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International Journal of Modern Physics E

EXACTLY SOLVABLE PAIRING HAMILTONIANS

J. Dukelsky, B. Errea, Sergio Lerma-Hernández, Stuart Pittel

Published Feb 1, 2007• 2 citations
Publisher

Abstract

We review the development of the microscopic BCS theory of superconductivity and the exact solution of the pairing Hamiltonian given by Richardson in 1963. We then introduce the generalized Richardson-Gaudin exactly-solvable pairing models for SU(2) (pairing between like particles), SO(5) ( T =1 pairing) and SO(8) ( T =0,1 pairing) algebras.

DOI
10.1142/s0218301307005545

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We review the development of the microscopic BCS theory of superconductivity and the exact solution of the pairing Hamiltonian given by Richardson in 1963. We then introduce the generalized Richardson-Gaudin exactly-solvable pairing models for SU(2) (pairing between like particles), SO(5) ( T =1 pairing) and SO(8) ( T =0,1 pairing) algebras.
The exact solution of the BCS pairing Hamiltonian was found by Richardson in 1963. While little attention was paid to this exactly solvable model in the remainder of the 20th century, there was a burst of work at the beginning of this century focusing on its applications in different areas of quantum physics. We review the history of this exact solution and discuss recent developments related to the Richardson-Gaudin class of integrable models, focussing on the role of these various models in nuclear physics.
The use of exactly solvable Richardson-Gaudin models to describe the physics of systems with strong pair correlations is reviewed. The article begins with a brief discussion of Richardson's early work, which demonstrated the exact solvability of the pure pairing model, and then shows how that work has evolved recently into a much richer class of exactly solvable models. The Richardson solution leads naturally to an exact analogy between these quantum models and classical electrostatic problems in two dimensions. This analogy is then used to demonstrate formally how BCS theory emerges as the large-NNN limit of the pure pairing Hamiltonian. Several applications to problems of relevance to condensed-matter physics, nuclear physics, and the physics of confined systems are considered. Some of the interesting effects that are discussed in the context of these exactly solvable models include (i) the crossover from superconductivity to a fluctuation-dominated regime in small metallic grains; (ii) the role of the nucleon Pauli principle in suppressing the effects of high-spin bosons in interacting boson models of nuclei, and (iii) the possibility of fragmentation in confined boson systems. Interesting insight is also provided into the origin of the superconducting phase transition both in two-dimensional electronic systems and in atomic nuclei, based on the electrostatic image of the corresponding exactly solvable quantum pairing models.