Spectrum of quantum transfer matrices via classical many-body systems

Abstract

In this paper we clarify the relationship between inhomogeneous quantum spin chains and classical integrable many-body systems. It provides an alternative (to the nested Bethe ansatz) method for computation of spectra of the spin chains. Namely, the spectrum of the quantum transfer matrix for the inhomogeneous gl n -invariant XXX spin chain on N sites with twisted boundary conditions can be found in terms of velocities of particles in the rational N -body Ruijsenaars-Schneider model. The possible values of the velocities are to be found from intersection points of two Lagrangian submanifolds in the phase space of the classical model. One of them is the Lagrangian hyperplane corresponding to fixed coordinates of all N particles and the other one is an N -dimensional Lagrangian submanifold obtained by fixing levels of N classical Hamiltonians in involution. The latter are determined by eigenvalues of the twist matrix. To support this picture, we give a direct proof that the eigenvalues of the Lax matrix for the classical Ruijsenaars-Schneider model, where velocities of particles are substituted by eigenvalues of the spin chain Hamiltonians, calculated through the Bethe equations, coincide with eigenvalues of the twist matrix, with certain multiplicities. We also prove a similar statement for the gl n Gaudin model with N marked points (on the quantum side) and the Calogero-Moser system with N particles (on the classical side). The realization of the results obtained in terms of branes and supersymmetric gauge theories is also discussed.

Separation of variables and scalar products at any rank

“…As a natural extension, it would be interesting to generalize our results to other types of spin chains based on B n , C n , D n Lie algebras, and especially to the supersymmetric case (particularly relevant for AdS/CFT applications), and also to various deformations, including trigonometric or even elliptic models, Gaudin models and boundary problems. It would be interesting to explore the implications of this construction for various classical/quantum and spectral dualities between integrable models [62][63][64].…”

Scalar Products in Twisted XXX Spin Chain. Determinant Representation

“…Recall that we use the shorthand notation for the products (2.12) in representations (3.1) and (3.2). The proof of the equivalence of these representations can be found in [19]. It is based on the recursive property (A.6).…”

Relativistic classical integrable tops and quantum R-matrices

“…• Relations between the Ruijsenaars-Schneider (RS) systems and quantum integrable chains appeared recently in the context of the Quantum-Classical duality using the Bethe ansatz approach [33] or the τ -function approach [3,4]. This duality, in particular, implies the substitution η = into the Lax matrix of the RS model and provides an alternative (to the algebraic Bethe ansatz) method for computation of spectrum of the quantum spin chains transfer-matrices.…”