  Materials can be classified by the topological character of their electronic
structure and, in this perspective, global attributes immune to local
deformations have been discussed in terms of Berry curvature and Chern numbers.
Except for instructional simple models, linear response theories have been
ubiquitously employed in calculations of topological properties of real
materials. Here we propose a completely different and versatile approach to get
the topological characteristics of materials by calculating physical
observables from the real-time evolving Bloch states: the cell-averaged current
density reveals the anomalous velocities whose integration leads to the
conductivity quantum. Results for prototypical cases are shown, including a
spin-frozen valley-Hall and a quantum anomalous Hall insulator. The advantage
of this method is best illustrated by the example of a quantum spin Hall
insulator: the quantized spin Hall conductivity is straightforwardly obtained
irrespective of the non-Abelian nature in its Berry curvature. Moreover, the
method can be extended to the description of real observables in
non-equilibrium states of topological materials.
