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Symmetry and topology are two fundamental aspects of many quantum states ofmatter. Recently, new topological materials, higher-order topologicalinsulators, were discovered, featuring, e.g., bulk-edge-corner correspondencethat goes beyond the conventional topological paradigms. Here, we discoverexperimentally that the nonsymmorphic $p4g$ acoustic metacrystals host asymmetry-protected hierarchy of topological multipoles: the lowest band gap hasa quantized Wannier dipole and can mimic the quantum spin Hall effect, whilethe second band gap exhibits quadrupole topology with anomalous Wannier bands.Such a topological hierarchy allows us to observe experimentally distinct,multiplexing topological phenomena and to reveal a topological transitiontriggered by the geometry-transition from the $p4g$ group to the $C_{4v}$ groupwhich demonstrates elegantly the fundamental interplay between symmetry andtopology. Our study demonstrates an instance that classical systems withcontrollable geometry can serve as powerful simulators for the discovery ofnovel topological states of matter and their phase transitions.

Symmetry-protected hierarchy of anomalous multipole topological band gaps in nonsymmorphic metacrystals