Interacting topological Dirac magnons

In this work, we study the magnon-magnon interaction effect in typical honeycomb ferromagnets consisting of van der Waals--bonded stacks of honeycomb layers, e.g., chromium trihalides $\mathrm{Cr}X3 (X=\mathrm{F},\mathrm{Cl},\mathrm{Br}, \mathrm{and} \mathrm{I})$, that display two spin-wave modes (Dirac magnon). Using Green's function formalism with the presence of the Dzyaloshinskii--Moriya interaction, we obtain a spinor Dyson equation up to the second-order approximation by the cluster expansion method. Numerical calculations show prominent renormalizations of the single-particle spectrum. Furthermore, we propose a tunable renormalization effect using a parametric magnon amplification scheme. By amplifying the magnon population at different $k$ points, the enabled renormalization effect not only reshapes the band structure but also modifies the Berry curvature distribution. Our work demonstrates the interplay between band geometry, interactions, and the external light field in the bosonic system and can potentially lead to new insights into the properties of magnon-based spintronic devices.