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arXiv:cond-mat/0010349 · Journal of Physics A Mathematical and General

The BCS model and the off-shell Bethe ansatz for vertex models

Luigi Amico A, G. Falci, Rosario Fazio

Published Aug 10, 2001• 38 citations
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Abstract

We study the connection between the BCS pairing model and the inhomogeneous vertex model. The two spectral problems coincide in the quasi-classical limit of the off-shell Bethe ansatz of the disordered six-vertex model. The latter problem is transformed into an auxiliary spectral problem which corresponds to the diagonalization of the integrals of motion of the BCS model. A generating functional whose quasi-classical expansion leads to the constants of motion of the BCS model, and in particular the Hamiltonian, is identified.

DOI
10.1088/0305-4470/34/33/307

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We study the connection between the BCS pairing model and the inhomogeneous vertex model. The two spectral problems coincide in the quasi-classical limit of the off-shell Bethe ansatz of the disordered six-vertex model. The latter problem is transformed into an auxiliary spectral problem which corresponds to the diagonalization of the integrals of motion of the BCS model. A generating functional whose quasi-classical expansion leads to the constants of motion of the BCS model, and in particular the Hamiltonian, is identified.
Abstract. We study the connection between the BCS pairing model and the inhomogeneous vertex model. The two spectral problems coincide in the quasi– classical limit of the off–shell Bethe Ansatz of the disordered six vertex model. The latter problem is transformed into an auxiliary spectral problem which corresponds to the diagonalization of the integrals of motion of the BCS model. A generating functional whose quasi classical expansion leads to the constants of motion of the BCS model and in particular the Hamiltonian, is identified. PACS: 03.65.Fd, 74.20.Fg
We show in detail how Richardson's exact solution of a discrete-state BCS (DBCS) model can be recovered as a special case of an algebraic Bethe-ansatz solution of the inhomogeneous XXX vertex model with twisted boundary conditions: by implementing the twist using Sklyanin's K\mathbf{K}K-matrix construction and taking the quasiclassical limit, one obtains a complete set of conserved quantities Hi{\mathbf{H}}_{i}Hi​ from which the DBCS Hamiltonian can be constructed as a second order polynomial. The eigenvalues and eigenstates of the Hi{\mathbf{H}}_{i}Hi​ (which reduce to the Gaudin Hamiltonians in the limit of infinitely strong coupling) are exactly known in terms of a set of parameters determined by a set of on-shell Bethe ansatz equations, which reproduce Richardson's equations for these parameters. We thus clarify that the integrability of the DBCS model is a special case of the integrability of the twisted inhomogeneous XXX vertex model. Furthermore, by considering the twisted inhomogeneous XXZ model and/or choosing a generic polynomial of the Hi{\mathbf{H}}_{i}Hi​'s as Hamiltonian, more general exactly solvable models can be constructed. To make the paper accessible to readers that are not Bethe-ansatz experts, the introductory sections include a self-contained review of those of its feature which are needed here.
We find an integrable generalization of the BCS model with nonuniform Coulomb and pairing interaction. The Hamiltonian is integrable by construction since it is a functional of commuting operators; these operators, which therefore are constants of motion of the model, contain the anisotropic Gaudin Hamiltonians. The exact solution is obtained diagonalizing them by means of Bethe ansatz. Uniform pairing and Coulomb interaction are obtained as the "isotropic limit" of the Gaudin Hamiltonians. We discuss possible applications of this model to a single grain and to a system of few interacting grains.