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Exploring new topological phases and phenomena has become a vital topic in condensed matter physics and materials sciences. Recent studies reveal that a braided colliding nodal pair can be stabilized in a multi-gap system with $$PT$$ P T or $${C}_{2z}T$$ C 2 z T symmetry. This exemplifies non-abelian topological charges beyond the scope of conventional single-gap abelian band topology. Here, we construct ideal acoustic metamaterials to realize non-abelian braiding with the fewest band nodes. By emulating the time with a sequence of acoustic samples, we experimentally observe an elegant but nontrivial nodal braiding process, including nodes creation, braiding, collision, and repulsion (i.e., impossible to annihilate), and measure the mirror eigenvalues to elucidate the braiding consequence. The latter, at the level of wavefunctions, is of prime importance since essentially braiding physics aims to entangle multi-band wavefunctions. Furthermore, we experimentally unveil the highly intricate correlation between the multi-gap edge responses and the bulk non-abelian charges. Our findings pave the way for developing non-abelian topological physics that is still in its infancy.

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Symmetry plays a key role in classifying topological phases. Recent theory shows that in the presence of gauge fields, the algebraic structure of crystalline symmetries needs to be projectively represented, which enables unprecedented topological band physics. Here, by exploiting the concept of projective symmetry, we report an acoustic realization of mirror Chern insulators that are widely believed to be impossible in spinless systems. More specifically, we introduce a simple but universal recipe for constructing projective mirror symmetry, and conceive a minimal model for achieving the projective symmetry-enriched mirror Chern insulators. Based on our selective-excitation measurements, we demonstrate unambiguously the projective mirror eigenvalue-locked topological nature of the bulk states and associated chiral edge states. More importantly, we extract the non-abelian Berry curvature and identify the mirror Chern number directly, as conclusive experimental evidence for this exotic topological phase. All experimental results agree well with the theoretical predictions. Our findings will shine new light on the topological systems equipped with gauge fields.