Interplay between superconductivity and non-Fermi liquid behavior at a quantum-critical point in a metal. V. The γ model and its phase diagram. The case γ=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(ωm)=(g¯/|ωm|)γ (the γ 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 γ <2. We have shown that the onset temperature for the pairing Tp is finite, of order 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 Ec has an infinite number of minima. The spectrum of Ec is discrete, but becomes more dense as γ increases. Here we consider the case γ=2. The γ=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 ωD vanishes. We show that for γ=2, the spectrum of Ec becomes continuous. We argue that the associated gapless "longitudinal"fluctuations destroy superconducting phase coherence at a finite T, such that at 0<T<Tp, the system displays pseudogap behavior of preformed pairs. 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 Tp=0.1827g¯, obtained in earlier studies, marks the onset of the pseudogap behavior, while the actual superconducting Tc vanishes at ωD→0.