Phonon Hall thermal conductivity from the Green-Kubo formula

We derive an exact formula for the thermal-conductivity tensor of a ballistic phonon Hall model. It is found that, although the diagonal elements of the conductivity tensor diverge to infinite, the off-diagonal elements are finite, antisymmetric, and odd in magnetic field. The off-diagonal elements are nonzero only if the dynamic matrix of the phonon system breaks mirror-reflection symmetry. The results are obtained without perturbative treatment of the spin-phonon interactions.