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Journal of Physics A Mathematical and Theoretical

Classicalr-matrices and integrable BCS models with many types of fermions

T. Skrypnyk

Published Oct 3, 2012• 2 citations
Publisher

Abstract

Based on generalized Gaudin models and Lie algebras so (2 m ), we construct two integrable cases of the pairing BCS Hamiltonians containing m types of ‘spinless’ fermions. They are the so-called s -type and p x + i p y -type BCS Hamiltonians. In the particular case of the Lie algebra so (8), the constructed models can be viewed as N = Z proton–neutron integrable models. We find the spectrum of the constructed Hamiltonians in terms of solutions of Bethe-type equations.

DOI
10.1088/1751-8113/45/41/415203

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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.
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.