In this paper we continue with our analysis of the interplay between the pairing and the non-Fermi liquid behavior in a metal for a set of quantum-critical (QC) systems with an effective dynamical electron-electron interaction V(ωm)∝1/|ωm|γ, mediated by a critical massless boson (the γ-model). In previous papers we considered the cases 0<γ<1 and γ≈1. We argued that the pairing by a gapless boson is fundamentally different from BCS/Eliashberg pairing by a massive boson as for the former there exists not one but an infinite discrete set of topologically distinct solutions for the gap function Δn(ωm) at T=0 (n=0,1,2,...), each with its own condensation energy Ec,n. Here we extend the analysis to larger 1<γ<2. We argue that the discrete set of solutions survives, and the spectrum of Ec,n get progressively denser as γ increases towards 2 and eventually becomes continuous at γ→2. This increases the strength of "longitudinal"gap fluctuations, which tend to reduce the actual superconducting Tc compared to the onset temperature for the pairing and give rise to a pseudogap region of preformed pairs. We also detect two features on the real axis, which develop at γ>1 and also become critical at γ→2. First, the density of states evolves towards a set of discrete δ-functions. Second, an array of dynamical vortices emerges in the upper frequency half plane, near the real axis. We argue that these two features come about because on a real axis, the real part of the dynamical electron-electron interaction, V′(ω)∝cos(πγ/2)/|ω|γ, becomes repulsive for γ>1, and the imaginary V′′(ω)∝sin(πγ/2)/|ω|γ, gets progressively smaller at γ→2. We speculate that the features on the real axis are consistent with the development of a continuum spectrum of the condensation energy, for which we used Δn(ωm) on the Matsubara axis. We consider the case γ=2 separately in the next paper.
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We thank I. Aleiner, B. Altshuler, E. Berg, R. Combescot, 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. Pepin, 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 Y. Wang, for useful discussions. The work by Y.-M.W. and A.V.C. was supported by the NSF DMR-1834856. Y.-M.W., S.-S.Z., and A.V.C. acknowledge the hospitality of KITP at UCSB, where part of the work has been conducted. The research at KITP is supported by the National Science Foundation under Grant No. NSF PHY-1748958. A.V.C. also acknowledges the hospitality of Stanford University, where some results of this work have been obtained. His stay at Stanford has been supported through the Gordon and Betty Moore Foundation's EPiQS Initiative, Grants GBMF4302 and GBMF8686.
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