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.
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