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International Journal of Modern Physics E

THE EXACTLY SOLVABLE RICHARDSON MODEL IN THE BCS-to-BEC CROSSOVER

J. Dukelsky, Gerardo Ortíz

Published Feb 20, 2006• 3 citations
Publisher

Abstract

The exact Richardson solution of the reduced BCS Hamiltonian is reviewed with special emphasis on the structure of the many-body wavefunction. A clear definition of the Cooper pairs in the superconducting or Fermi superfluid medium emerges. The structure of the Cooper pair wavefunction is general enough to describe large overlapping pair resonances in the weak coupling limit (BCS) as well as a Bose-Einstein condensate (BEC) of quasi-bound molecules in the strong coupling limit.

DOI
10.1142/s0218301306004168

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The exact Richardson solution of the reduced BCS Hamiltonian is reviewed with special emphasis on the structure of the many-body wavefunction. A clear definition of the Cooper pairs in the superconducting or Fermi superfluid medium emerges. The structure of the Cooper pair wavefunction is general enough to describe large overlapping pair resonances in the weak coupling limit (BCS) as well as a Bose-Einstein condensate (BEC) of quasi-bound molecules in the strong coupling limit.
The exact Richardson solution of the reduced BCS Hamiltonian is used to study the BCS-to-BEC crossover, as well as the nature of Cooper pairs, in superconducting and Fermi superfluid media. Based on the exact eigenstate we will discuss the Cooper-pair concept proposing a scenario for the BCS-to-BEC crossover in which a mixture of quasifree fermions and pair resonances (BCS) evolves to a system of weakly bound molecules (BEC). In this single unified scenario the Cooper-pair wavefunction has a unique functional form. We propose a new definition of the condensate fraction which, within the limits of the BCS model, gives a qualitative description of recent experiments in ultracold atomic Fermi gases. Finally, we will introduce a new integrable model for asymmetric superfluid systems able to describe different homogeneous and inhomogeneous competing phases such as, breached superconductivity, deformed Fermi superfluidity, and the elusive Larkin-Ovchinnikov-Flde-Ferrell (LOFF) state.
The BCS-to-BEC crossover, as well as the nature of Cooper pairs, in superconducting and Fermi superfluid media is studied from the exact ground state wave function of the reduced BCS Hamiltonian. As the strength of the interaction increases, the ground state continuously evolves from a mixed system of quasifree fermions and pair resonances (BCS), to pair resonances and quasibound molecules (pseudogap), and finally to a system of quasibound molecules (BEC). A single unified scenario arises where the Cooper-pair wave function has a unique functional form. Several exact analytic expressions such as the binding energy and condensate fraction are derived. We compare our results with recent experiments in ultracold atomic Fermi gases.
This chapter presents the crossover from the Bardeen-Cooper-Schrieffer (BCS) state of weakly-correlated pairs of fermions to the Bose-Einstein condensation (BEC) of diatomic molecules in the atomic Fermi gas. Our aim is to provide a pedagogical review of the BCS-BEC crossover, with an emphasis on the basic concepts, particularly those that are not generally known or are difficult to find in the literature. We shall not attempt to give an exhaustive survey of current research in the limited space here; where possible, we will direct the reader to more extensive reviews.
This article gives a detailed review of our current theoretical understanding of the crossover from cooperative Cooper pairing to independent bound-state formation and Bose–Einstein condensation in Fermi systems with increasing attractive interactions. Introduction There are two well-known paradigms for understanding the phenomena of superconductivity and superfluidity: (I) BCS theory [1], in which the normal state is a degenerate Fermi liquid that undergoes a pairing instability at a temperature T c ≪ ε f , the degeneracy scale. The formation of Cooper pairs and their condensation (macroscopic occupation of a single quantum state) both occur simultaneously at the transition temperature T c . (II) Bose–Einstein condensation (BEC) of bosons at a T c of the order of their degeneracy temperature. At a fundamental level these bosons (for example, 4 He or excitons) are invariably composite objects made up of an even number of fermions. The composite particles form at some very high temperature scale of the order of their dissociation temperature T dissoc , and these “pre-formed” bosons then condense at the BEC T c ≪ T dissoc . In most cases of experimental interest the system under consideration clearly falls into one category or the other. For instance, 3 He is a Fermi superfluid described by (I) whereas 4 He is a Bose superfluid (II). Essentially all cases of superconductivity in metals that we understand reasonably well are much closer to (I) than to (II), as evidenced by the existence of fermionic quasiparticles above T c .
The two-particle spectrum in a dense Fermion system is treated using a thermodynamic Green function approach. A self-consistent description of possible bound states and a superfluid condensate in a correlated medium is given. A detailed understanding of superfluidity and superconductivity in correlated Fermion systems, especially the transition from the Cooper-paired state (weak-coupling limit) to the Bose condensed state of tightly bound pairs of Fermions (strong coupling limit) [1], is of great interest for very different physical systems. The problem of a unified treatment of Bose–Einstein condensation (BEC) and the Bardeen–Cooper–Schrieffer (BCS) phase arises not only in describing the electron structure of strongly correlated electron superfluids such as superconductors [2], the electronhole system in semiconductors [3], spin-polarized hydrogen and liquid He [4], but also in the theory of nuclear matter [5] and quark–gluon systems [6]. Recently, there have been several new approaches to this stimulating problem. A Monte-Carlo simulation of a finite He system at zero temperature has been performed in Ref. [7]. The microscopic theory of strongly coupled quantum fluids has been treated within the Jastrow approximation (cf. Ref. [8]) to obtain the ground state and low-lying excited states of strongly correlated boson quantum fluids. The crossover from weak to strong coupling superconductivity has been considered using a functional integral representation [2] (see also Ref. [7]).