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

Abstract

Significance It was established by Thouless, Kohmoto, Nightingale, and den Nijs in 1982 that the topology of the solid-state wavefunctions leads to quantization of transverse electrical conductivity of an insulator. This recognition has led to the development of the new field of topological materials characterized by symmetry-protected quantum numbers. Here, we propose a general and computationally efficient framework enabling one to unveil and predict materials-topological invariants in terms of physical observables, such as the bulk time-dependent current. We show how the quantized charge and spin Hall effect appears even for materials with a non-Abelian Berry phase. This dynamical approach is not necessarily restricted to density functional theory, but can be extended to other schemes and to other methods dealing with correlations explicitly. 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 used in calculations of topological properties of real materials. Here we propose a completely different and versatile approach to obtain 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 that lead 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 nonequilibrium states of topological materials.