  Beyond the well-known topological band theory for single-particle systems, it
is a great challenge to characterize the topological nature of interacting
multi-particle quantum systems. Here, we uncover the relation between
topological invariants defined through the twist boundary condition (TBC) and
the center-of-mass (c.m.) momentum state in multi-particle systems. We find
that the Berry phase defined through TBC can be equivalently obtained from the
multi-particle Wilson loop formulated by c.m. momentum states. As the Chern
number can be written as the winding of the Berry phase, we consequently prove
the equivalence of Chern numbers obtained via TBC and c.m. momentum state
approaches. As a proof-of-principle example, we study topological properties of
the Aubry-Andr{\'e}-Harper (AAH) model. Our numerical results show that the TBC
approach and c.m. approach are well consistent with each other for both
many-body case and few-body case. Our work lays a concrete foundation and
provides new insights for exploring multi-particle topological states.
