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 ${\lambda }_E=\lambda {\omega }_D/{\varepsilon }_F\ll 1$, 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.

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• Quantum criticality
• High-temperature superconductors
• Strongly correlated systems
• Eliashberg theory
• Sachdev-Ye-Kitaev model