Topological defects in Floquet systems: Anomalous chiral modes and topological invariant

Backscattering-immune chiral modes arise along certain line defects in three-dimensional materials. Here, we study the Floquet chiral modes along Floquet topological defects, namely, the defects come entirely from spatial modulations of periodic driving. We define a precise topological invariant that counts the number of Floquet chiral modes, which is expressed as an integral on a five-dimensional torus parametrized by $({k}_x,{k}_y,{k}_z,\ensuremath{\theta},t)$. This work demonstrates the possibility of creating chiral modes in three-dimensional bulk materials by modulated driving.