  Magnon spin Hall systems could hardly show experimentally observable particle
and thermal transport phenomena intrinsically due to the spin cancellation.
Here we demonstrated that the magnon spin Hall systems can exhibit magnon
Nernst effect and thermal Hall effect under external magnetic field by
considering two typical systems, i.e. the antiferromagnetically (AFM) coupled
bilayer honeycomb ferromagnets and monolayer collinear honeycomb
antiferromagnet. The both systems experience magnetic phase transitions from
AFM phase to a field-polarized phase via a spin-flop (SF) phase or directly. In
both systems, there exist magnon Nerst effect and also thermal Hall effect
under a longitudinal temperature gradient, which can be regarded as the
indicator of the magnetic phase transitions, with the Hall conductivity
dependence on magnetic field consistent with the order of magnetic phase
transition.
