  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 Dzyanshikii-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 non-magnetic atoms in a
systems hosting a magnon Chern insulator in the clean limit puts all the three
phases within experimental reach.
