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We study the temperature evolution of the quasiparticle bands of the FeSe monolayer on the SrTiO3 (STO) substrate from 10 to 300 K by applying the anisotropic, multiband and full-bandwidth Eliashberg theory. To achieve this, we extend this theory by self-consistently coupling the chemical potential to the full set of Eliashberg equations. In this way, the electron filling can accurately be kept at a constant level at any temperature. Solving the coupled equations self-consistently, and with focus on the interfacial electron-phonon coupling, we compute a nearly constant Fermi surface with respect to temperature and predict a non-trivial temperature evolution of the global chemical potential. This evolution includes a total shift of 5 meV when increasing temperature from 10 to 300 K and a hump-like dependence followed by a kink at the critical temperature Tc. We argue that the latter behavior indicates that superconductivity in FeSe/SrTiO3 is near to the BCS-BEC crossover regime. Calculating the temperature dependent Angle Resolved Photoemission Spectroscopy (ARPES) spectra, we suggest a new route to determine the energy scale of the interfacial phonon mode by measuring the energy position of second-order replica bands. Further, we reexamine the often used symmetrization procedure applied to such ARPES curves and demonstrate substantial asymmetric deviations. Lastly, our results reveal important aspects for the experimental determination of the momentum anisotropy of the superconducting gap.

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We investigate phonon-mediated Cooper pairing in flat electronic band systems by solving the full-bandwidth multiband Eliashberg equations for superconductivity in magic angle twisted bilayer graphene using a realistic tight-binding model. We find that Cooper pairing away from the Fermi level contributes decisively to superconductivity by enhancing the critical temperature and ensures a robust finite superfluid density. We show that this pairing yields particle-hole asymmetric superconducting domes in the temperature-gating phase diagram and gives rise to distinct spectroscopic signatures in the superconducting state. We predict several such features in tunneling and angle resolved photoemission spectra for future experiments.

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We present a method to considerably improve the numerical performance for solving Eliashberg-type coupled equations on the imaginary axis. Instead of the standard practice of introducing a hard numerical cutoff for treating the infinite summations involved, our scheme allows for the efficient calculation of such sums extended formally up to infinity. The method is first benchmarked with isotropic Migdal-Eliashberg theory calculations and subsequently applied to the solution of the fullbandwidth, multiband and anisotropic equations focusing on the FeSe/SrTiO3 interface as a case study. Compared to the standard procedure, we reach similarly well converged results with less than one fifth of the number of frequencies for the anisotropic case, while for the isotropic set of equations we spare approximately ninety percent of the complexity. Since our proposed approximations are very general, our numerical scheme opens the possibility of studying the superconducting properties of a wide range of materials at ultra-low temperatures.

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