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

In this paper, we present an in-depth analysis of the condensation energy Ec 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 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(ωm)=g¯γ/|ωm|γ, where γ is model-specific (the γ model). In previous papers, we 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, Δn(ωm), where an integer n, running between 0 and infinity, is the number of zeros of Δn(ωm) on the positive Matsubara axis. This gives rise to the set of extrema of Ec at Ec,n, of which Ec,0 is the global minimum. The spectrum Ec,n is discrete for a generic γ<2 but becomes continuous at γ=2-0. Here, we discuss in more detail the profile of the condensation energy near each Ec,n and the transformation from a discrete to a continuous spectrum at γ→2. We also discuss the free energy and the specific heat of the γ model in the normal state.