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Journal of Statistical Mechanics Theory and Experiment

Generalized Gaudin systems in an external magnetic field and reflection equation algebras

T. Skrypnyk

Published Jun 28, 2010• 3 citations
Publisher

Abstract

We construct a generalized Gaudin system in an external magnetic field corresponding to non-skew-symmetric classical r -matrices r K ( u , v ) associated with the reflection equation algebras. We show that an external magnetic field for which the system stays integrable is constructed with the help of the ‘boundary’ matrix K + ( u , η) satisfying the dual reflection equation, while the corresponding non-skew-symmetric classical r -matrices r K ( u , v ) are constructed with the help of the other ‘boundary matrix’ K ( u ) satisfying the reflection equation.

DOI
10.1088/1742-5468/2010/06/p06028

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We construct a generalized Gaudin system in an external magnetic field corresponding to non-skew-symmetric classical r -matrices r K ( u , v ) associated with the reflection equation algebras. We show that an external magnetic field for which the system stays integrable is constructed with the help of the ‘boundary’ matrix K + ( u , η) satisfying the dual reflection equation, while the corresponding non-skew-symmetric classical r -matrices r K ( u , v ) are constructed with the help of the other ‘boundary matrix’ K ( u ) satisfying the reflection equation.
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We construct integrable cases of generalized classical and quantum Gaudin spin chains in an external magnetic field. For this purpose, we generalize the 'shift of argument method' onto the case of classical and quantum integrable systems governed by an arbitrary -valued, non-dynamical classical r -matrix with spectral parameters. We consider several examples of the obtained construction for the cases of skew-symmetric, 'twisted' non-skew-symmetric and 'anisotropic' non-skew-symmetric classical r -matrices. We show, in particular, that in a general case in order for the Gaudin system in a magnetic field to be integrable, the corresponding magnetic field should be non-homogeneous.
We consider generalized quantum Gaudin systems in an external magnetic field associated with non-skew-symmetric sl (2)-valued classical r -matrices. We calculate spectra of the generating function of the corresponding Hamiltonians using the algebraic Bethe ansatz. We apply these results to the construction of integrable fermionic Hamiltonians of a generalized BCS type. We investigate the special cases when the corresponding integrable Hamiltonians contain only a pairing interaction term and consider an example of such a situation associated with a special non-skew-symmetric r -matrix.
We consider quantum integrable systems associated with the Lie algebra gl(n) and Cartan-invariant non-dynamical non-skew-symmetric classical r-matrices. We describe the sub-class of Cartan-invariant non-skew-symmetric r-matrices for which exists the standard procedure of the nested Bethe ansatz associated with the chain of embeddings gl(n)⊃gl(n−1)⊃gl(n−2)⊃⋯⊃gl(1). We diagonalize the corresponding quantum integrable systems by its means. We illustrate the obtained results by the examples of the generalized Gaudin systems with and without external magnetic field associated with three classes of non-dynamical non-skew-symmetric classical r-matrices.
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We construct quantum integrable systems associated with non-skew-symmetric gl(2)-valued classical r-matrices. We find a new explicit multiparametric family of such the non-skew-symmetric classical r-matrices. We consider two classes of examples of the corresponding integrable systems, namely generalized Gaudin systems with and without an external magnetic field. In the case of arbitrary r-matrices diagonal in a standard gl(2)-basis, we calculate the spectrum of the corresponding quantum integrable systems using the algebraic Bethe ansatz. We apply these results to a construction of integrable fermionic models and obtain a wide class of integrable Bardeen–Cooper–Schrieffer (BCS)-type fermionic Hamiltonians containing the pairing and electrostatic interaction terms. We also consider special cases when the corresponding integrable Hamiltonians contain only pairing interaction term and are exact analogs of the “reduced BCS Hamiltonian” of Richardson.