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

We formulate an efficient scheme to perform a 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 Green's 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 obtain a superconducting transition temperature of elemental Nb (similar to 10 K), which is consistent with experiment. The present result indicates that our approach has a superior performance for many superconductors for which T-c is lower than O(10) K.