Ward identity preserving approach for investigation of phonon spectrum with self-energy and vertex corrections

We propose an approach for investigating the many-body effects on the phonon spectrum in an electron-phonon coupled system by taking into account the electron self-energy and vertex corrections which respect the Ward identity representing the conservation of electronic charge. Our approach provides a systematic diagrammatic expansion of the phonon self-energy ${\mathrm{\ensuremath{\Pi}}}_{\mathbf{q}}(\ensuremath{\omega})$ in powers of the electron-phonon scattering strength squared $({g}^{2})$. In this approach the many-body corrections to the phonon spectrum vanish identically for the $\mathbf{q}=0$ mode due to an exact cancellation between the contributions arising from electron self-energy and vertex corrections. This cancellation holds irrespective of the microscopic details of the system such as dimensionality, adiabaticity, strength of electron-phonon scattering, etc. Our results demonstrate that the contributions of electron self-energy and vertex corrections are not only comparable but they also tend to cancel each other so that the phonon spectrum remains nearly unaffected due to many-body effects in the regime of long-wavelength excitations ($\mathbf{q}\ensuremath{\rightarrow}0$). Our results provide another constraint for the applicability of the Migdal theorem.