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Topological phases of matter span a wide area of research shaping fundamentalpursuits and offering promise for future applications. While a significantfraction of topological materials has been characterized using symmetryrequirements of wave functions, the past two years have witnessed the rise ofnovel multi-gap dependent topological states, the properties of which go beyondthese approaches and are yet to be fully explored. Thriving upon theseinsights, we report on uncharted anomalous phases and properties that can onlyarise in out-of-equilibrium Floquet settings. In particular, we identifyFloquet-induced non-Abelian braiding mechanisms, which in turn lead to a phasecharacterized by an anomalous Euler class, the prime example of a multi-gaptopological invariant. Most strikingly, we also retrieve the first example ofan `anomalous Dirac string phase'. This gapped out-of-equilibrium phasefeatures an unconventional Dirac string configuration that physically manifestsitself via anomalous edge states on the boundary. Our results therefore notonly provide a stepping stone for the exploration of intrinsically dynamicaland experimentally viable multi-gap topological phases, but also demonstrate apowerful way to observe these non-Abelian processes notably in quantumsimulators.

Floquet multi-gap topology: Non-Abelian braiding and anomalous Dirac string phase