Efficient ab initio Migdal-Eliashberg calculation considering the retardation effect in phonon-mediated superconductors

We formulate an efficient scheme to perform Migdal-Eliashberg calculation considering the retardation effect from first principles. While the conventional approach requires a huge number of Matsubara frequencies, we show that the intermediate representation of the Greens function [H. Shinaoka et al., Phys. Rev. B 96, 035147 (2017)] dramatically reduces the numerical cost to solve the linearized gap equation. Without introducing any empirical parameter, we demonstrate that we can successfully reproduce the experimental superconducting transition temperature of elemental Nb ($sim 10$ K) very accurately. The present result indicates that our approach has a superior performance for many superconductors for which $T_{rm c}$ is lower than ${mathcal O}(10)$ K