We develop a theory for the phonon magnetic moment in doped Dirac materials, treating phonons as emergent gauge and gravitational fields coupled to Dirac fermions in curved space. By classifying electron-phonon coupling into angular momentum channels of Fermi surface deformation, we show that the phonon moment arises from two mechanisms: proportional to the electron Hall conductivity through the emergent gauge field coupling, and to the Hall viscosity through the frame field coupling. Applying our theory to ${\mathrm{Cd}}_3{\mathrm{As}}_2$ with first-principles calculations, we find order-of-magnitude agreement with experiment. Our results reveal a general mechanism for dynamically generating large phonon magnetism in metals and suggest a route for probing Hall viscosity via phonon dynamics.

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• Dirac fermions
• Electron-phonon coupling
• Phonons
• Topological materials
• Magnetic moment
• Viscosity
• First-principles calculations