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

Symmetry and topology are two fundamental aspects of many quantum states of matter. Recently, new topological materials, higher-order topological insulators, were discovered, featuring, e.g., bulk-edge-corner correspondence that goes beyond the conventional topological paradigms. Here, we discover experimentally that the nonsymmorphic $p4g$ acoustic metacrystals host a symmetry-protected hierarchy of topological multipoles: the lowest band gap has a quantized Wannier dipole and can mimic the quantum spin Hall effect, while the 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 transition triggered by the geometry-transition from the $p4g$ group to the $C_{4v}$ group which demonstrates elegantly the fundamental interplay between symmetry and topology. Our study demonstrates an instance that classical systems with controllable geometry can serve as powerful simulators for the discovery of novel topological states of matter and their phase transitions.