Trigonometric version of quantum–classical duality in integrable systems

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.

Quantum-classical duality for Gaudin magnets with boundary

“…We understand that the results obtained in this paper need further generalizations in several directions, including the extension to higher rank Gaudin magnets solved by means of the nested Bethe ansatz method, the supersymmetric extension (see [24] for the A N −1 case), the trigonometric version of the duality [2] and the generalization to the level of spin chains and related Ruijsenaars-Schneider-van Diejen relativistic many-body systems.…”

On factorized Lax pairs for classical many-body integrable systems

“…The latter follows from (2.2) by the transposition (denoted by T ) and changing q → −q. Curiously, both factorization (for L RS and L RS ′ ) emerge in the framework of the quantum-classical correspondence [41,82,15]. They emerge for two possible values of the Z 2 -grading parameter in the supersymmetric spin chains.…”

“…In the last Section we propose factorization formulae of type (1.18) for the rational Calogero models related to root systems B, C, D. This study is inspired by possible application to quantumclassical duality [41,82,15].…”

Supersymmetric extension of qKZ-Ruijsenaars correspondence

“…Using the asymptotics of solutions to the (q)KZ equations [15] it was also argued in [18,19] that the qKZ-Ruijsenaars correspondence can be viewed as a quantization of the quantum-classical duality [1,7,2] (see also [13,5]), which relates the generalized inhomogeneous quantum spin chains and the classical Ruijsenaars-Schneider model. Consider the classical Kbody Ruijsenaars-Schneider model, where the positions of particles {x i } are identified with the inhomogeneity parameters of the spin chain which is described by its transfer matrix…”