Upper bound on $T_c$ in a strongly coupled electron-boson superconductor

Migdal-Eliashberg theory of boson-mediated superconductivity contains a $\sqrt{\lambda}$ divergence in the critical temperature $T_c$ for strong electron-boson coupling $\lambda$. In the conventional Migdal-Eliashberg theory, the strong-coupling regime can be accessed only in the limit that $\lambda_E = \lambda \, \omega_D/\varepsilon_F\ll1$, where $\omega_D$ is the Debye frequency and $\varepsilon_F$ is the Fermi energy. Here we go beyond this restriction in the context of the two-dimensional Yukawa-SYK (Y-SYK) model, which is solvable for arbitrary values of $\lambda_E$. We find that $T_c\approx 0.18 \,\omega_D \sqrt{\lambda}$ for large $\lambda$, provided $\lambda_E$ remains small, and crosses over to a universal value of $T_c \approx 0.04\, \varepsilon_F$ for large $\lambda_E$. The saturation of $T_c$ is due to a self-consistent account of the boson dynamics for large $\lambda_E$ and remains valid provided the vertex corrections are negligible. Depending on the value of $\lambda$, this self-consistent approach leads to pairing that describes multiple classes of quantum critical electronic systems. These results demonstrate how the $\sqrt{\lambda}$ growth of $T_c$ in Migdal-Eliashberg theory saturates to a universal value independent of $\lambda$ and $\omega_D$, providing an upper bound on the critical temperature at strong electron-boson coupling.