Classification of materials with phonon angular momentum and microscopic origin of angular momentum

We group materials into five symmetry classes and determine in which of these classes will phonons generically carry angular momentum. In some classes of materials, phonons acquire angular momentum via the forces induced by relative displacements of atoms out of their equilibrium positions. However, for other materials, such as ferromagnetic iron, phonon angular momentum arises from the forces induced by relative velocities of atoms. These latter effects are driven by the spin-orbit interaction.