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

Backscattering-immune chiral modes arise along certain line defects in three-dimensional materials. In this paper, we study 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 parameterized by $(k_x,k_y,k_z,\theta ,t)$. This work demonstrates the possibility of creating chiral modes in three-dimensional bulk materials by modulated driving. We hope that it will stimulate further studies of Floquet topological defects.