  The study of magnonic thermal Hall effect has recently attracted attention
because this effect can be associated with topological phases activated by
Dzyaloshinskii-Moriya interaction, which acts similar to a spin-orbital
coupling in an electronic system. A topological phase transition may arise when
there exist two or more distinct topological phases, and this transition is
often revealed by a gap closing. In this work, we consider a ferromagnetic
honeycomb lattice described by a Hamiltonian that contains Heisenberg exchange
interaction, Dzyaloshinskii-Moriya interaction, and an applied Zeeman field.
When expanding the spin operators in the Hamiltonian using Holstein-Primakoff
(HP) transformation to the order of $S^{1/2}$, where $S$ is the magnitude of
spin, the thermal Hall conductivity stays negative for all values of parameters
such as the strength of Zeeman interaction and temperature. However, we
demonstrate in this work that by including the next order, $S^{-1/2}$, in HP
transformation to take into account magnon-magnon interaction, the Hartree type
of interaction gives rise to topological phase transitions driven by
temperature. When the temperature increases, we find that the gap of the
magnonic energy spectrum closes at Dirac points at a critical temperature,
$T_c$, and the gap-closing is indeed the signature for a topological phase
transition as confirmed by showing that the Chern numbers are distinct above
and below $T_c$. Finally, our analysis points out that thermal Hall
conductivity exhibits sign reversal at the same temperature. This phenomenon
can be used in experiments to verify the topological nature of magnons in
honeycomb magnets.
