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