Bethe Ansatz and Classical Hirota Equation

TL;DR

This paper explores the deep connection between quantum integrable models and classical soliton equations with discretized time, revealing that spectral data of quantum systems can be derived from classical tau-functions and dynamics of zeros.

Contribution

It establishes a novel link between quantum spectral characteristics and classical tau-functions, providing a new perspective on quantum integrable systems through classical equations.

Findings

Quantum transfer matrix eigenvalues are identified with classical tau-functions.

Bethe ansatz equations correspond to the dynamics of zeros in classical models.

Classical discrete Sine-Gordon equations are related to elliptic solutions of the Bethe ansatz.

Abstract

We discuss an interrelation between quantum integrable models and classical soliton equations with discretized time. It appeared that spectral characteristics of quantum integrable systems may be obtained from entirely classical set up. Namely, the eigenvalues of the quantum transfer matrix and the scattering $S$-matrix itself are identified with a certain $\tau$-functions of the discrete Liouville equation. The Bethe ansatz equations are obtained as dynamics of zeros. For comparison we also present the Bethe ansatz equations for elliptic solutions of the classical discrete Sine-Gordon equation. The paper is based on the recent study of classical integrable structures in quantum integrable systems, hep-th/9604080.