Comparison of different pairing fluctuation approaches to BCS–BEC crossover

The subject of BCS–Bose–Einstein condensation (BEC) crossover is particularly exciting because of its realization in ultracold atomic Fermi gases and its possible relevance to high temperature superconductors. In this paper we review the body of theoretical work on this subject, which represents a natural extension of the seminal papers by Leggett and by Nozières and Schmitt-Rink (NSR). The former addressed only the ground state, now known as the “BCS-Leggett” wave-function, and the key contributions of the latter pertain to calculations of the superfluid transition temperature TcTc. These two papers have given rise to two main and, importantly, distinct, theoretical schools in the BCS–BEC crossover literature. The first of these extends the BCS-Leggett ground state to finite temperature and the second extends the NSR scheme away from TcTc both in the superfluid and normal phases. It is now rather widely accepted that these extensions of NSR produce a different ground state than that first introduced by Leggett. This observation provides a central motivation for the present paper which seeks to clarify the distinctions in the two approaches. Our analysis shows how the NSR-based approach views the bosonic contributions more completely but treats the fermions as “quasi-free”. By contrast, the BCS-Leggett based approach treats the fermionic contributions more completely but treats the bosons as “quasi-free”. In a related fashion, the NSR-based schemes approach the crossover between BCS and BEC by starting from the BEC limit and the BCS-Leggett based scheme approaches this crossover by starting from the BCS limit. Ultimately, one would like to combine these two schemes. There are, however, many difficult problems to surmount in any attempt to bridge the gap in the two theory classes. In this paper we review the strengths and weaknesses of both approaches. The flexibility of the BCS-Leggett based approach and its ease of handling make it widely used in T=0T=0 applications, although the NSR-based schemes tend to be widely used at T≠0. To reach a full understanding, it is important in the future to invest effort in investigating in more detail the T=0T=0 aspects of NSR-based theory and at the same time the T≠0T≠0 aspects of BCS-Leggett theory.