On the Crossover from BCS Superconductivity to Bose Condensation

We outline a microscopic approach to the superconducting fluctuations and pairing correlations in the attractive Hubbard model above T$\text{}_{c}$, using the functional integral method. A crossover from BCS superconductivity to Bose condensation of preformed pairs is studied by constructing the appropriate Ginzburg-Landau functionals. The differences between the lattice and the continuum models are discussed. The case of quasi-two-dimensional superconductors as well as the model with non-local pairing interaction are also examined. The effects of Gaussian fluctuations of the order parameter are analyzed in the T-matrix approach, the self-consistent Hartree approach and finally within the Ginzburg-Landau theory. Similarities with a paramagnon theory of itinerant-electron magnetism are pointed out.