Non-Abelian Floquet braiding and anomalous Dirac string phase in periodically driven systems

Abstract While a significant fraction of topological materials has been characterized using symmetry requirements 1–4 , the past two years have witnessed the rise of novel multi-gap dependent topological states 5–9 , the properties of which go beyond these approaches and are yet to be fully explored. Although already of active interest at equilibrium 10–15 , we show that the combination of out-of-equilibrium processes and multi-gap topological insights galvanize a new direction within topological phases of matter. We show that periodic driving can induce anomalous multi-gap topological properties that have no static counterpart. In particular, we identify Floquet-induced non-Abelian braiding, which in turn leads to a phase characterized by an anomalous Euler class, being the prime example of a multi-gap topological invariant. Most strikingly, we also retrieve the first example of an ‘anomalous Dirac string phase’. This gapped out-of-equilibrium phase features an unconventional Dirac string configuration that physically manifests itself via anomalous edge states on the boundary. Our results not only provide a stepping stone for the exploration of intrinsically dynamical and experimentally viable multi-gap topological phases, but also demonstrate periodic driving as a powerful way to observe these non-Abelian braiding processes notably in quantum simulators.