  Topological magnon insulators (TMI) are ordered magnets supporting chiral
edge magnon excitations. These edge states are envisioned to serve as
topologically protected information channels in low-loss magnonic devices. The
standard description of TMI is based on linear spin-wave theory (LSWT), which
approximates magnons as free non-interacting particles. However, magnon
excitations of TMI are genuinely interacting even at zero temperature, calling
into question descriptions based on LSWT alone. Here we perform a detailed
non-linear spin-wave analysis to investigate the stability of chiral edge
magnons. We identify three general breakdown mechanisms: (1) The edge magnon
couples to itself, generating a finite lifetime that can be large enough to
lead to a spectral annihilation of the chiral state; (2) The edge magnon
hybridizes with the extended bulk magnons and, as a consequence, delocalizes
away from the edge; (3) Due to a bulk-magnon mediated edge-to-edge coupling,
the chiral magnons at opposite edges hybridize. We argue that, in general,
these breakdown mechanisms may invalidate predictions based on LSWT and violate
the notion of topological protection. We discuss strategies how the breakdown
of chiral edge magnons can be avoided, e.g. via the application of large
magnetic fields. Our results highlight a challenge for the realization of
chiral edge states in TMI and in other bosonic topological systems without
particle number conservation.
