Pair Excitations, Collective Modes and Gauge Invariance in the BCS -- Bose-Einstein Crossover Scenario

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Abstract
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 bosonic-like 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^2_{pg} = \Delta^2 - \Delta_{sc}^2 into the description of superconductivity. This parameter is governed by the pair excitations and is associated with particle-hole asymmetry effects which are important for sufficiently 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 superconducting 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 there is a significant physical distinction between this BCS-BEC crossover approach (in which g is the essential variable which determines \Delta_{pg}) and the widely discussed (Coulomb-modulated) phase fluctuation scenario in which the plasma frequency is the tuning parameter.

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