Interplay between superconductivity and non-Fermi liquid at a quantum-critical point in a metal: V. The $gamma$ model and its phase diagram. The case $gamma =2$

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 $V(Omega_m) propto 1/|Omega_m|^gamma$ (the $gamma$-model). In previous papers we studied the cases $01$ which become critical at $gamma o 2$. 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. These two features come about because on a real axis, the real part of the interaction, $V' (Omega) propto cos(pi gamma/2)/|Omega|^gamma$, becomes repulsive for $gamma >1$, and the imaginary $V^{''} (Omega) propto sin(pi gamma/2)/|Omega|^gamma$, gets progressively smaller at $gamma o 2$. The features on the real axis are consistent with the development of a continuum spectrum of $E_{c,n}$ obtained using $Delta_n (omega_m)$ on the Matsubara axis. We consider the case $gamma =2$ separately in the next paper.

Interplay between superconductivity and non-Fermi liquid at a quantum-critical point in a metal: V. The $gamma$ model and its phase diagram. The case $gamma =2$ Institutional Repository Document