Effective action for superconductors and BCS-Bose crossover

A standard perturbative expansion around the mean-field solution is used to derive the low-energy effective action for superconductors at $T=0.$ Taking into account the density fluctuations at the outset we get the effective action where the density ρ is the conjugated momentum to the phase θ of the order parameter. In the hydrodynamic regime, the dynamics of the superconductor is described by a time-dependent nonlinear Schr\"odinger (TDNLS) equation for the field Ψ(x)=√ρ/2e^iθ. The evolution of the density fluctuations in the crossover from weak-coupling (BCS) to strong-coupling (Bose condensation of localized pairs) superconductivity is discussed for the attractive Hubbard model. In the bosonic limit, the TDNLS equation reduces to the Gross-Pitaevskii equation for the order parameter, as in the standard description of superfluidity. The conditions under which a phase-only action can be derived in the presence of a long-range interaction to describe the physics of the superconductivity of ``bad metals'' are discussed.