First-principles theory of phonon renormalization from nonlinear electron-phonon interactions
Electron-phonon interactions renormalize phonon frequencies and lifetimes and are central to the dynamical properties of solids. While these effects are usually described within linear electron-phonon coupling, the role of nonlinear electron-phonon interactions for phonon properties remains largely unexplored. In this work, we study phonon renormalization arising from the long-range linear one-electron-one-phonon and the nonlinear one-electron-two-phonon interactions within a diagrammatic framework. We derive the corresponding self-energy diagrams, which depend on the chemical potential and temperature, and evaluate them from first principles for the two polar semiconductors LiF and KTaO$_3$. In both materials, the two interaction channels renormalize the phonon spectrum in qualitatively distinct ways. The linear contribution is sharply localized near the Brillouin-zone center, whereas the nonlinear process couples an incoming phonon to other branches throughout the spectrum. As a result, it renormalizes phonons across the entire Brillouin-zone, with a pronounced temperature dependence governed by the thermal occupation of those branches. This behavior provides a clean experimental signature of the one-electron-two-phonon coupling. While the nonlinear phonon renormalization is small in LiF, it is somewhat larger in KTaO$_3$, which we attribute to its greater number of thermally populated phonon branches at room temperature. Our results establish a general framework to assess nonlinear electron-phonon effects on the phonon properties in materials with stronger lattice fluctuations, including soft semiconductors such as lead-halide perovskites.