Unraveling materials Berry curvature and Chern-Simons numbers from real-time evolution of Bloch states

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.