  We present computations of the thermal Hall coefficient of phonons scattering
off a defect with multiple energy levels. Using a microscopic formulation based
on the Kubo formula, we find that the leading contribution perturbative in the
phonon-defect coupling is proportional to the phonon lifetime, and has a
`side-jump' interpretation. Consequently, the thermal Hall angle is independent
of the phonon lifetime. The contribution to the thermal Hall coefficient is at
resonance when the phonon energy equals a defect level spacing. Our results are
obtained for three different defect models, which apply to different correlated
electron materials. For the pseudogap regime of the cuprates, we propose a
model of phonons coupled to an impurity quantum spin in the presence of
quasi-static magnetic order with an isotropic Zeeman coupling to the applied
field, and without spin-orbit interaction.
