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Comparative Study of BCS-BEC Crossover Theories above $T_c$: the Nature of the Pseudogap in Ultra-Cold Atomic Fermi Gases

This paper presents a comparison of two finite-temperature BCS-Bose Einstein condensation (BEC) crossover theories above the transition temperature: Nozieres Schmitt-Rink (NSR) theory and finite $T$-extended BCS-Leggett theory. The comparison is cast in the form of numerical studies of the behavior of the fermionic spectral function both theoretically and as constrained by (primarily) radio frequency (RF) experiments. Both theories include pair fluctuations and exhibit pseudogap effects, although the nature of this pseudogap is very different. The pseudogap in finite $T$-extended BCS-Leggett theory is found to follow a BCS-like dispersion which, in turn, is associated with a broadened BCS-like self energy, rather more similar to what is observed in high temperature superconductors (albeit, for a d-wave case). The fermionic quasi-particle dispersion is different in NSR theory and the damping is considerably larger. We argue that the two theories are appropriate in different temperature regimes with the BCS-Leggett approach more suitable nearer to condensation. There should, in effect, be little difference at higher $T$ as the pseudogap becomes weaker and where the simplifying approximations used in the BCS-Leggett approach break down. On the basis of momentum-integrated radio frequency studies of unpolarized gases, it would be difficult to distinguish which theory is the better. A full comparison for polarized gases is not possible since there is claimed to be inconsistencies in the NSR approach (not found in the BCS-Leggett scheme). Future experiments along the lines of momentum resolved experiments look to be very promising in distinguishing the two theories.