Temperature-filling phase diagram of the two-dimensional Holstein model in the thermodynamic limit by self-consistent Migdal approximation

We study the temperature-filling phase diagram of the single-band Holstein model in two dimensions using the self-consistent Migdal approximation, where both the electron and phonon self-energies are treated on an equal footing. By employing an efficient numerical algorithm utilizing fast Fourier transforms to evaluate momentum and Matsubara frequency summations, we determine the charge-density-wave (CDW) and superconducting transition temperatures in the thermodynamic limit using lattice sizes that are sufficient to eliminate significant finite-size effects present at lower temperatures. We obtain the temperature-filling phase diagrams for a range of coupling strengths and phonon frequencies for the model defined on a square lattice with and without next-nearest-neighbor hopping. We find the appearance of a superconducting dome with a critical temperature that decreases before reaching the q(max) = (pi, pi) CDW phase boundary. For very low phonon frequencies, we also find an incommensurate CDW phase with the ordering vector q(max approximate to) (pi, pi) appearing between the commensurate CDW and superconducting phases. Our numerical implementation can be easily extended to treat momentum-dependent electron-phonon coupling, as well as dispersive phonon branches, and has been made available to the public.