Micromotion area as proxy for anomalous Floquet topological systems

Driven Floquet systems can realize topological phases with no static counterparts. These so-called anomalous Floquet topology breaks the bulk-boundary correspondence based on the Chern number. The number of edge modes in each band gap is instead determined by another integer index, a winding number, which is calculated from the time evolution operator of the bulk states within one driving period. While in the non-driven system, Chern markers provide a useful local proxy for the Chern number in the bulk, so far no such local bulk indicator is known for the winding number in Floquet systems. Here we consider two-band models and show that the area enclosed during a Floquet period by an initially localized particle signals the presence of an anomalous phase when it approaches half the unit cell area. In general, we show that at the fine-tuned point of dispersionless dynamics during the micromotion, the enclosed area is quantized and an exact proportionality relation exists between the area and the winding number. Direct detection of anomalous topology in real space could be realized in several quantum simulation platforms, and could be useful for systems with disorder or interactions. Building on the connection between area and winding number, we also show a way to realize arbitrarily high winding numbers.