Interplay between superconductivity and non-Fermi liquid at a quantum critical point in a metal. I. The γ model and its phase diagram at T=0. The case 0<γ<1

Near a quantum critical point in a metal, a strong fermion-fermion interaction, mediated by a soft boson, acts in two different directions: it destroys fermionic coherence and it gives rise to an attraction in one or more pairing channels. The two tendencies compete with each other. We analyze a class of quantum critical models, in which momentum integration and the selection of a particular pairing symmetry can be done explicitly, and the competition between non-Fermi liquid and pairing can be analyzed within an effective model with dynamical electron-electron interaction V(ωm) 1/|ωm|γ (the γ model). In this paper, the first in the series, we consider the case T=0 and 0<γ<1. We argue that tendency to pairing is stronger, and the ground state is a superconductor. We argue, however, that a superconducting state is highly nontrivial as there exists a discrete set of topologically distinct solutions for the pairing gap Δn(ωm) (n=0,1,2,...,∞). All solutions have the same spatial pairing symmetry, but differ in the time domain: Δn(ωm) changes sign n times as a function of Matsubara frequency ωm. The n=0 solution Δ0(ωm) is sign preserving and tends to a finite value at ωm=0, like in BCS theory. The n=∞ solution corresponds to an infinitesimally small Δ(ωm), which oscillates down to the lowest frequencies as Δ(ωm) |ωm|γ/2cos[2βlog(|ωm|/ω0)], where β=O(1) and ω0 is of order of fermion-boson coupling. As a proof, we obtain the exact solution of the linearized gap equation at T=0 on the entire frequency axis for all 0<γ<1, and an approximate solution of the nonlinear gap equation. We argue that the presence of an infinite set of solutions opens up a new channel of gap fluctuations. We extend the analysis to the case where the pairing component of the interaction has additional factor 1/N and show that there exists a critical Ncr>1, above which superconductivity disappears, and the ground state becomes a non-Fermi liquid. We show that all solutions develop simultaneously once N gets smaller than Ncr.