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 $gamma =2$

This paper is a continuation and a partial summary of our analysis of the pairing at a quantum-critical point (QCP) in a metal for a set of quantum-critical systems, whose low-energy physics is described by an effective model with dynamical electron-electron interaction $V(Omega_m) = ({bar g}/|Omega_m|)^gamma$ (the $gamma$-model). Examples include pairing at the onset of various spin and charge density-wave and nematic orders and pairing in SYK-type models. In previous papers, we analyzed the physics for $gamma <2$. We have shown that the onset temperature for the pairing $T_p$ is finite, of order ${bar g}$, yet the gap equation at $T=0$ has an infinite set of solutions within the same spatial symmetry. As the consequence, the condensation energy $E_c$ has an infinite number of minima. The spectrum of $E_c$ is discrete, but becomes more dense as $gamma$ increases. Here we consider the case $gamma =2$. The $gamma=2$ model attracted special interest in the past as it describes the pairing by an Einstein phonon in the limit when the dressed phonon mass $omega_D$ vanishes. We show that for $gamma =2$, the spectrum of $E_c$ becomes continuous. We argue that the associated gapless longitudinal fluctuations destroy superconducting phase coherence at a finite $T$, such that at $0<T< T_p$ the system displays pseudogap behavior. We show that for each gap function from the continuum spectrum, there is an infinite array of dynamical vortices in the upper half-plane of frequency. For the electron-phonon case, our results show that $T_p =0.1827 {bar g}$, obtained in earlier studies, marks the onset of the pseudogap behavior of preformed pairs, while the actual superconducting $T_c$vanishes at $omega_D to 0$.