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Nuclear Physics B

Classical r-matrices, “elliptic” BCS and Gaudin-type hamiltonians and spectral problem

T. Skrypnyk

Published Feb 19, 2019• 27 citations
Publisher

Abstract

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.

DOI
10.1016/j.nuclphysb.2019.02.018

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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.
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.
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 We consider the elliptic Gaudin-type model in an external magnetic field (Skrypnyk T 2005 Phys. Lett. A 334 390–9; Skrypnyk T 2005 Phys. Lett. A 347 266–7; Skrypnyk T 2006 J. Geom. Phys. 57 53–67; Skrypnyk T 2006 J. Math. Phys. 47; Skrypnyk T 2007 J. Phys. A 40 1611–23; Skrypnyk T 2019 Nucl. Phys. B 941 225–48) associated with non-skew-symmetric elliptic r-matrix (Skrypnyk T 2005 Phys. Lett. A 334 390–9; Skrypnyk T 2005 Phys. Lett. A 347 266–7; Skrypnyk T 2006 J. Geom. Phys. 57 53–67; Skrypnyk T 2006 J. Math. Phys. 47). Using them we construct a new integrable fermion Hamiltonian of the Richardson type. We use the modified algebraic Bethe ansatz obtained for integrable models with the considered elliptic r-matrix in (Skrypnyk T 2023 Nucl. Phys. B 988 116102) and find the spectrum of the obtained Richardson-type Hamiltonian in terms of solutions of the modified Bethe equations. The obtained results generalize our previous results on Richardson-type models (Skrypnyk T 2022 Nucl. Phys. B 975 115679).
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 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 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.