Phonon Hall Viscosity and Berry Phase in Magnetic Insulators

Phonon Hall viscosity may generate the Berry phase of phonons, and be observed experimentally in acoustic Faraday effect and thermal Hall transport. We present a systematic procedure to obtain the phonon Hall viscosity induced by phonon-magnon interaction in magnetic insulators under an external magnetic field. We first obtain a general symmetry criterion that leads to non-zero phonon Berry phase, and clarify the interplay between lattice symmetry, spin-orbit-coupling, external magnetic field and magnetic ordering. Next, by constructing the general symmetry allowed effective action that describes the spin dynamics and spin-lattice coupling, and then integrating out the spin fluctuations, the phonon Hall viscosity terms in the phonon effective action that break time-reversal symmetry can be obtained. The analysis of the square lattice antiferromagnet for a cuprate Mott insulator, Sr₂CuO₂Cl₂, is presented explicitly, and the procedure described here can be readily generalized to other magnetic insulators.