Study of the superconducting <i>T_c</i> dome in the phase diagram of the two-dimensional Holstein model

T_c dome, i.e., the nonmonotonic superconducting T_c dependence as a function of parameters such as doping or pressure, is ubiquitous in the phase diagrams of unconventional superconductors, where the pairing interaction has a magnetic and electronic origin. By contrast, T_c dome is rarely seen in conventional electron-phonon superconductors. We report the T_c dome in the phase diagram of the two-dimensional Holstein model, where electrons and phonons interact through a local coupling. By solving the model using strong coupling Migdal-Eliashberg formalism and self-consistently renormalizing both the electrons and phonons, a T_c dome as function of filling n arises in the phase diagram, alongside a charge-density-wave phase around the half-filling. The dome is tied to the competition of three renormalized quantities as the filling increases: the increasing coupling constant λ, the decreasing ω_log, and the nonmonotonically varying density of states at the Fermi-level. Interestingly, we also find the effective electron-electron interaction becomes nonlocal in r-space and even repulsive at neighboring lattice sites. Our discovery could shed light on the high T_c of the strongly electron-phonon coupled system such as the dense hydrogen-rich compounds and the monolayer FeSe on SrTiO₃.