Phase diagram of the spin- 12 Yukawa–Sachdev-Ye-Kitaev model: Non-Fermi liquid, insulator, and superconductor

We analyze the phase diagram of the spin-$\frac{1}{2}$ Yukawa--Sachdev-Ye-Kitaev model, which describes complex spin-$\frac{1}{2}$ fermions randomly interacting with real bosons via a Yukawa coupling, at finite temperatures and varying fermion density. A recent work [Wang and Chubukov, Phys. Rev. Res. 2, 033084 (2020).] showed that, upon varying the filling or chemical potential, a first-order quantum phase transition exists between a non-Fermi-liquid (NFL) phase and an insulating phase. Here we show that in such a model with time-reversal symmetry this quantum phase transition is preempted by a pairing phase that develops as a low-temperature instability. As a remnant of the would-be NFL-insulator transition, the superconducting critical temperature rapidly decreases beyond a certain chemical potential. On the other hand, depending on the parameters, the first-order quantum phase transition extends to finite temperatures and terminates at a thermal critical point, beyond which the NFL and the insulator become the same phase, similar to that of the liquid-gas and metal-insulator transition in real materials. We determine the pairing phase boundary and the location of the thermal critical point via combined analytic and quantum Monte Carlo numeric efforts. Our results provide a model realization of the transition of NFLs towards superconductivity and insulating states and therefore offer a controlled platform for future investigations of the generic phase diagram that hosts a NFL, insulator, and superconductor and their phase transitions.