Interplay between superconductivity and non-Fermi liquid at a quantum critical point in a metal. I. The. 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 (Omega(m)) alpha 1 vertical bar Omega(m)vertical bar. (the gamma model). In this paper, the first in the series, we consider the case T = 0 and 0 <. < 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 Delta(n)(omega(m)) (n = 0, 1, 2,..., infinity). All solutions have the same spatial pairing symmetry, but differ in the time domain: Delta(n) (omega(m)) changes sign n times as a function of Matsubara frequency omega(m). The n = 0 solution Delta(0) (Delta(m)) is sign preserving and tends to a finite value at omega(m) = 0, like in BCS theory. The n=infinity solution corresponds to an infinitesimally small Delta(omega(m)), which oscillates down to the lowest frequencies as Delta(omega(m)) proportional to vertical bar omega(m)vertical bar/(gamma 2) cos[2 beta log(vertical bar omega(m)vertical bar/omega(0))], where beta = O(1) and.0 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 < gamma < 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 N-cr > 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 N-cr.