The spectial role of the first Matsubara frequency for superconductivity near a quantum critical point: the non linear gap equation below Tc and spectral properties in real frequency axis.

We consider a model of itinerant fermions near a quantum critical point, where the effective interaction is mediated by order parameter fluctuations. This interaction makes fermions ncoherent and, at the same time, gives rise to attraction in at least one pairing channel. These two tendencies compete with each other. We argue that the situation is more tricky because the self-energy without thermal fluctuation piece,
which one needs for the calculation of T_c, vanishes at the first two Matsubara frequencies(±πT), while the pairing interacion between fermions with these two frequencies remains strong. We investigate how this affects the system behavior below T_c, in particular the density of states and the spectral function. We argue that there is a crossover at some T_cross <T_c between conventional behavior at low T, when frequency variation occurs at a scale of the gap Delta, and a new behavior at higher T, when frequency variation occurs at a scale set by T.
We argue that our theory naturally explains the crossover from "gap closing" to "gap filling", observed in cuprate and other superconductors.