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.

arXiv comment14 pages

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• On factorized Lax pairs for classical many-body integrable systems
• Dualities in quantum integrable many-body systems and integrable probabilities -- I
• 3d Mirror Symmetry for Instanton Moduli Spaces
• Lax representation of the hyperbolic van Diejen dynamics with two coupling parameters
• Self-dual form of Ruijsenaars-Schneider models and ILW equation with discrete Laplacian
• QKZ-Ruijsenaars correspondence revisited
• Supersymmetric extension of qKZ-Ruijsenaars correspondence
• KZ-Calogero correspondence revisited
• Generalized Devil's staircase and RG flows
• Quantum-classical duality for Gaudin magnets with boundary
• The Zoo of Opers and Dualities
• Self-duality and scattering map for the hyperbolic van Diejen systems with two coupling parameters (with an appendix by S. Ruijsenaars)
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• Quantum-classical correspondence for gl(1|1) supersymmetric Gaudin magnet with boundary
• Integrable many-body systems of Calogero-Ruijsenaars type
• Interrelations between dualities in classical integrable systems and classical-classical version of quantum-classical duality