  We present a quantum theory of phonon magnetic moment in doped Dirac
semimetals. Our theory is based on an emergent gauge field approach to the
electron-phonon coupling, applicable to both gapless and gapped systems. We
find that the magnetic moment is directly proportional to the electrical Hall
conductivity through the phonon Hall viscosity. Our theory is combined with the
first-principles calculations, allowing us to quantitatively implement it to
realistic materials. Magnetic moments are found to be on the order of Bohr
magneton for certain phonon modes in graphene and $\text{Cd}_3 \text{As}_2$.
Our results provide practical guidance for the dynamical generation of large
magnetization in the topological quantum materials.
