  The Phonon Hall Viscosity is the leading term evincing time-reversal symmetry
breaking in the low energy description of lattice phonons. It may generate
phonon Berry curvature, and can be observed experimentally through the acoustic
Faraday effect and thermal Hall transport. We present a systematic procedure to
obtain the phonon Hall viscosity induced by phonon-magnon interactions in
magnetic insulators under an external magnetic field. We obtain a general
symmetry criterion that leads to non-zero Faraday rotation and Hall
conductivity, and clarify the interplay between lattice symmetry,
spin-orbit-coupling, external magnetic field and magnetic ordering. The
symmetry analysis is verified through a microscopic calculation. 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 leading order time-reversal breaking term in the phonon
effective action, i.e. the phonon Hall viscosity, can be obtained. The analysis
of the square lattice antiferromagnet for a cuprate Mott insulator,
Sr$_2$CuO$_2$Cl$_2$, is presented explicitly, and the procedure described here
can be readily generalized to other magnetic insulators.
