Topological Phase Transitions of Dirac Magnons in Honeycomb Ferromagnets

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.