A bound on the superconducting transition temperature

Abstract It is notoriously difficult to make quantitative theoretical predictions of the superconducting transition temperature, 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 one 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 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 A_max is a dimensionless number of order one that we estimate to be A_max ≈ 1/10.