  Quite recently, magnon Hall effect of spin excitations has been observed
experimentally on the kagome and pyrochlore lattices. Thermal Hall conductivity
$\kappa^{xy}$, changes sign as a function of magnetic field or temperature on
the kagome lattice, and $\kappa^{xy}$ changes sign upon reversing the sign of
the magnetic field on the pyrochlore lattice. Motivated by these recent
exciting experimental observations, we theoretically propose a simple
realization of magnon Hall effect in a two-band model on the honeycomb lattice.
The magnon Hall effect of spin excitations arises in the usual way via the
breaking of inversion symmetry of the lattice, however, by a
next-nearest-neighbour Dzyaloshinsky-Moriya (DM) interaction. We find that
$\kappa^{xy}$ has a fixed sign for all parameter regimes considered. These
results are in contrast to the Lieb, kagome and pyrochlore lattices. We further
show that the low-temperature dependence on the magnon Hall conductivity
follows a $T^{2}$ law, as opposed to the kagome and pyrochlore lattices. These
results suggest an experimental procedure to measure thermal Hall conductivity
within a class of 2D honeycomb quantum magnets and ultracold atoms trapped in
honeycomb optical lattice.
