Finite-temperature topological magnons in honeycomb ferromagnets with sublattice asymmetries

The Comment [Y.-M. Li, B. Wei, and K. Chang, Phys. Rev. Lett. 132, 219601 (2024)] pointed out that it is incorrect to predict the temperature-driven topological phase transition of Dirac magnons in honeycomb ferromagnets with Dzyaloshinskii-Moriya interactions based on the theory in Lu et al. [Y.-S. Lu, J.-L. Li, and C.-T. Wu, Phys. Rev. Lett. 127, 217202 (2021)]. Here we propose that by breaking the sublattice symmetries in honeycomb ferromagnets, increasing temperature could induce topological transitions from the trivial phase at zero temperature based on the linear spin wave theory to the Chern insulating phase above a critical temperature without changing any spin-spin interactions. The key to the finite-temperature topological magnons is considering the magnon-magnon interactions (MMIs) at a mean-field level. A self-consistently renormalized spin wave theory is employed to include self-energy corrections from MMIs, guaranteeing that the critical temperatures for topological transitions are below the Curi\'e temperatures. Across the critical temperatures, the magnon band gap closes and reopens at $\mathbf{K}$ or ${\mathbf{K}}^{\ensuremath{'}}$ points in the Brillouin zone, accompanied by nontrivial Berry curvature transitions. However, in stark contrast to the work of Lu et al. [Phys. Rev. Lett. 127, 217202 (2021)], the topological transitions cannot be revealed by the thermal Hall effect of magnons. Our work provides a realistic scheme for achieving a finite-temperature topological phase in honeycomb ferromagnets.