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 $V\left({\mathrm{\Omega }}_m\right)\propto 1/{\left|{\mathrm{\Omega }}_m\right|}^{\gamma }$, mediated by a critical massless boson (the $\gamma$-model). In previous papers we considered the cases $0<\gamma <1$ and $\gamma \approx 1$. 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 not one but an infinite discrete set of topologically distinct solutions for the gap function ${\mathrm{\Delta }}_n\left({\omega }_m\right)$ at $T=0$ ($n=0,1,2,...$), each with its own condensation energy $E_{c,n}$. Here we extend the analysis to larger $1<\gamma <2$. We argue that the discrete set of solutions survives, and the spectrum of $E_{c,n}$ get progressively denser as $\gamma$ increases towards 2 and eventually becomes continuous at $\gamma \to 2$. This increases the strength of “longitudinal” gap fluctuations, which tend to reduce the actual superconducting $T_c$ 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 $\gamma >1$ and also become critical at $\gamma \to 2$. First, the density of states evolves towards a set of discrete $\delta$-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, $V^{\prime }\left(\mathrm{\Omega }\right)\propto cos(\pi \gamma /2)/{\left|\mathrm{\Omega }\right|}^{\gamma }$, becomes repulsive for $\gamma >1$, and the imaginary $V^{\prime \prime }\left(\mathrm{\Omega }\right)\propto sin(\pi \gamma /2)/{\left|\mathrm{\Omega }\right|}^{\gamma }$, gets progressively smaller at $\gamma \to 2$. 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 ${\mathrm{\Delta }}_n\left({\omega }_m\right)$ on the Matsubara axis. We consider the case $\gamma =2$ separately in the next paper.

• Received 22 September 2020
• Revised 20 December 2020
• Accepted 4 January 2021

DOI:https://doi.org/10.1103/PhysRevB.103.024522