Pair excitations, collective modes, and gauge invariance in the BCS-Bose-Einstein crossover scenario

In this paper we study the BCS Bose-Einstein condensation (BEC) crossover s cenario within the superconducting state, using a T-matrix approach which y ields the ground state proposed by Leggett. Here we extend this ground stat e analysis to finite temperatures T and interpret the resulting physics. We find two types of bosoniclike excitations of the system: long lived, incoh erent pair excitations and collective modes of the superconducting order pa rameter, which have different dynamics. Using a gauge invariant formalism, this paper addresses their contrasting behavior as a function of T and supe rconducting coupling, constant g. At a more physical level, our paper empha sizes how, at finite T, BCS-BEC approaches introduce an important parameter Delta(pg)(2) = Delta(2) - Delta(sc)(2) into the description of superconduc tivity. This parameter is governed by the pair excitations and is associate d with particle-hole asymmetry effects that are significant for sufficientl y large g. In the fermionic regime, Delta(pg)(2), represents the difference between the square of the excitation gap Delta(2) and that of the supercon ducting order parameter Delta(sc)(2). The parameter Delta(pg)(2), which is necessarily zero in the BCS (mean field) limit increases monotonically with the strength of the attractive interaction g. It follows that then is a si gnificant physical distinction between this BCS-BEC crossover approach (in which g is the essential variable which determines Delta(pg)) and the widel y discussed phase fluctuation scenario in which the plasma frequency is the tuning parameter. Finally, we emphasize that in the strong coupling limit, there are important differences between the composite bosons that arise in crossover theories and the usual bosons of the (interacting) Bose liquid. Because of constraints imposed on the fermionic excitation gap and chemical potential, in crossover theories, the fermionic degrees of freedom can nev er be fully removed from consideration.