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

Abstract

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 Bole-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 gaps of the model 2Δ 0 /k B T g ∼4 is almost independent on the pairing strength ‖U‖/t, whereas 2Δ 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.