  Search for higher-order topological insulators, characterized by
topologically protected gapless boundary states of codimension higher than one,
in bosonic systems has attracted growing interest. Here, we establish twisted
bilayer honeycomb magnets as a new platform for hosting second-order
topological magnon insulators (SOTMIs) without fine-tuning. We employ a simple,
minimal Heisenberg spin model to describe misaligned bilayer sheets of
honeycomb ferromagnetic magnets with a large commensurate twist angle. We found
that the higher-order topology in this bilayer system shows a significant
dependence on the interlayer exchange coupling. The SOTMI, featuring
topologically protected magnon corner states, appears for ferromagnetic
interlayer couplings, while the twisted bilayer exhibits a nodal phase in the
case of antiferromagnetic interlayer coupling.
