Superconductivity near a quantum critical point in the extreme retardation regime

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