  By revealing an underlying relation between the Dzyaloshinskii-Moriya
interaction (DMI) and the scalar spin chirality, we develop the theory of
magnon thermal Hall effects in antiferromagnetic systems. The dynamic
fluctuation of the scalar chirality is shown to directly respond to the
nontrivial topology of magnon bands. In materials such as the jarosites
compounds KFe$_3$(OH)$_6$(SO$_4$)$_2$ and veseignite BaCu$_3$V$_2$O$_8$(OH)$_2$
in the presence of in-plane DMI, the time-reversal symmetry can be broken by
the fluctuations of scalar chirality even in the case of coplanar
$\mathbf{q}=0$ magnetic configuration. The spin-wave Hamiltonian is influenced
by a fictitious magnetic flux determined by the in-plane DMI. Topological
magnon bands and corresponding nonzero Chern numbers are presented without the
need of a canted non-coplanar magnetic ordering. The canting angle dependence
of thermal Hall conductivity is discussed in detail as well. These results
offer a clear principle of chirality-driven topological effects in
antiferromagnetically coupled systems.
