  Condensed matter systems admit topological collective excitations above a
trivial ground state, an example being Chern insulators formed by Dirac bosons
with a gap at finite energies. However, in contrast to electrons, there is no
particle-number conservation law for collective excitations. This gives rise to
particle number-nonconserving many-body interactions whose influence on
single-particle topology is an open issue of fundamental interest in the field
of topological quantum materials. Taking magnons in ferromagnets as an example,
we uncover topological magnon insulators that are stabilized by interactions
through opening Chern-insulating gaps in the magnon spectrum. This can be
traced back to the fact that the particle-number nonconserving interactions
break the effective time-reversal symmetry of the harmonic theory. Hence,
magnon-magnon interactions are a source of topology that can introduce chiral
edge states, whose chirality depends on the magnetization direction.
Importantly, interactions do not necessarily cause detrimental damping but can
give rise to topological magnons with exceptionally long lifetimes. We identify
two mechanisms of interaction-induced topological phase transitions---one
driven by an external field, the other by temperature---and show that they
cause unconventional sign reversals of transverse transport signals, in
particular of the thermal Hall conductivity. We identify candidate materials
where this many-body mechanism is expected to occur, such as the metal-organic
kagome-lattice magnet Cu(1,3-benzenedicarboxylate), the van der Waals
honeycomb-lattice magnet CrI$_3$, and the multiferroic kamiokite
(Fe$_2$Mo$_3$O$_8$). Our results demonstrate that interactions can play an
important role in generating nontrivial topology.
