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

Abstract

In previous theoretical studies of phonon-mediated superconductors, the electron-phonon coupling is treated by solving the Migdal-Eliashberg equations under the bare vertex approximation, whereas the effect of Coulomb repulsion is incorporated by introducing one single pseudopotential parameter. These two approximations become unreliable in low carrier density superconductors in which the vertex corrections are not small and the Coulomb interaction is poorly screened. Here, we go beyond these two approximations and employ the Dyson-Schwinger equation approach to handle the interplay of the electron-phonon interaction and Coulomb interaction in a self-consistent way. We first derive the exact Dyson-Schwinger integral equation of the full electron propagator. Such an equation contains several unknown single-particle propagators and fermion-boson vertex functions, and thus seems to be intractable. To solve this difficulty, we further derive a number of identities satisfied by all the relevant propagators and vertex functions and then use these identities to show that the exact Dyson-Schwinger equation of the electron propagator is actually self-closed. This self-closed equation takes into account not only all the vertex corrections, but also the mutual influence between electron-phonon interaction and Coulomb interaction. Solving it by using proper numerical methods leads to the superconducting temperature Tc and other quantities. As an application of the approach, we compute the Tc of the interfacial superconductivity realized in the one-unit-cell FeSe/SrTiO3 system. We find that Tc can be strongly influenced by the vertex corrections and the competition between phonon-mediated attraction and Coulomb repulsion.

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