  We propose the velocity field approach to characterize topological invariants
of quantum states. We introduce the indexes of the velocity field flow based on
the zero modes of the velocity field and find that these zero modes play the
role of effective topological charges or defects linking to Euler
characteristic by the Poincar\'{e}-Hopf theorem. The global property of the
indexes is topological invariants against the parameter deformation. We
demonstrate this approach by the quantum torus model and compare the
topological invariant with that obtained from the Chern number. We find that
the physical mechanism of the topological invariant based on the zero modes of
the velocity field is different from that of the topological invariant by the
Chern number. The topological invariant characterized by the velocity field
describes a homeomorphic topological invariant associated with the zero modes
on the submanifold of the base manifold of the SU(2)-fibre bundle for quantum
torus, whereas the Chern number characterizes a homotopy invariant associated
with the exceptional points in the Brillouin zone. We also propose the
generalized winding number in terms of the velocity field for both Hermitian
and non-Hermitian systems. This gives a connection between the zero mode and
winding number in the velocity space. These results enrich the topological
invariants of quantum states and promises us a novel insight to understanding
topological invariants of quantum states as well as expected to be further
applied in more generic models.
