Interplay between superconductivity and non-Fermi liquid at a quantum critical point in a metal. IV. The γ model and its phase diagram at 1<γ<2

Abstract

In this paper we continue with our analysis of the interplay between the pairing and the non-Fermi liquid behavior in a metal for a set of quantum-critical (QC) systems with an effective dynamical electron-electron interaction V(Ωm)∝1/|Ωm|γ, mediated by a critical massless boson (the γ-model). In previous papers we considered the cases 0<γ<1 and γ≈1. We argued that the pairing by a gapless boson is fundamentally different from BCS/Eliashberg pairing by a massive boson as for the former there exists not one but an infinite discrete set of topologically distinct solutions for the gap function Δn(ωm) at T=0 (n=0,1,2,...), each with its own condensation energy Ec,n. Here we extend the analysis to larger 1<γ<2. We argue that the discrete set of solutions survives, and the spectrum of Ec,n get progressively denser as γ increases towards 2 and eventually becomes continuous at γ→2. This increases the strength of “longitudinal” gap fluctuations, which tend to reduce the actual superconducting Tc compared to the onset temperature for the pairing and give rise to a pseudogap region of preformed pairs. We also detect two features on the real axis, which develop at γ>1 and also become critical at γ→2. First, the density of states evolves towards a set of discrete δ-functions. Second, an array of dynamical vortices emerges in the upper frequency half plane, near the real axis. We argue that these two features come about because on a real axis, the real part of the dynamical electron-electron interaction, V′(Ω)∝cos(πγ/2)/|Ω|γ, becomes repulsive for γ>1, and the imaginary V′′(Ω)∝sin(πγ/2)/|Ω|γ, gets progressively smaller at γ→2. We speculate that the features on the real axis are consistent with the development of a continuum spectrum of the condensation energy, for which we used Δn(ωm) on the Matsubara axis. We consider the case γ=2 separately in the next paper.

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