Classical facets of quantum integrability

TL;DR

This review clarifies how quantum integrable models solvable by Bethe ansatz can be reinterpreted through classical soliton theory via the master $T$-operator and the modified KP hierarchy, establishing a quantum–classical duality with Ruijsenaars–Schneider dynamics. The core methodology replaces Bethe-ansatz diagonalization with an operator-valued tau-function framework, where eigenvalues of transfer matrices are tau-functions of the mKP hierarchy and the nested Bethe equations emerge from discrete Bäcklund/dressing transformations. It shows that spectral data of quantum chains correspond to Lax spectra of classical many-body systems, enabling a classical inverse spectral approach to the quantum spectrum. The discussion covers polynomial/quasi-polynomial solutions, co-derivative representations, $Q$-operators, and extensions to supersymmetric and trigonometric cases, with open directions toward elliptic models and broader dualities. Overall, the work provides a deep, constructive bridge between quantum integrability and classical soliton theory, with potential practical impact on diagonalizing quantum Hamiltonians via classical dynamics.

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.