  Thermal transport in topologically-ordered phases of matter provides valuable
insights as it can detect the charge-neutral quasiparticles that would not
directly couple to electromagnetic probes. An important example is edge heat
transport of Majorana fermions in a chiral spin liquid, which leads to a
half-quantized thermal Hall conductivity. This signature is precisely what has
recently been measured in $\alpha$-RuCl$_3$ under external magnetic fields. The
plateau-like behavior of the half-quantized thermal Hall conductivity as a
function of external magnetic field, and the peculiar sign change depending on
the magnetic field orientation, has been proposed as strong evidence for the
non-Abelian Kitaev spin liquid. Alternatively, for in-plane magnetic fields, it
was theoretically shown that such a sign structure can also arise from
topological magnons in the field-polarized state. In this work, we investigate
the full implications of topological magnons as heat carriers on thermal
transport measurements. We first prove analytically that for any commensurate
order with a finite magnetic unit cell, reversing the field direction leads to
a sign change in the magnon thermal Hall conductivity in two-dimensional
systems. We verify this proof numerically with nontrivial magnetic orders as
well as the field-polarized state in Kitaev magnets subjected to an in-plane
field. In the case of a tilted magnetic field, whereby there exist both finite
in-plane and out-of-plane field components, we find that the plateau-like
behavior of the thermal Hall conductivity and the sign change upon reversing
the in-plane component of the magnetic field arise in the partially-polarized
state, as long as the in-plane field contribution to the Zeeman energy is
significant. While these results are consistent with the experimental
observations, we comment on other aspects requiring investigation in future
studies.
