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

Abstract

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 ${\mathbf{q}}_{\text{max}}=(\ensuremath{\pi},\ensuremath{\pi})$ CDW phase boundary. For very low phonon frequencies, we also find an incommensurate CDW phase with the ordering vector ${\mathbf{q}}_{\text{max}}\ensuremath{\approx}(\ensuremath{\pi},\ensuremath{\pi})$ 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.