  Spontaneous magnon decay is a generic feature of the magnetic excitations of
anisotropic magnets and isotropic magnets with non-collinear order. In this
paper, we argue that the effect of interactions on one-magnon states can, under
many circumstances, be treated in terms of an effective, energy independent,
non-Hermitian Hamiltonian for the magnons. In the vicinity of Dirac or Weyl
touching points, we show that the spectral function has a characteristic
anisotropy arising from topologically protected exceptional points or lines in
the non-Hermitian spectrum. Such features can, in principle, be detected using
inelastic neutron scattering or other spectroscopic probes. We illustrate this
physics through a concrete example: a honeycomb ferromagnet with
Dzyaloshinskii-Moriya exchange. We perform interacting spin wave calculations
of the structure factor and spectral function of this model, showing good
agreement with results from a simple effective non-Hermitian model for the
splitting of the Dirac point. Finally, we argue that the zoo of known
topological protected magnon band structures may serve as a nearly ideal
platform for realizing and exploring non-Hermitian physics in solid-state
systems.
