Superconductivity near a quantum critical point in the extreme retardation regime

We study fermions at quantum criticality with extremely retarded interactions of the form $V(\omega_l)=(g/|\omega_l|)^\gamma$, where $\omega_l$ is the transferred Matsubara frequency. This system undergoes a normal-superconductor phase transition at a critical temperature $T=T_c$. The order parameter is the frequency-dependent gap function $\Delta(\omega_n)$ as in the Eliashberg theory. In general, the interaction is extremely retarded for $\gamma\gg 1$, except at low temperatures $\gamma>3$ is sufficient. We evaluate the normal state specific heat, $T_c$, the jump in the specific heat, $\Delta(\omega_n)$ near $T_c$, and the Landau free energy. Our answers are asymptotically exact in the limit $\gamma\to\infty$. 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 $T\to0$ and $\gamma>3$. We also uncover and investigate an instability of the $\gamma$ model: negative specific heat at $T\to0$ and just above $T_c$.