Superconductivity near a quantum critical point in the extreme retardation regime

We study fermions at quantum criticality with extremely retarded interactions of the form V (omega l) = (g/k omega lk)gamma , where omega l is the transferred Matsubara frequency. This system undergoes a normal-superconductor phase transition at a critical temperature T = Tc. The order parameter is the frequency-dependent gap function (omega n) as in the Eliashberg theory. In general, the interaction is extremely retarded for gamma >> 1, except at low temperatures gamma > 3 is sufficient. We evaluate the normal state specific heat Tc, the jump in the specific heat (omega n) near Tc, and the Landau free energy. Our answers are asymptotically exact in the limit gamma - oo. At low temperatures, we prove that the global minimum of the free energy is nondegenerate and determine the order parameter, the free energy, and the specific heat. These answers are exact for T 0 and gamma > 3. We also uncover and investigate an instability of the gamma model: Negative specific heat at T 0 and just above Tc.