  It is still an outstanding challenge to characterize and understand the
topological features of strongly interacting states such as bound-states in
interacting quantum systems. Here, by introducing a cotranslational symmetry in
an interacting multi-particle quantum system, we systematically develop a
method to define a Chern invariant, which is a generalization of the well-known
Thouless-Kohmoto-Nightingale-den Nijs invariant, for identifying strongly
interacting topological states. As an example, we study the topological
multi-magnon states in a generalized Heisenberg XXZ model, which can be
realized by the currently available experiment techniques of cold atoms [Phys.
Rev. Lett. \textbf{111}, 185301 (2013); Phys. Rev. Lett. \textbf{111}, 185302
(2013)]. Through calculating the two-magnon excitation spectrum and the defined
Chern number, we explore the emergence of topological edge bound-states and
give their topological phase diagram. We also analytically derive an effective
single-particle Hofstadter superlattice model for a better understanding of the
topological bound-states. Our results not only provide a new approach to
defining a topological invariant for interacting multi-particle systems, but
also give insights into the characterization and understanding of strongly
interacting topological states.

Quantum Gases;Quantum Physics