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Journal of Physics A Mathematical and Theoretical

Separation of variables in anisotropic models: anisotropic Rabi and elliptic Gaudin model in an external magnetic field

T. Skrypnyk

Published Jun 6, 2017• 11 citations
Publisher

Abstract

Abstract We study the problem of separation of variables for classical integrable Hamiltonian systems governed by non-skew-symmetric non-dynamical s o ( 3 ) ⊗ s o ( 3 ) -valued elliptic r -matrices with spectral parameters. We consider several examples of such models, and perform separation of variables for classical anisotropic one- and two-spin Gaudin-type models in an external magnetic field, and for Jaynes–Cummings–Dicke-type models without the rotating wave approximation.

DOI
10.1088/1751-8121/aa7784

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Abstract We study the problem of separation of variables for classical integrable Hamiltonian systems governed by non-skew-symmetric non-dynamical s o ( 3 ) ⊗ s o ( 3 ) -valued elliptic r -matrices with spectral parameters. We consider several examples of such models, and perform separation of variables for classical anisotropic one- and two-spin Gaudin-type models in an external magnetic field, and for Jaynes–Cummings–Dicke-type models without the rotating wave approximation.
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We construct generalized Gaudin systems in an external magnetic field corresponding to arbitrary so(3)-valued non-skew-symmetric r-matrices with spectral parameters and non-homogeneous external magnetic fields. In the case of r-matrices diagonal in the sl(2) basis we calculate the spectrum and the eigen-values of the corresponding generalized Gaudin hamiltonians using the algebraic Bethe ansatz. We explicitly consider several one-parametric families of non-skew-symmetric classical r-matrices and the corresponding generalized Gaudin systems in a magnetic field. We apply these results to fermionic systems and obtain a wide class of new integrable fermionic BCS-type hamiltonians.
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We consider the limiting j3→∞ case of the elliptic Gaudin-type model in an external magnetic field [13], [14], [15], [17], [27] and of the elliptic BCS-Richardon model with one type of fermions [27] associated with non-skew-symmetric elliptic r-matrix defined on 4:1 unramified covering of the Weierstrass cubic y2=(u+j1)(u+j2)(u+j3). We develop the algebraic Bethe ansatz for the considered elliptic r-matrix in the j3→∞ limit and find the spectrum of its Gaudin-type and Richardson-type Hamiltonians in terms of solutions of the modified Bethe equations.
Abstract. In this work we generalise previous results connecting (rational) Gaudin magnet models and classical separation of variables. It is shown that the connection persists for the case of linear r-matrix algebra which corresponds to the trigonometric 4 × 4 r-matrix (of the XXZ type). We comment also on the corresponding problem for the elliptic (XYZ) r-matrix. A prescription for obtaining integrable systems associated with multiple poles of an L-operator is given. 1. Introduction. Separation of variables for the Hamilton-Jacobi and Schrödinger equations have long been known as methods for explicit solution of these equations in appropriate circumstances. The technical requirements for this method of solution have quite fully developed in recent years (see [1–8]). In particular the relationship between the separable systems and the Gaudin magnet [4,9] integrable
In this article we make a brief review of the generalized Gaudin spin chains with [21] and without [18], [19] external magnetic field in the simplest case of g=so(3) and provide the r-matrix interpretation and generalization of the recent results of [38], [39], [40], on their exact solution. We present the classification theorem for the r-matrices that are diagonal in the natural basis and for the “shift elements” playing the role of the interaction strengths and external magnetic fields in the corresponding Gaudin-type models. We apply these results to the description, classification and solution of BCS-Richardson-type models. In particular, we introduce “elliptic” BCS-Richardson hamiltonian corresponding to non-skew-symmetric elliptic r-matrix and calculate its spectrum. We show that the “closed” [23], [30] and the “open” [36] BCS-Richardson hamiltonian of p+ip-type coincide with its trigonometric degenerations.