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 E2), 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 Tc 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 Tg due to a state with bound pairs but without long–range phase coherence. Furthermore, we demonstrate that the reduced gaps of the model 2Δ0/kBTg∼4 is almost independent on the pairing strength |U|/t, whereas 2Δ0/kBTc 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.

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