Evolution from BCS superconductivity to Bose-Einstein condensation: Current correlation function in the broken-syemmetry phase

We consider the current correlation function for a three-dimensional system of fermions embedded in a homogeneous background and mutually interacting via an attractive short-range potential, below the ~superconducting! critical temperature. Diagrammatic contributions in the broken-symmetry phase are identified, which yield for the ~wave-vector and frequency-dependent! current correlation function the fermionic BCS approximation in the weak-coupling limit and the bosonic Bogoliubov approximation in the strong-coupling limit ~whereby composite bosons form as bound-fermion pairs!. The temperature dependence of the superfluid density ~from the BCS exponential behavior at weak coupling to a power-law behavior at strong coupling! and the form of the Pippard-like kernel at zero temperature are explicitly obtained from weak to strong coupling. Quite generally, it is shown that the Pippard-like kernel is the sum of a local ~London-like! term and of a nonlocal component, the local term being dominant in the strong-coupling limit and the nonlocal component in the BCS ~weak-coupling! limit. It is also shown that the range of the nonlocal component is determined by the coherence length measuring the spatial correlations of the amplitude of the order parameter, namely, the correlations among different Cooper pairs ~or composite bosons!, rather than between the fermionic partners of a given pair. In addition, a prescription is provided for mapping the fermionic onto the bosonic diagrammatic theories in the broken-symmetry phase, thus complementing what has been already done in the normal phase.