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What limits the value of the superconducting transition temperature ($T_c$)is a question of great fundamental and practical importance. Various heuristicupper bounds on $T_c$ have been proposed, expressed as fractions of the Fermitemperature, $T_F$, the zero-temperature superfluid stiffness, $\rho_s(0)$, ora characteristic Debye frequency, $\omega_0$. We show that while these boundsare physically motivated and are certainly useful in many relevant situations,none of them serve as a fundamental bound on $T_c$. To demonstrate this, weprovide explicit models where $T_c/T_F$ (with an appropriately defined $T_F$),$T_c/\rho_s(0)$, and $T_c/\omega_0$ are unbounded.

Heuristic bounds on superconductivity and how to exceed them