  We analyze the phase diagram of the Yukawa-Sachdev-Ye-Kitaev model, which
describes complex fermions randomly interacting with real bosons via a Yukawa
coupling, at finite temperatures and varying fermion density. In a recent work
[Phys. Rev. Research 2, 033084 (2020)] it has been shown that, upon varying
filling or chemical potential, there exists a first-order quantum phase
transition 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 parameters the first-order
quantum phase transition extend to finite-temperatures and terminate 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 the model realization of the transition of nFL's
towards superconductivity and insulating states, therefore offer a controlled
platform for future investigations of the generic phase diagram that hosts nFL,
insulator and superconductor and their phase transitions.
