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Workshop on Integrable Theories, Solitons and Duality

Exact results for BCS systems

Xi‐Wen Guan, Angela Foerster, Jon Links, Huan-Qiang Zhou

Published Dec 1, 2002• 3 citations
Publisher

Abstract

In this paper, we discuss the construction of integrable BCS models in the framework of the quantum inverse scattering method.Specifically, the reduced BCS model, which is based on the su(2) algebra, a model of two coupled BCS systems, based on the su(4) algebra and an extended pairing model associated to the u(2|2) symmetry are presented.* Speaker.

DOI
10.22323/1.008.0016

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In this paper, we discuss the construction of integrable BCS models in the framework of the quantum inverse scattering method.Specifically, the reduced BCS model, which is based on the su(2) algebra, a model of two coupled BCS systems, based on the su(4) algebra and an extended pairing model associated to the u(2|2) symmetry are presented.* Speaker.
Superconducting pairing of electrons in nanoscale metallic particles with discrete energy levels and a fixed number of electrons is described by the reduced Bardeen, Cooper, and Schrieffer model Hamiltonian. We show that this model is integrable by the algebraic Bethe ansatz. The eigenstates, spectrum, conserved operators, integrals of motion, and norms of wave functions are obtained. Furthermore, the quantum inverse problem is solved, meaning that form factors and correlation functions can be explicitly evaluated. Closed form expressions are given for the form factors and correlation functions that describe superconducting pairing.
Introduction to the quantum inverse scattering method The quantum inverse scattering method is the modern algebraic theory of exactly solvable quantum systems. It arose [404, 410, 411] as an attempt to carry over the concepts of the inverse scattering method for classical non-linear evolution equations [2, 134] into quantum mechanics. As a result, our understanding of both the theory of integrable partial differential equations and the theory of exactly solvable quantum systems changed, and the algebraic roots of the exact solvability became apparent. These roots originate from the Yang-Baxter equation and its classical counterpart. Before turning to our actual subject, which is the application of the quantum inverse scattering method to the Hubbard model, we give a brief general introduction. We shall limit our exposition basically to the material which is needed later for the understanding of the algebraic structure of the Hubbard model. The reader who is interested in the general scope of the method and in the history of its development is referred to the excellent books and review articles [131, 270, 276, 277, 407]. Integrability As a motivation for the definition of the Yang-Baxter algebra in the following subsection we shall first recall the concept of integrability in classical mechanics. Then, by considering the elementary example of the harmonic oscillator, we shall see that this concept does not directly apply to quantum mechanical systems and needs to be extended.
We demonstrate the occurrence of free quasi-particle excitations obeying generalized exclusion statistics in a BCS model with asymmetric pair scattering. The results are derived from an exact solution of the Hamiltonian, which was obtained via the algebraic Bethe ansatz utilizing the representation theory of an underlying Yangian algebra. The free quasi-particle excitations are associated with highest weight states of the Yangian algebra, corresponding to a class of analytic solutions of the Bethe ansatz equations.