Breakdown of the Mean Field Theory in the Superconducting Transition Region

We consider a dirty superconducting material above its transition temperature. Using a perturbation expansion, we study the effect of the fluctuations of the order parameter on the electron Green's function G as well as on the fluctuation Green's function D. Our main results are: (a) G and D can be determined separately. (b) The behavior of D can be analyzed using the Ginzburg-Landau functional where the only nonlinear term present is the lowest-order one. (c) As a result of this analysis, the corrections to the mean field theory are generally small as far as D is concerned. They become important only at the onset of the critical region, where our perturbation approach breaks down. We derive a criterion for the onset of this critical region and compare our estimate to previous ones. (d) On the other hand, G is well described by taking only the lowest-order correction to the electron self-energy due to the fluctuations, even inside the critical region. (e) We propose a scheme for the analysis of the critical region. (f) We establish in this way the complete equivalence between the critical behavior of the interacting Bose system and that of a superconductor.