Superconducting phase coherence and pairing gap in the three-dimensional attractive Hubbard model

We consider a lattice version of the "negative-U" Hubbard model in three dimensions, to study the crossover from a superconducting phase transition (dominated by phase fluctuation) to a regime where the amplitude of the superconducting order parameter controls the critical temperature. Starting from a fermionic Hamiltonian, we give a microscopic derivation of the effective action in terms of the relevant physical fields: the modulus and phase of the superconducting order parameter. Furthermore, by employing to the resulting coarse-grained quantum XY model the phase fluctuation algebra between number and phase operators (given by the Euclidean group E-2), we map the "phase-only" action onto a solvable quantum spherical model. We establish the self-consistent theory (involving both fermionic and bosonic degrees of freedom), and calculate the superconducting phase coherence transition temperature T-c as a function of the coupling strength \U\, exhibiting a maximum where the behavior crosses over from BCS to Bose-Einstein (BE) condensation. We examine a pseudogap in the normal state which emerges naturally as a precursor of superconductivity at the second characteristic temperature T-g due to a state with bound pairs but without long-range phase coherence. Furthermore, we demonstrate that the reduced gap, of the model 2Delta(0)/k(B)T(g)similar to4 is almost independent on the pairing strength \U\/t, whereas 2Delta(0)/k(B)T(c) increases with increasing \U\-in striking analogy with the phenomenology of the high-temperature cuprate superconductors when interpreted in terms of the BCS-BC crossover scenario.