This short note is a review of the intriguing connection between the quantum Gaudin model and the classical KP hierarchy recently established in [1]. We construct the generating function of integrals of motion for the quantum Gaudin model with twisted boundary conditions (the master T-operator) and show that it satisfies the bilinear identity and Hirota equations for the classical KP hierarchy. This implies that zeros of eigenvalues of the master $T$ -operator in the spectral parameter have the same dynamics as the Calogero-Moser system of particles.This short note is a review of the intriguing connection between the quantum Gaudin model and the classical KP hierarchy recently established in [1]. We construct the generating function of integrals of motion for the quantum Gaudin model with twisted boundary conditions (the master T-operator) and show that it satisfies the bilinear identity and Hirota equations for the classical KP hierarchy. This implies that zeros of eigenvalues of the master $T$-operator in the spectral parameter have the same dynamics as the Calogero-Moser system of particles.

Jan 1, 20130 citations

The Master T-Operator for Inhomogeneous XXX Spin Chain and mKP Hierarchy

Anton Vladimirovich Zabrodin

we show how to construct the master T -operator for the quantum inhomogeneous GL(N ) XXX spin chain with twisted boundary conditions.It satisfies the bilinear identity and Hirota equations for the classical mKP hierarchy.We also characterize the class of solutions to the mKP hierarchy that correspond to eigenvalues of the master T -operator and study dynamics of their zeros as functions of the spectral parameter.This implies a remarkable connection between the quantum spin chain and the classical Ruijsenaars-Schneider system of particles.we show how to construct the master T -operator for the quantum inhomogeneous GL(N ) XXX spin chain with twisted boundary conditions.It satisfies the bilinear identity and Hirota equations for the classical mKP hierarchy.We also characterize the class of solutions to the mKP hierarchy that correspond to eigenvalues of the master T -operator and study dynamics of their zeros as functions of the spectral parameter.This implies a remarkable connection between the quantum spin chain and the classical Ruijsenaars-Schneider system of particles.

arXiv:1310.6988Symmetry Integrability and Geometry Methods and ApplicationsJan 11, 201410 citations

Quantum Gaudin model, spin chains, and universal characters

Chuanzhong Li, Bao Shou

In this paper, we will construct the connection between the quantum Gaudin model and the universal character hierarchy by a newly defined coupled master T-operator. The coupled master T-operator of the quantum Gaudin model satisfies the bilinear identity of the universal character hierarchy, which is an extension of the Kadomtsev–Petviashvili hierarchy. In addition, for the generalized quantum integrable spin chains with rational GL(N)-invariant R-matrices, we construct its coupled master T-operator, which represents a generating function for two-folds commuting quantum transfer matrices. The functional relations for the transfer matrices are equivalent to an infinite set of Hirota bilinear equations of the modified universal character hierarchy.In this paper, we will construct the connection between the quantum Gaudin model and the universal character hierarchy by a newly defined coupled master T-operator. The coupled master T-operator of the quantum Gaudin model satisfies the bilinear identity of the universal character hierarchy, which is an extension of the Kadomtsev–Petviashvili hierarchy. In addition, for the generalized quantum integrable spin chains with rational GL(N)-invariant R-matrices, we construct its coupled master T-operator, which represents a generating function for two-folds commuting quantum transfer matrices. The functional relations for the transfer matrices are equivalent to an infinite set of Hirota bilinear equations of the modified universal character hierarchy.

