Effects of dilution in a two-dimensional topological magnon insulator

We study the effect of diluting a two-dimensional ferromagnetic insulator hosting a topological phase in the clean limit. By considering the ferromagnetic Heisenberg model in the honeycomb lattice with second-nearest-neighbor Dzyaloshinskii-Moriya interaction, and working in the linear spin-wave approximation, we establish the topological phase diagram as a function of the fraction $p$ of diluted magnetic atoms. The topological phase with Chern number $C=1$ is robust up to a moderate dilution ${p}_{1}^{*}$, while above a higher dilution ${p}_{2}^{*}>{p}_{1}^{*}$ the system becomes trivial. Interestingly, both ${p}_{1}^{*}$ and ${p}_{2}^{*}$ are below the classical percolation threshold ${p}_{c}$ for the honeycomb lattice, which gives physical significance to the obtained phases. In the topological phase for $p<{p}_{1}^{*}$, the magnon spectrum is gapless but the states filling the topological, clean-limit gap region are spatially localized. For energies above and below the region of localized states, there are windows composed of extended states. This is at odds with standard Chern insulators, where extended states occur only at single energies. For dilutions ${p}_{1}^{*}<p<{p}_{2}^{*}$, the two regions of extended states merge and a continuum of delocalized states appears around the middle of the magnon spectrum. For this range of dilutions the Chern number seems to be ill defined in the thermodynamic limit, and only for $p>{p}_{2}^{*}$, when all states become localized, the system shows $C=0$ as expected for a trivial phase. Replacing magnetic with nonmagnetic atoms in a system hosting a magnon Chern insulator in the clean limit puts all the three phases within experimental reach.