A bound on the superconducting transition temperature

It is notoriously difficult to make quantitative theoretical predictions of the superconducting transition temperature, Tc, 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 Tc depends exponentially on a subtle interplay between different interactions so that small uncertainties in microscopic processes can lead to order one effects on Tc. However, in some circumstances, it may be possible to determine (approximate) bounds on Tc. 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 EF is the Fermi energy. Specifically, drawing on both empirical results (shown in Fig. 2 below) and recent results1 of determinant quantum Monte Carlo (DQMC) studies of the paradigmatic Holstein model, we propose that \(k_BT_c \le A_{\max }\hbar \bar \omega\), where Amax is a dimensionless number of order one that we estimate to be Amax？≈？1/10