrete Nonlinear Equations

TL;DR

This paper establishes a deep connection between quantum integrable models and classical discrete nonlinear equations, providing new solutions and determinant formulas for eigenvalues of transfer matrices.

Contribution

It identifies the functional relations of quantum transfer matrices with classical Hirota's bilinear difference equation, revealing new solutions and determinant representations.

Findings

Elliptic solutions of Hirota's equation give complete eigenvalues of quantum transfer matrices.

Eigenvalues of Baxter's Q-operator are solutions to classical Hirota's linear problems.

New determinant formula for eigenvalues of quantum transfer matrices.

Abstract

Functional relation for commuting quantum transfer matrices of quantum integrable models is identified with classical Hirota's bilinear difference equation. This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries. The standard objects of quantum integrable models are identified with elements of classical nonlinear integrable difference equation. In particular, elliptic solutions of Hirota's equation give complete set of eigenvalues of the quantum transfer matrices. Eigenvalues of Baxter's $Q$-operator are solutions to the auxiliary linear problems for classical Hirota's equation. The elliptic solutions relevant to Bethe ansatz are studied. The nested Bethe ansatz equations for $A_{k-1}$-type models appear as discrete time equations of motions for zeros of classical $\tau$-functions and Baker-Akhiezer functions. Determinant representations of the general solution to bilinear discrete Hirota's equation and a new determinant formula for eigenvalues of the quantum transfer matrices are obtained.