Superconductivity out of a non-Fermi liquid. Free energy analysis.

We present an in-depth analysis of the condensation energy for a superconductor in a situation when superconductivity emerges out of a non-Fermi liquid due to pairing mediated by a massless boson. This is the case for electronic-mediated pairing near a quantum-critical point in a metal, for pairing in SYK-type models, and for phonon-mediated pairing in the properly defined limit, when the dressed Debye frequency vanishes. We consider a subset of these quantum-critical models, in which the pairing in a channel with a proper spatial symmetry is described by an effective 0+1 dimensional model with the effective dynamical interaction V(Ω_m)=1/[Ω_m]^γ, where γ is model-specific (the γ-model). We have argued that the pairing in the γ-model is qualitatively different from that in a Fermi liquid, and the gap equation at T=0 has an infinite number of topologically distinct solutions, Δ_n(ω_m), where an integer n, running between 0 and infinity, is the number of zeros of Δ_n(ω_m) on the positive Matsubara axis. This gives rise to the set of extrema of the condensation energy at each solution, of which the sign-preserving solution is the global minimum. The condensation spectrum is discrete for a generic γ<2, but becomes continuous at γ= 2-0. We discuss the profile of the condensation energy near each saddle point and the transformation from a discrete to a continuous spectrum as γ approaches 2.