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Interplay between superconductivity and non-Fermi liquid at a quantum critical point in a metal. VI. Theγmodel and its phase diagram at2<γ<3
Physical Review B  (IF4.036),  Pub Date : 2021-10-25, DOI: 10.1103/physrevb.104.144509
Shang-Shun Zhang, Yi-Ming Wu, Artem Abanov, Andrey V. Chubukov

In this paper, the sixth in series, we continue our analysis of the interplay between non-Fermi liquid and pairing in the effective low-energy model of fermions with singular dynamical interaction $V\left({\mathrm{\Omega }}_{m}\right)={\overline{g}}^{\gamma }/{|{\mathrm{\Omega }}_{m}|}^{\gamma }$ (the $\gamma$ model). The model describes low-energy physics of various quantum-critical metallic systems at the verge of an instability towards density or spin order, pairing of fermions at the half-filled Landau level, color superconductivity, and pairing in SYK-type models. In previous papers I–V, we analyzed the $\gamma$ model for $\gamma \le 2$ and argued that the ground state is an ordinary superconductor, but there is an infinite number of local minima of the condensation energy. We further argued that the condensation energy spectrum becomes continuous and gapless, and superconducting ${T}_{c}$ vanishes due to critical longitudinal gap fluctuations. In this paper, we consider larger $2<\gamma <3$. We show that the system moves away from criticality in that the condensation energy spectrum again becomes discrete and gapless, and ${T}_{c}$ becomes finite. Yet, we show that the gap functions for $\gamma >2$ and $\gamma <2$ are topologically different as they live on different sheets of the Riemann surface. This makes $\gamma =2$ a topological quantum-critical point. We further show that the fermionic excitation spectrum for $\gamma >2$ acquires a new feature—a bound state at the edge of a continuum, with a macroscopic degeneracy, which is a fraction of the total number of states in the system. We obtain the phase diagram on the $\left({\omega }_{D},\gamma \right)$ plane, where ${\omega }_{D}$ is a mass of a pairing boson, and on the $\left(T,\gamma \right)$ plane. The latter consists of two distinct superconducting phases at $\gamma <2$ and $\gamma >2$ and the intermediate pseudogap state of preformed pairs in between.