Numerical studies of the s-wave pseudogap state and related Tc:: the ''pairing approximation'' theory

We investigate the pseudogap onset temperature T*, the superconducting transition temperature T-c and the general nature of the pseudogap phase using a diagrammatic BCS/Bose-Einstein cross-over theory. This scheme is based on the "pairing approximation" of Kadanoff and Martin, further extended by Patron (KMP). Our consideration of the KMP "pairing approximation'' is driven by the objective to obtain BCS like behavior in the weak coupling limit. Two coupled equations, corresponding to those for the single particle and pair propagators, must be solved numerically, along with the number equation constraint. The variation of small to large coupling constant g is explored, whereby the system is found to cross-over from BCS to Bose-Einstein behavior. Our numerical calculations proceed in two stages: first, we investigate the "lowest order theory", which is appropriate at temperatures well above T-c. We use this theory to determine where the Fermi Liquid state first breaks down. This breakdown, which occurs at T* and is associated with intermediate values of the coupling, corresponds to a splitting of the single peaked (Fermi liquid) electronic spectral function into two peaks well separated by a gap, as might be expected for the pseudogap phase. Indeed, our calculations provide physical insight into the pseudogap state which is characterized by the presence of metastable pairs or "resonances", which occupy states around the Fermi energy; in this way, they effectively reduce the single partile density of states. The superconducting instability T-c is evaluated in the second stage of our calculations. Here, we introduce ''mode coupling'' effects, in which the long lived pairs are affected by the single particle pseudogap states and vice versa. Our T-c equations, which turn out to be rather simple as a result of the KMP scheme, reveal a rich structure as a function of g in which the pseudogap is found to compete with superconductivity. Our results are compared with alternate theories in the literature. (C) 1999 Elsevier Science B.V. All rights reserved.