Exactly solvable model of superconducting magnetic alloys

A model describing the Anderson impurity in the Bardeen-Cooper-Schriffer superconductor is proven to exhibit hidden integrability and is diagonalized exactly by the Bethe ansatz. Typeset using REVTEX 1 The basic theoretical models describing magnetic impurities in nonmagnetic normal metals, such as the s-d (Kondo) model, the Anderson model, etc., are integrable under two additional conditions [1]: (i) an electron-impurity coupling is assumed to be energy independent, and (ii) a band electron dispersion ε(k) can be linearized around the Fermi level, ε(k) ≃ vF (k − kF ), where kF and vF are the Fermi momentum and velocity, respectively. Only under these conditions, both an electron-impurity scattering and an effective electronelectron coupling are described in terms of discontinuous jumps in the Bethe ansatz wave functions. Therefore a linear dispersion of particles and a pointlike particle-impurity coupling are considered now as the necessary mathematical conditions for integrability of the “impurity” models. Because a carrier dispersion in superconducting metals cannot be linearized, the “linear dispersion approximation” (LDA) is clear to eliminate superconductivity from an exact analysis of the behaviour of magnetic alloys [1]. It has recently been discerned [2] that LDA is not necessary for an exact diagonalization of the basic impurity models of quantum optics, describing a system of Bose particles with a nonlinear dispersion coupled to two-level atoms [3]. Making use of some mathematical analogies between “magnetic” and “optical” models, it can be shown [4] that the degenerate Anderson model with a nonlinear band electron dispersion also exhibits hidden integrability [2] and is diagonalized exactly by the Bethe ansatz. One of the most exciting applications of the approach developed [2–4] could be an exact treatment of the superconductivity problem in the presence of magnetic impurities. Since the pioneering work of Abrikosov and Gor’kov [5], the problem has been the subject of many theoretical and experimental studies [6] but still remains one of the most intriguing puzzles of modern physics. Therefore an extension of the Bethe ansatz technique to superconducting magnetic models accounting for a gap dispersion of charge carriers is of fundamental physical interest. In the present letter, we report first results for a model describing the Anderson impurity placed within the Bardeen-Cooper-Schriffer (BCS) superconductor. To diagonalize the model Hamiltonian, we introduce auxiliary particles and show that the multiparticle scattering process is factorized into two-particle ones, despite a nonlocal effective particle2 impurity coupling. The continuous multielectron wave functions result from an integral “dressing” of the discontinuous Bethe ansatz wave functions of auxiliary particles, the information about the electron dispersion being contained into a dressing function. Imposing the periodic boundary conditions on the multielectron wave functions, we derive the Bethe ansatz equations (BAE) for the cases of a rare-earth and a transition metal impurity with infinitely large Coulomb repulsion on the impurity orbital. To derive the model under consideration, one can start with the Hamiltonian involving the Hamiltonians of the BCS and Anderson models, H = ∑