  Strongly interacting topological states in multi-particle quantum systems
pose great challenges to both theory and experiment. Recently, bound states of
elementary spin waves (magnons) in quantum magnets have been experimentally
observed in quantum Heisenberg chains comprising ultracold Bose atoms in
optical lattices. Here, we explore a strongly interacting topological state
called topological magnon bound-state in the quantum Heisenberg chain under
cotranslational symmetry. We find that the cotranslational symmetry is the key
to the definition of a topological invariant for multi-particle quantum states,
which enables us to characterize the topological features of multi-magnon
excitations. We calculate energy spectra, density distributions, correlations
and Chern numbers of the two-magnon bound-states and show the existence of
topological protected edge bound-states. Our study not only opens a new
prospect to pursue topological magnon bound-states, but also gives insights
into the characterization and understanding of strongly interacting topological
states.
