Strange metals and bad metals constitute exotic but ubiquitous metallic phases, endowed with anomalous thermodynamic and spectroscopic properties that do not comply with the conventional Landau Fermi-liquid (FL) paradigm [1,2]. Functional materials such as heavy fermions, pnictides, and high-temperature superconducting cuprates all host strange/bad metallic states of strongly interacting electrons; specifically, theses states occur in the crossover region between distinct stable phases, as a function of a non-thermal tuning parameter like chemical doping or pressure. Such tunability suggests the presence of quantum critical points (QCPs) or extended quantum critical phases, i.e., zero-temperature phase transitions in the phase diagram, which engender non-Fermi liquid (NFL) physics at finite temperature through strong fluctuations of an associated soft bosonic mode (e.g., charge/spin density, nematic, or magnetic fluctuations)[1].

Within this context, the two-dimensional (2D) spatially disordered Yukawa-Sachdev-Ye-Kitaev (YSYK) theory provides an exactly solvable platform to analyze non-Fermi liquid states and their associated phenomenology, microscopically rooted in quantum criticality [3-8]. Specifically, this model entails 2D dispersive fermions and bosons, coupled through spatially random contact interactions. It qualitatively reproduces several unconventional properties of strange metals: a linear-in-temperature (T-linear) DC resistivity [9-12], which crosses over into the bad-metal regime when the resistivity becomes larger than the 2D quantum unit h/e^2; universal ω/T scalings of the optical conductivity as a function of frequency ω [13], due to a marginal Fermi liquid (MFL) ground state; superconductivity below a critical temperature Tc, maximal when the system is tuned to quantum criticality; a finite superfluid phase stiffness and associated coherence peaks in the superconducting spectral functions, born out of incoherent, NFL normal-state spectra [14-16].

In this work we focus on the thermodynamic and spectral properties of the mean-field superconducting phase, in particular the critical temperature and the superfluid phase stiffness, across all strange-metal and bad-metal regimes of the model. We find a nonmonotonic evolution of the phase stiffness as a function of distance from the QCP, and we analyze the maximum critical temperature attainable in the strange-metal and bad-metal phases.

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