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 scenario within the superconducting state, using a T-matrix approach which yields the ground state proposed by Leggett. Here we extend this ground state analysis to finite temperatures T and interpret the resulting physics. We find two types of bosoniclike excitations of the system: long lived, incoherent pair excitations and collective modes of the superconducting order parameter, which have different dynamics. Using a gauge invariant formalism, this paper addresses their contrasting behavior as a function of T and superconducting coupling constant g. At a more physical level, our paper emphasizes how, at finite T, BCS-BEC approaches introduce an important parameter ${\Delta }_{\mathrm{pg}}^2={\Delta }^2-{\Delta }_{\mathrm{sc}}^2$ into the description of superconductivity. This parameter is governed by the pair excitations and is associated with particle-hole asymmetry effects that are significant for sufficiently large g. In the fermionic regime, ${\Delta }_{\mathrm{pg}}^2$ represents the difference between the square of the excitation gap ${\Delta }^2$ and that of the superconducting order parameter ${\Delta }_{\mathrm{sc}}^2.$ The parameter ${\Delta }_{\mathrm{pg}}^2,$ which is necessarily zero in the BCS (mean field) limit increases monotonically with the strength of the attractive interaction g. It follows that there is a significant physical distinction between this BCS-BEC crossover approach (in which g is the essential variable which determines ${\Delta }_{\mathrm{pg}})$ and the widely 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 never be fully removed from consideration.