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

TL;DR

This paper introduces a Ward identity-preserving method to analyze many-body effects on phonon spectra, showing that electron self-energy and vertex corrections cancel out at long wavelengths, leaving the phonon spectrum nearly unchanged.

Contribution

It presents a systematic diagrammatic approach that incorporates electron self-energy and vertex corrections while respecting charge conservation, revealing their canceling effects on phonon spectrum at long wavelengths.

Findings

Many-body corrections vanish for long-wavelength phonons due to cancellation.

Electron self-energy and vertex corrections are comparable and tend to cancel.

Phonon spectrum remains nearly unaffected by many-body effects at small q.

Abstract

We propose an approach for investigating the many body effects on 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 $\Pi(\mathbf{q},\omega)$ in powers of the electron-phonon scattering strength $(g^2)$. In this approach the many body corrections to phonon spectrum vanish identically in the regime of long-wavelength excitations ($\mathbf{q} \rightarrow 0$) due to an exact cancellation between the contributions arising from electron self-energy and vertex corrections. Our results demonstrate that the contributions of electron self-energy and vertex corrections are not only comparable but they also tend to cancel each oher so that the phonon spectrum remains nearly unaffected due to many body effects in the regime of long-wavelength excitations ($\mathbf{q} \rightarrow 0$)