Journal of Mathematical PhysicsOct 1, 20205 citations

Quantum-classical duality for Gaudin magnets with boundary

M. Vasilyev, Anton Vladimirovich Zabrodin, Andrei Vladimirovich Zotov

We establish a remarkable relationship between the quantum Gaudin models with boundary and the classical many-body integrable systems of Calogero-Moser type associated with the root systems of classical Lie algebras (B, C and D). We show that under identification of spectra of the Gaudin Hamiltonians HjG with particles velocities q˙j of the classical model all integrals of motion of the latter take zero values. This is the generalization of the quantum-classical duality observed earlier for Gaudin models with periodic boundary conditions and Calogero-Moser models associated with the root system of the type A.We establish a remarkable relationship between the quantum Gaudin models with boundary and the classical many-body integrable systems of Calogero-Moser type associated with the root systems of classical Lie algebras (B, C and D). We show that under identification of spectra of the Gaudin Hamiltonians HjG with particles velocities q˙j of the classical model all integrals of motion of the latter take zero values. This is the generalization of the quantum-classical duality observed earlier for Gaudin models with periodic boundary conditions and Calogero-Moser models associated with the root system of the type A.

arXiv:1911.11792Nuclear Physics BJan 21, 20204 citations

Quantum spin chains and classical integrable systems

Zabrodin Anton

Abstract This chapter is a review of the recently established quantum-classical correspondence for integrable systems based on the construction of the master T-operator. For integrable inhomogeneous quantum spin chains with gl(N)-invariant R-matrices in finite-dimensional representations, the master T-operator is a sort of generating function for the family of commuting quantum transfer matrices depending on an infinite number of parameters. Any eigenvalue of the master T-operator is the tau-function of the classical modified KP hierarchy. It is a polynomial in the spectral parameter which is identified with the 0th time of the hierarchy. This implies a remarkable relation between the quantum spin chains and classical many-body integrable systems of particles of the Ruijsenaars-Schneider type. As an outcome, a system of algebraic equations can be obtained for the spectrum of the spin chain Hamiltonians.Abstract This chapter is a review of the recently established quantum-classical correspondence for integrable systems based on the construction of the master T-operator. For integrable inhomogeneous quantum spin chains with gl(N)-invariant R-matrices in finite-dimensional representations, the master T-operator is a sort of generating function for the family of commuting quantum transfer matrices depending on an infinite number of parameters. Any eigenvalue of the master T-operator is the tau-function of the classical modified KP hierarchy. It is a polynomial in the spectral parameter which is identified with the 0th time of the hierarchy. This implies a remarkable relation between the quantum spin chains and classical many-body integrable systems of particles of the Ruijsenaars-Schneider type. As an outcome, a system of algebraic equations can be obtained for the spectrum of the spin chain Hamiltonians.

Aug 10, 20170 citations

Supersymmetric quantum spin chains and classical integrable systems

Zengo Tsuboi, Anton Zabrodin, Andrei Vladimirovich Zotov

For integrable inhomogeneous supersymmetric spin chains (generalized graded magnets) constructed employing Y(gl(N|M))-invariant R-matrices in finite-dimensional representations we introduce the master T-operator which is a sort of generating function for the family of commuting quantum transfer matrices. Any eigenvalue of the master T-operator is the tau-function of the classical mKP hierarchy. It is a polynomial in the spectral parameter which is identified with the 0-th time of the hierarchy. This implies a remarkable relation between the quantum supersymmetric spin chains and classical many-body integrable systems of particles of the Ruijsenaars-Schneider type. As an outcome, we obtain a system of algebraic equations for the spectrum of the spin chain Hamiltonians.For integrable inhomogeneous supersymmetric spin chains (generalized graded magnets) constructed employing Y(gl(N|M))-invariant R-matrices in finite-dimensional representations we introduce the master T-operator which is a sort of generating function for the family of commuting quantum transfer matrices. Any eigenvalue of the master T-operator is the tau-function of the classical mKP hierarchy. It is a polynomial in the spectral parameter which is identified with the 0-th time of the hierarchy. This implies a remarkable relation between the quantum supersymmetric spin chains and classical many-body integrable systems of particles of the Ruijsenaars-Schneider type. As an outcome, we obtain a system of algebraic equations for the spectrum of the spin chain Hamiltonians.

arXiv:1412.2586Journal of High Energy PhysicsMay 1, 201528 citations

T

-operator in the spectral parameter have the same dynamics as the Calogero-Moser system of particles.

T

-operator in the spectral parameter have the same dynamics as the Calogero-Moser system of particles.