A bound on superconducting $T_c$'s

It is notoriously difficult to make quantitative theoretical predictions of the superconducting $T_c$, either from first-principles or even from a knowledge of normal state properties. Ultimately, this reflects the fact that the energy scales involved in the superconducting state are extremely small in natural units, and that $T_c$ depends exponentially on a subtle interplay between different interactions so that small uncertainties in microscopic processes can lead to order 1 effects on $T_c$. However, in some circumstances, it may be possible to determine (approximate) bounds on $T_c$. Here, we propose such a bound for the conventional phonon-mediated mechanism of pairing with strongly retarded interactions, i.e. in the case in which $\hbar\bar \omega \ll E_F$ where $\bar \omega$ is an appropriate characteristic phonon frequency and $E_F$ is the Fermi energy. Specifically, drawing on both empirical results (shown in Figure 2 below) and recent results[1] of determinant quantum Monte Carlo (DQMC) studies of the paradigmatic Holstein model, we propose that \begin{equation} k_B T_c \leq A_{max} \ \hbar \bar \omega \end{equation} where $A_{max}$ is a dimensionless number of order one that we estimate to be \begin{equation} A_{max} \approx 1/10. \end{equation}