Upper bound on T c in a strongly coupled electron-boson superconductor

Abstract

Migdal-Eliashberg theory of boson-mediated superconductivity contains a $\sqrt{\ensuremath{\lambda}}$ divergence in the critical temperature ${T}_{c}$ for strong electron-boson coupling $\ensuremath{\lambda}$. In the conventional Migdal-Eliashberg theory, the strong-coupling regime can be accessed only in the limit that ${\ensuremath{\lambda}}_{E}=\ensuremath{\lambda}\phantom{\rule{0.16em}{0ex}}{\ensuremath{\omega}}_{D}/{\ensuremath{\varepsilon}}_{F}\ensuremath{\ll}1$, where ${\ensuremath{\omega}}_{D}$ is the Debye frequency and ${\ensuremath{\varepsilon}}_{F}$ 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 ${\ensuremath{\lambda}}_{E}$. We find that ${T}_{c}\ensuremath{\approx}0.18\phantom{\rule{0.16em}{0ex}}{\ensuremath{\omega}}_{D}\sqrt{\ensuremath{\lambda}}$ for large $\ensuremath{\lambda}$, provided ${\ensuremath{\lambda}}_{E}$ remains small, and crosses over to a universal value of ${T}_{c}\ensuremath{\approx}0.04\phantom{\rule{0.16em}{0ex}}{\ensuremath{\varepsilon}}_{F}$ for large ${\ensuremath{\lambda}}_{E}$. The saturation of ${T}_{c}$ is due to a self-consistent account of the boson dynamics for large ${\ensuremath{\lambda}}_{E}$ and remains valid provided the vertex corrections are negligible. Depending on the value of $\ensuremath{\lambda}$, this self-consistent approach leads to pairing that describes multiple classes of quantum critical electronic systems. These results demonstrate how the $\sqrt{\ensuremath{\lambda}}$ growth of ${T}_{c}$ in Migdal-Eliashberg theory saturates to a universal value independent of $\ensuremath{\lambda}$ and ${\ensuremath{\omega}}_{D}$, providing an upper bound on the critical temperature at strong electron-boson coupling.