  We present a systematic theory of the phonon Hall effect in a ballistic
crystal lattice system, and apply it on the kagome lattice which is ubiquitous
in various real materials. By proposing a proper second quantization for the
non-Hermite Hamiltonian in the polarization-vector space, we obtain a new heat
current density operator with two separate contributions: the normal velocity
responsible for the longitudinal phonon transport, and the anomalous velocity
manifesting itself as the Hall effect of transverse phonon transport. As
exemplified in kagome lattices, our theory predicts that the direction of Hall
conductivity at low magnetic field can be reversed by tuning temperatures,
which we hope can be verified by experiments in the future. Three
phonon-Hall-conductivity singularities induced by phonon-band-topology change
are discovered as well, which correspond to the degeneracies at three different
symmetric center points, \Gamma, K, X, in the wave-vector space of the kagome
lattice.
