  In this Letter we present a comprehensive study of topological magnon bands
and thermal Hall effect in non-collinear antiferromagnetic systems on the
honeycomb lattice with an intrinsic Dzyaloshinskii-Moriya interaction. We
theoretically show that the system possesses topological magnon bands with
Chern number protected edge modes accompanied by a nonzero thermal magnon Hall
effect. These features result from non-collinearity of the magnetic moments due
to an applied out-of-plane magnetic field. Our results provide an experimental
clue towards the realization of topological magnon transports in honeycomb
antiferromagnetic compounds such as XPS$_3$ (X=Mn,Fe) and
$\alpha$-Cu$_2$V$_2$O$_7$.
