Materials' specific theory expounding high-temperature superconductivity in monolayer FeSe/SrTiO$_3$
Superconductivity in monolayer-thick FeSe on SrTiO$_3$ reaches amazingly high transition temperatures of typically T$_{\rm c}$=50-70 K [1-6] and up to 100 K [7], much higher than the 8 K value of bulk FeSe[8]. A coupling between SrTiO$_3$ phonons and FeSe electrons occurs at the FeSe/SrTiO$_3$ interface, which manifests itself as electron replica bands [6]. This coupling is commonly believed to enhance T$_{\rm c}$ moderately but not be enough to explain it [6,9,10]. Here, we examine the impact of interfacial phonons on the superconducting state of FeSe/SrTiO$_3$ developing full bandwidth, multiband Eliashberg theory. Our selfconsistent calculations prove that the interfacial electron-phonon interaction which is hidden behind a seemingly weak coupling constant, $\lambda_m$=0.4 [6,10], fully explains the high-T$_{\rm c}$ as well as puzzling experimental observations like the s-wave symmetry and replica bands. We find that deep Fermi-sea Cooper pairing has a T$_{\rm c}$ enhancing effect whereas replica-band formation has a decreasing effect. We predict a strong coupling dip-hump signature in the tunneling spectra due to the interfacial coupling.