Non-Abelian braiding of topological singularities and their signatures

Recent years have witnessed the rise of novel multi-gap dependent topological states which arise from intricate non-Abelian braiding procedures involving multiple gaps in a system and go beyond the now-well-established symmetry-based classification schemes. While their properties are yet to be fully explored both in and out of equilibrium, the first implementations have already started to appear in experimental quantum simulation settings. We here consider periodically driven multi-gap systems and introduce novel anomalous non-Abelian phases that can only arise out of equilibrium and via braiding processes, which require new invariants to be defined to capture these phases. The underlying non-Abelian charges can be however quite elusive, and consequently the direct measurement of their charges as well as of the Euler invariant is still missing. For this, we here construct concrete interferometry methods to experimentally probe these non-Abelian braiding processes and charges in ultracold atomic systems.