  Understanding correlation effects in topological phases of matter is at the
forefront of current research in condensed matter physics. Here we try to
clarify some subtleties in studying topological behaviors of interacting Weyl
semimetals. It is well-known that there exist two topological invariants
defined to identify their topological character. One is the many-body Chern
number, which can be directly linked to the Hall conductivity and thus to the
two-particle correlations. The other is the topological index constructed from
the single-particle Green's functions. Because the information of Green's
functions is easier to be achieved than the many-body wavefunctions, usually
only the latter is employed in the literature. However, the approach based on
the single-particle Green's function can break down in the strongly correlated
phase. For illustration, an exactly solvable two-orbital model with
momentum-local two-body interactions is discussed, in which both topological
invariants can be calculated analytically. We find that the topological index
calculated from the Green's function formalism can be nonzero even for a
non-topological strongly correlated phase with vanishing many-body Chern
number. In addition, we stress that the physical surface states implied by
nonzero many-body Chern numbers should be the edge modes of particle-hole
collective excitations, rather than those of quasiparticle nature derived from
the Green's function formalism. Our observations thus demonstrate the
limitation of the validity of Green's function formalism in the investigations
of interacting topological materials.
