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arXiv:1412.2586 · Journal of High Energy Physics

Supersymmetric quantum spin chains and classical integrable systems

Zengo Tsuboi, Anton Zabrodin, Andrei Vladimirovich Zotov

Published May 1, 2015• 28 citations
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Abstract

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.

DOI
10.1007/jhep05(2015)086

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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.
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.
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.
Abstract For an arbitrary generalized quantum integrable spin chain we introduce a “masterT-operator” which represents a generating function for commuting quantum transfer matrices constructed by means of the fusion procedure in the auxiliary space. We show that the functional relations for the transfer matrices are equivalent to an infinite set of model-independent bilinear equations of the Hirota form for the masterT-operator, which allows one to identify it withτ-function of an integrable hierarchy of classical soliton equations. In this paper we consider spin chains with rationalGL(N)-invariantR-matrices but the result is independent of a particular functional form of the transfer matrices and directly applies to quantum integrable models with more general (trigonometric and elliptic)R-matrices and to supersymmetric spin chains.
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.
The master T-operator is a generating function for transfer matrices and a tau-function of the modified KP (mKP) hierarchy. It is conventionally introduced through a Schur-function expansion whose coefficients are fused transfer matrices satisfying the Cherednik--Bazhanov--Reshetikhin determinant formula. For rational inhomogeneous gl(M) spin chains, we give an alternative definition: we realize gl(M) on an auxiliary Fock space generated by finitely many families of Schwinger bosons, form a monodromy matrix from the resulting L-operator, and define the master T-operator as its trace over the Fock space. Howe duality then reproduces the Schur-function expansion. This realization relates two constructions of Baxter Q-operators. In the construction of [arXiv:1112.3310] (see also [arXiv:1010.4022]), the Q-operators are obtained from residues of the master T-operator with respect to selected Miwa variables. In the construction of [arXiv:1010.3699], they are defined using degenerate Yangian L-operators in oscillator form. When selected Miwa variables approach the inverses of boundary-twist eigenvalues, the defining trace develops poles. After a suitable rescaling of this L-operator, only the terms contributing to the highest-order pole survive in the normalized limit; before the trace is taken, they form the degenerate Yangian L-operator of the second construction. The trace then gives an explicit residue formula for the corresponding Q-operator. Independently, a Holstein--Primakoff-type large-occupation-number contraction of this L-operator on subspaces with fixed total occupation numbers yields the same degenerate Yangian L-operator.
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.