Topological effects in chiral symmetric driven systems

Recent years have seen a strong interest in topological effects within periodically driven systems. In this work, we explore topological effects in two closely related 2-dimensional driven systems described by Floquet operators possessing chiral symmetry (CS). Our numerical and analytical results suggest the following. First, the CS is associated with the existence of the anomalous counterpropagating (ACP) modes reported recently. Specifically, we show that a particular form of CS protects the ACP modes crossing the quasienergy band gap at +/-pi. We also find that these modes are only present along selected boundaries, suggesting that they are a weak topological effect. Second, we find that CS can give rise to protected 0 and pi quasienergy modes, and that the number of these modes may increase without bound as we tune up certain system parameters. Like the ACP modes, these 0 and pi modes also appear only along selected boundaries and thus appear to be a weak topological effect. This work represents a detailed study of weak topological effects in periodically driven systems. Our findings add to the still-growing knowledge on driven topological systems.