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In previous theoretical studies of phonon-mediated superconductors, theelectron-phonon coupling is treated by solving the Migdal-Eliashberg equationsunder the bare vertex approximation, whereas the effect of Coulomb repulsion isincorporated by introducing one single pseudopotential parameter. These twoapproximations become unreliable in low carrier-density superconductors inwhich the vertex corrections are not small and the Coulomb interaction ispoorly screened. Here, we shall go beyond these two approximations and employthe Dyson-Schwinger equation approach to handle the interplay ofelectron-phonon interaction and Coulomb interaction in a self-consistent way.We first derive the exact Dyson-Schwinger integral equation of the fullelectron propagator. Such an equation contains several unknown single-particlepropagators and fermion-boson vertex functions, and thus seems to beintractable. To solve this difficulty, we further derive a number of identitiessatisfied by all the relevant propagators and vertex functions and then usethese identities to show that the exact Dyson-Schwinger equation of electronpropagator is actually self-closed. This self-closed equation takes intoaccount not only all the vertex corrections, but also the mutual influencebetween electron-phonon interaction and Coulomb interaction. Solving it byusing proper numerical methods leads to the superconducting temperature $T_{c}$and other quantities. As an application of the approach, we compute the $T_{c}$of the interfacial superconductivity realized in the one-unit-cellFeSe/SrTiO$_{3}$ system. We find that $T_{c}$ can be strongly influenced by thevertex corrections and the competition between phonon-mediated attraction andCoulomb repulsion.

Theoretical study of phonon-mediated superconductivity beyond Migdal-Eliashberg approximation and Coulomb pseudopotential