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We study the temperature-filling phase diagram of the single-band Holsteinmodel in two dimensions using the self-consistent Migdal approximation, whereboth the electron and phonon self-energies are treated on an equal footing. Byemploying an efficient numerical algorithm utilizing fast Fourier transforms toevaluate momentum and Matsubara frequency summations, we determine thecharge-density-wave (CDW) and superconducting transition temperatures in thethermodynamic limit using lattice sizes that are sufficient to eliminatesignificant finite size effects present at lower temperatures. We obtain thetemperature-filling phase diagrams for a range of coupling strengths and phononfrequencies for the model defined on a square lattice with and withoutnext-nearest neighbor hopping. We find the appearance of a superconducting domewith a critical temperature that decreases before reaching the$\mathbf{q}_{\text{max}} = (\pi,\pi)$ CDW phase boundary. For very low phononfrequencies, we also find an incommensurate CDW phase with the ordering vector$\mathbf{q}_{\text{max}} \approx (\pi,\pi)$ appearing between the commensurateCDW and superconducting phases. Our numerical implementation can be easilyextended to treat momentum-dependent electron-phonon coupling, as well asdispersive phonon branches, and has been made available to the public.

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