Multiband full-bandwidth anisotropic Eliashberg theory of interfacial electron-phonon coupling and high-T$_c$ superconductivity in FeSe/SrTiO$_3$

We examine the impact of interfacial phonons on the superconducting state of FeSe/SrTiO$_3$ developing a material's specific multiband, full bandwidth, and anisotropic Eliashberg theory for this system. Our self-consistent calculations highlight the importance of the interfacial electron-phonon interaction, which is hidden behind the seemingly weak-coupling constant {\lambda}$_m$ = 0.4, in mediating the high T$_c$, and explain other puzzling experimental observations, such as the s-wave symmetry and replica bands. We discover that the formation of replica bands has a T$_c$ decreasing effect that is nevertheless compensated by deep Fermi-sea Cooper pairing which has a T$_c$ enhancing effect. We predict a strong-coupling dip-hump signature in the tunneling spectra due to the interfacial coupling.