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Nuclear Physics B

Anisotropic BCS-Richardson model and algebraic Bethe ansatz

T. Skrypnyk

Published Jan 25, 2022• 10 citations
Publisher

Abstract

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.

DOI
10.1016/j.nuclphysb.2022.115679

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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.
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We consider the elliptic Gaudin-type model in an external magnetic field [11], [12], [13], [16], [15], [22] associated with non-skew-symmetric elliptic r-matrix [11] defined on 4:1 unramified covering of the Weierstrass cubic y2=(u+j1)(u+j2)(u+j3). We develop a modified algebraic Bethe ansatz for the considered elliptic r-matrix and for the algebra generated by the entries of the corresponding Lax operator with the aim of obtaining the spectra of the relevant Gaudin-type Hamiltonians in terms of solutions of modified Bethe equations. The applications of the obtained result to the diagonalization of the anisotropic quantum Euler top, quantum Zhukovsky-Volterrra top, quantum Steklov and Rubanovsky tops are given.
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In the present paper we develop the algebraic Bethe ansatz approach to the case of non-skew-symmetric gl(2)⊗gl(2)-valued Cartan-non-invariant classical r-matrices with spectral parameters. We consider the two families of these r-matrices, namely, the two non-standard rational r-matrices twisted with the help of second order automorphisms and realize the algebraic Bethe ansatz method for them. We study physically important examples of the Gaudin-type and BCS-type systems associated with these r-matrices and obtain explicitly the Bethe vectors and the spectrum for the corresponding quantum hamiltonians in terms of solutions of Bethe equations.
We show that the Lipkin-Meshkov-Glick 2N-fermion model is a particular case of one-spin Gaudin-type model in an external magnetic field corresponding to a limiting case of non-skew-symmetric elliptic r-matrix and to an external magnetic field directed along one axis. We propose an exactly-solvable generalization of the Lipkin-Meshkov-Glick fermion model based on the Gaudin-type model corresponding to the same r-matrix but arbitrary external magnetic field. This model coincides with the quantization of the classical Zhukovsky-Volterra gyrostat. We diagonalize the corresponding quantum Hamiltonian by means of the modified algebraic Bethe ansatz. We explicitly solve the corresponding Bethe-type equations for the case of small fermion number N=1,2.
Abstract In this work we construct the eigenstates of the most general spin-1/2 Richardson–Gaudin model integrable in an external magnetic field. This includes the possibility for fully anisotropic XYZ coupling such that the S i x S j x , S i y S j y and S i z S j z terms all have distinct coupling strengths. While insuring that integrability is maintained in the presence of an external field excludes the elliptic XYZ model which is only integrable at zero field, this work still covers a wide class of fully anisotropic (XYZ) models associated with non skew-symmetric r -matrices. The eigenstates, as constructed here, do not require any usable Bethe ansatz and therefore: no proper pseudo-vacuum, Bethe roots, or generalised spin raising (Gaudin) operators have to be defined. Indeed, the eigenstates are generically built only through the conserved charges which define the model of interest and the specification of the set of eigenvalues defining the particular eigenstate. Since these eigenvalues are, in general, solutions to a simple set of quadratic equations, the proposed approach is simpler to implement than any Bethe ansatz and, moreover, it remains completely identical independently of the symmetries of the model. Indeed, the construction removes any distinction between XYZ models and XXZ/XXX models and, generically, that between models with or without U (1) symmetry so that any difficulties associated with the use of a Bethe ansatz in any of these cases are avoided.