Superconductivity out of a non-Fermi liquid: Free energy analysis
In this paper, we present an in-depth analysis of the condensation energy E c 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(Ω m ) = g¯ γ /|Ω m | 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, 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 E c at E c,n , of which E c,0 is the global minimum. Here, the spectrum E c,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 E c,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.