Berry curvature and the phonon Hall effect

We show that an effective magnetic field acting on phonons naturally emerges in the phonon dynamics of magnetic solids, giving rise to the phonon Hall effect. A general formula for the intrinsic phonon Hall conductivity is derived by using the corrected Kubo formula with the energy magnetization contribution incorporated properly. We thus establish a direct connection between the phonon Hall effect and the intrinsic phonon band structure, i.e., the phonon Berry curvature and phonon dispersion. Based on the formalism, we predict that phonons could also display the quantum Hall effect in certain topological phonon systems. In the low-temperature regime, we predict that the phonon Hall conductivity is proportional to T-3 for ordinary phonon systems, while that for the topological phonon system has a linear T dependence with a quantized temperature coefficient.