Heuristic bounds on superconductivity and how to exceed them

What limits the value of the superconducting transition temperature ($T_c$) is a question of great fundamental and practical importance. Various heuristic upper bounds on $T_c$ have been proposed, expressed as fractions of the Fermi temperature, $T_F$, the zero-temperature superfluid stiffness, ${\rho }_s(0)$, or a characteristic Debye frequency, ${\omega }_0$. We show that while these bounds are physically motivated and are certainly useful in many relevant situations, none of them serve as a fundamental bound on $T_c$. To demonstrate this, we provide explicit models where $T_c\mathrm{/}{T}_F$ (with an appropriately defined $T_F$), $T_c\mathrm{/}{\rho }_s(0)$, and $T_c\mathrm{/}{\omega }_0$ are unbounded.