Integrability of the pairing hamiltonian

Abstract

We show that a many-body Hamiltonian that corresponds to a system of fermions interacting through a pairing force is an integrable problem, i.e. it has as many constants of the motion as degrees of freedom. At the classical level this implies that the Time-dependent Hartree-Fock- Bogoliubov dynamics is integrable and at the quantum level that there are conserved operators of two-body character which reduce to the number operators when the pairing strength vanishes. We display these operators explicitly and study in detail the three-level example.Comment: 14 pages, latex, 2 figures, to be published in Nuclear Physics

Spectroscopy of discrete energy levels in ultrasmall metallic grains

“…Finally, let us mention that Cambiaggio, Rivas and Saraceno have recently shown that the discrete BCS model is integrable and have constructed explicit expressions for all its constants of the motion [117]. The latter's relation to Richardson's solution was clarified by Sierra [118], who has also explored possible connections between the exact solution and conformal field theory.…”

“…Richardson published his solution in the context of nuclear physics in a series of papers between 1963 and 1977 [46][47][48][49][50][51][52][53][54] which seem to have completely escaped the attention of the condensed matter community. Very recently, the model was also shown to be integrable [117,118]. The revival of this remarkably simple exact solution after such a long and undeserved period of neglect is perhaps one of the most important consequences of RBT's experimental breakthrough: Richardson's solution allows the elucidation and illustration by exact means of many important conceptual ingredients of the standard BCS theory of superconductivity, such as the nature of pairing correlations, the importance of phase coherence, the validity of using a mean-field approximation and a grand-canonical formulation for bulk systems, and the limitations of the latter approaches for ultrasmall systems.…”

“…It would be very interesting if progress in this direction could be made by exploiting the integrability [117,118] of the model, using Bethe Ansatz techniques. For the FD regime, another possibility would be to develop a finite-T generalization of the self-consistent RPA approach of Dukelsky and Schuck [41].…”

Eigenvalue-based method and form-factor determinant representations for integrable XXZ Richardson-Gaudin models

“…The reduced BCS Hamiltonian can be found from the rational model [3], the p x + ip y pairing Hamiltonian can be derived from the hyperbolic model [4,5], and the central spin model Hamiltonian is identical to one of the constants of motion of the rational model [6]. By introducing a bosonic degree of freedom, the inhomogeneous Dicke model can be found as a limiting case of the trigonometric model [9].…”

“…One class of these systems is the class of Richardson-Gaudin (RG) integrable systems, which can be derived from a generalized Gaudin algebra [1,2]. The pairing model in the reduced BCS approximation, used to describe superconductivity, has been shown to be RG integrable [3], as has the p x + ip y pairing Hamiltonian [4,5], the central spin model [6], factorizable pairing models in heavy nuclei [7], an extended d + id pairing Hamiltonian [8], and several atom-molecule Hamiltonians such as the inhomogeneous Dicke model [9,10]. For these models, diagonalizing the Hamiltonian in an exponentially scaling Hilbert space can be reduced to solving a set of nonlinear equations scaling linearly with system size.…”