  We introduce the velocity field of the Bloch electrons and propose the
velocity field approach to characterize the topological invariants of quantum
states. We find that the zero modes of the velocity field flow play the roles
of effective topological charges or defects. A key global property of the zero
modes is topological invariant against the parameter deformation. These can be
characterized by the Euler characteristic based on the Poincar\'{e}-Hopf
theorem. We demonstrate the validity of this approach by using the quantum
sphere and torus models. The topological invariants of the velocity field in
the quantum sphere and torus are consistent with the mathematical results of
the vector fields in the manifolds of the sphere and torus, Euler
characteristic $\chi=2$ for sphere and $\chi=0$ for torus. We also discuss the
non-Hermitian quantum torus model and compare differences in the topological
invariants obtained using the velocity field and Chern number methods. The
topological invariant characterized by the velocity field is homeomorphic in
the Brillouin zone and the subbase manifold of the SU(2)-bundle of the system,
whereas the Chern number characterizes a homotopic invariant that is associated
with the exceptional points in the Brillouin zone. These results enrich the
topological invariants of quantum states and provide novel insights into the
topological invariants of quantum states.
