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

In this paper we continue 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 $V(Ω_m) \propto 1/|Ω_m|^γ$ (the $γ$-model). In previous papers we studied the cases $0<γ<1$ and $γ\approx 1$. 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 an infinite number of topologically distinct solutions for the gap function $Δ_n (ω_m)$ at $T=0$ ($n=0,1,2...$), each with its own condensation energy $E_{c,n}$. Here we extend the analysis to larger $1< γ<2$. We argue that the discrete set of solutions survives, and the spectrum of $E_{c,n}$ gets progressively denser as $γ$ approaches $2$ and eventually becomes continuous at $γ\to 2$. This increases the strength of "longitudinal" gap fluctuations, which tend to reduce the actual superconducting $T_c$ and give rise to a pseudogap region of preformed pairs. We also detect two features on the real axis for $γ>1$ which become critical at $γ\to 2$. 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. These two features come about because on a real axis, the real part of the interaction, $V' (Ω) \propto \cos(πγ/2)/|Ω|^γ$, becomes repulsive for $γ>1$, and the imaginary $V^{''} (Ω) \propto \sin(πγ/2)/|Ω|^γ$, gets progressively smaller at $γ\to 2$. The features on the real axis are consistent with the development of a continuum spectrum of $E_{c,n}$ obtained using $Δ_n (ω_m)$ on the Matsubara axis. We consider the case $γ=2$ separately in the next paper.

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