Geometric Theory of Phonon Magnetic Moment in Dirac Materials

We develop a theoretical framework for the phonon magnetic moment in Dirac materials. Utilizing the theory of Dirac fermions in curved space, we classify electron-phonon couplings into three angular momentum channels. The l=1 channel represents an emergent gauge field, while the l=0 and l=2 channels correspond to a frame field, functioning as an effective gravitational field. Through these gauge and frame fields, we demonstrate that the phonon magnetic moment is proportional to the electrical Hall conductivity and electron Hall viscosity, respectively. To apply our theory to real materials, we establish a quantitative scheme grounded in first-principles calculations, tight-binding models, and quantum field theory. The magnetic moments of both Raman and infrared-active modes are found to be on the order of the Bohr magneton. This mechanism, which is broadly applicable across a wide range of materials, offers a promising avenue for future exploration of metallic systems.