Superconductivity of incoherent electrons in Yukawa-SYK model

We explore the interplay between non-Fermi liquid and superconductivity in a strongly coupled, quantum-critical environment within a model of N fermions in a quantum dot coupled to M bosons by a disorder-induced complex Yukawa coupling (Yukawa-SYK model). We analyze the phase diagram for an arbitrary interaction and ratio of N/M with special focus on two non-Fermi-liquid regimes: an SYK-like regime with a power-law frequency dependence of the fermionic self-energy and an impurity-like regime with frequency independent self-energy. We show that the crossover between the two can be reached by varying either the strength of the fermion-boson coupling or the ratio M/N. In both regimes, the system is unstable to superconductivity if the strength of time-reversal-symmetry-breaking disorder is below a certain threshold, and we determine the behavior of the corresponding onset temperatures. We argue that the superconducting state is highly unconventional with an infinite set of minima of the condensation energy at T=0 corresponding to topologically different gap functions. We discuss similarities and differences between this model and the $\gamma$-model, which describes dispersion-full fermions tuned to a metallic quantum-critical point with an effective singular dynamical interaction.