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.
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We thank I. Aleiner, B. Altshuler, E. Berg, D. Chowdhury, L. Classen, K. Efetov, R. Fernandes, A. Finkelstein, E. Fradkin, A. Georges, S. Hartnol, S. Karchu, S. Kivelson, I. Klebanov, A. Klein, R. Laughlin, S.-S. Lee, G. Lonzarich, D. Maslov, F. Marsiglio, M. Metlitski, W. Metzner, A. Millis, D. Mozyrsky, C. Pepan, V. Pokrovsky, N. Prokofiev, S. Raghu, S. Sachdev, T. Senthil, D. Scalapino, Y. Schattner, J. Schmalian, D. Son, G. Tarnopolsky, A-M Tremblay, A. Tsvelik, G. Torroba, E. Yuzbashyan, J. Zaanen, and particularly R. Combescot, Y. Wang, and Y. Wu for useful discussions. The work by A.V.C. was supported by the NSF-DMR Grant No. 1834856.
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