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

TL;DR

This work shows that the Ward-Takahashi identity enforces current conservation in electron-phonon systems and redefines the leading mechanism for phonon-induced shift currents: the photocurrent arises from the electric polarization carried by optically excited phonons, not from conventional charge transport. By deriving a compact expression for the phonon-shift-current conductivity and developing a low-energy effective theory for phonons, the authors connect off-resonant polarization responses to resonance-enhanced photocurrents at the phonon frequency. The Rice-Mele model illustrates a measurable, phonon-resonant shift current, while the WT constraints explain cancellations in subleading diagrams and clarify the relative magnitudes of phonon vs electronic contributions. The results offer a unified, polarization-centric view of BPVE in phonon-coupled systems and a practical effective-theory description for low-energy phonon dynamics under light irradiation.

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.

Figures (2)

• Figure 1: Diagrams for phonon shift current. (a) Diagram for photocurrent response mediated by phonon excitation. Left and right electron loops $A$ describe a coupling of a phonon to the external electric field. The middle electron loop $B$ describes a photocurrent carried by a phonon. (b) Electron loop diagrams $A$ for the coupling of a phonon to an external electric field. A bubble diagram $A_b$ and a tadpole diagram $A_d$ contribute to the coupling between the phonon and the light. (c) Electron loop diagrams $B$ for the photocurrent carried by a phonon. A triangle diagram $B_t$ and a bubble diagram $B_b$ contribute to the current response.
• Figure 2: Phonon shift current conductivity $\sigma^\mathrm{ph}(\omega)$. The inset is a schematic for the Rice-Mele model. We used parameters, $\delta t/t=0.3, m/t=-1.5, \omega_0/t=0.1, \gamma/t=0.01$.