Trigonometric version of quantum-classical duality in integrable systems

TL;DR

This paper extends the quantum-classical duality to the trigonometric case, linking the Ruijsenaars-Schneider model with the inhomogeneous twisted XXZ spin chain, revealing spectrum splitting and connections to classical and quantum integrable systems.

Contribution

It introduces a trigonometric version of the quantum-classical duality, detailing the relationship between classical and quantum integrable models with new spectral properties.

Findings

Duality relates classical Ruijsenaars-Schneider model to XXZ spin chain.

Spectrum of action variables exhibits splitting in the trigonometric case.

Connections to Calogero-Sutherland and Gaudin models are established.

Abstract

We extend the quantum-classical duality to the trigonometric (hyperbolic) case. The duality establishes an explicit relationship between the classical N-body trigonometric Ruijsenaars-Schneider model and the inhomogeneous twisted XXZ spin chain on N sites. Similarly to the rational version, the spin chain data fixes a certain Lagrangian submanifold in the phase space of the classical integrable system. The inhomogeneity parameters are equal to the coordinates of particles while the velocities of classical particles are proportional to the eigenvalues of the spin chain Hamiltonians (residues of the properly normalized transfer matrix). In the rational version of the duality, the action variables of the Ruijsenaars-Schneider model are equal to the twist parameters with some multiplicities defined by quantum (occupation) numbers. In contrast to the rational version, in the trigonometric case there is a splitting of the spectrum of action variables (eigenvalues of the classical Lax matrix). The limit corresponding to the classical Calogero-Sutherland system and quantum trigonometric Gaudin model is also described as well as the XX limit to free fermions.