Dynamic winding number for exploring band topology

Topological invariants play a key role in the characterization of topological states. Because of the existence of exceptional points, it is a great challenge to detect topological invariants in non-Hermitian systems. We put forward a dynamic winding number, the winding of realistic observables in long-time average, for exploring band topology in both Hermitian and non-Hermitian two-band models via a unified approach. We build a concrete relation between dynamic winding numbers and conventional topological invariants. In one dimension, the dynamic winding number directly gives the conventional winding number. In two dimensions, the Chern number is related to the weighted sum of all the dynamic winding numbers of phase singularity points. This work opens a new avenue to measure topological invariants via time-averaged spin textures without requesting any prior knowledge of the system topology.