Quantum spin chains and integrable many-body systems of classical mechanics

TL;DR

This paper reviews the recent discovery of a deep connection between quantum spin chains and classical integrable many-body systems, highlighting how spectral problems in quantum models relate to inverse spectral problems in classical Lax matrices.

Contribution

It elucidates the relationship between quantum spin chains and classical integrable systems, focusing on the GL(2)-invariant models and their link to the Ruijsenaars-Schneider system.

Findings

Spectral problem for quantum Hamiltonians relates to inverse spectral problem of classical Lax matrices.

Quantum spin chains on N sites are connected to the Ruijsenaars-Schneider system.

Explicit analysis provided for the case N=2.

Abstract

This note is a review of the recently revealed intriguing connection between integrable quantum spin chains and integrable many-body systems of classical mechanics. The essence of this connection lies in the fact that the spectral problem for quantum Hamiltonians of the former models is closely related to a sort of inverse spectral problem for Lax matrices of the latter ones. For simplicity, we focus on the most transparent and familiar case of spin chains on N sites constructed by means of the GL(2)-invariant R-matrix. They are related to the classical Ruijsenaars-Schneider system of N particles, which is known to be an integrable deformation of the Calogero-Moser system. As an explicit example the case N=2 is considered in detail.