The interplay between superconductivity and non-Fermi liquid at a quantum-critical point in a metal

Near a quantum-critical point, a metal reveals two competing tendencies: destruction of fermionic coherence and attraction in one or more pairing channels. We analyze the competition within Eliashberg theory for a class of quantum-critical models with an effective dynamical electron-electron interaction $V(\Omega_m) \propto 1/|\Omega_m|^\gamma$ (the $\gamma$-model) for $0 < \gamma <1$. We argue that the two tendencies are comparable in strength, yet the one towards pairing is stronger, and the ground state is a superconductor. We show, however, that there exist two distinct regimes of system behavior below the onset temperature of the pairing $T_p$. In the range $T_{cr} < T < T_p$ fermions remain incoherent and the density of states $N(\omega)$ displays "gap filling" behavior in which the position of the maximum in $N(\omega)$ is set by temperature rather than the pairing gap. At lower $T < T_{cr}$, fermions acquire coherence, and $N(\omega)$ displays a conventional "gap closing" behavior. We argue that the existence of the two regimes comes about because of special behavior of fermions with frequencies $\omega = \pm \pi T$ along the Matsubara axis. Specifically, for these fermions, the component of the self-energy, which competes with the pairing, vanishes in the normal state. We further argue that the crossover at $T \sim T_{cr}$ comes about because Eliashberg equations allow an infinite number of topologically distinct solutions for the onset temperature of the pairing within the same gap symmetry. Only one solution, with the highest $T_p$, actually emerges, but other solutions are generated and modify the form of the gap function at $T \leq T_{cr}$. Finally, we argue that the actual $T_c$ is comparable to $T_{cr}$, while at $T_{cr} < T < T_{p}$ phase fluctuations destroy superconducting long-range order, and the system displays pseudogap behavior.