Classical facets of quantum integrability

TL;DR

This paper reviews the connection between quantum integrable models solvable by Bethe ansatz and classical soliton equations, offering new methods for spectral problems and insights into quantum-classical duality.

Contribution

It introduces an alternative approach to solving quantum spectral problems using classical soliton theory and provides a simplified proof of quantum-classical duality.

Findings

Classical soliton methods can solve quantum spectral problems.

A new, simpler proof of quantum-classical duality is proposed.

The study links quantum spin chains with classical integrable systems.

Abstract

This paper is a review of the works devoted to understanding and reinterpretation of the theory of quantum integrable models solvable by Bethe ansatz in terms of the theory of purely classical soliton equations. Remarkably, studying polynomial solutions of the latter by methods of classical soliton theory, one is able to develop a method of solving the spectral problem for the former which provides an alternative to the Bethe ansatz procedure. Our main examples are the generalized inhomogeneous spins chains with twisted boundary conditions on the quantum side and the modified Kadomtsev-Petviashvili hierarchy of nonlinear differential-difference equations on the classical side. In this paper, we restrict ourselves to quantum spin chains with rational $GL(n)$-invariant $R$-matrices (of the XXX type). Also, the connection of quantum spin chains with classical soliton equations implies a close interrelation between the spectral problem for spin chains and integrable many-body systems of classical mechanics such as Calogero-Moser and Ruijsenaars-Scheider models, which is known as the quantum-classical duality. Revisiting this topic, we suggest a simpler and more instructive proof of this kind of duality.