Ab Initio Theory of Phonon Magnetic Moment Induced by Electron-Phonon Coupling in Magnetic Materials

Circularly polarized phonons, characterized by nonzero angular momenta and magnetic moments, have attracted extensive attention. However, a long-standing critical issue in this field is the lack of an approach to accurately calculate phonon magnetic moments resulting from electron-phonon coupling (EPC) in realistic materials. Here, based on the linear response framework, we develop an ab initio theory for calculating EPC-induced magnetic properties of phonons, applicable to both insulating and metallic materials. Our method can precisely calculate phonon Zeeman splittings in magnetic metals with significant EPC, as demonstrated by the remarkable agreement with recent experimental observations of phonon Zeeman splitting in the ferromagnetic Weyl semimetal Co_{3}Sn_{2}S_{2}. In addition, the long-sought magnetic phonon spectra across the entire Brillouin zone are obtained, facilitating the study of magnetic phonon transport and topology. Specifically, by constructing an inertially decoupled lattice model, we propose candidate materials exhibiting intrinsic phonon Chern states with robust unidirectional edge phonon currents. Our Letter paves the way for investigating novel phonon phenomena in magnetic quantum materials.