Phonon Hall thermal conductivity from Green-Kubo formula

We derive a 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 non-zero only if the dynamic matrix of the phonon system breaks mirror reflection symmetry. The results are obtained without perturbative assumptions about the spin-phonon interactions.