Role of Ward-Takahashi identity in an electron-phonon coupled system -- Revisiting phonon shift current

TL;DR

This paper investigates how the Ward-Takahashi identity influences the bulk photovoltaic effect in electron-phonon systems, highlighting the role of phonon-induced polarization shifts in generating shift current.

Contribution

It revisits the phonon shift current mechanism by emphasizing the importance of gauge invariance and the Ward-Takahashi identity in analyzing electron-phonon interactions.

Findings

Leading order phonon shift current arises from optically excited phonon polarization.

The Ward-Takahashi identity is crucial for understanding current conservation in these systems.

Electric polarization change under optical excitation drives the shift current.

Abstract

We study bulk photovoltaic effects in electron-phonon coupled systems. The conservation of current or gauge invariance, manifested as the Ward-Takahashi identity, plays an essential role in the analysis of the Feynman diagrams, and the leading order contribution to the phonon shift current is identified accordingly. The leading order contribution essentially arises from the electric polarization carried by optically excited phonons, where the shift current is generated due to a change of electric polarization in the steady state under the optical excitation of phonons.