Hierarchical multipole topological insulators.

Topological insulators have stimulated tremendous interest in quantum materials with exotic bulk and edge states that may have applications in spintronics and quantum computation. Recently, it was predicted and observed that an insulator with a quantized quadrupole moment hosts gapped edge states and in-gap corner states, demonstrating a new paradigm of higher-order topology. Such quadrupole band topology, however, is exclusive to conventional band topology based on electric dipole polarizations. Here, we discover a new topological system, denoted as the hierarchical multipole topological insulator, which exhibits multiplexing topological phenomena with concurrent dipole and quadrupole band topologies, in sonic crystals with nonsymmorphic p4g symmetry. Unlike electronic systems where topological phenomena are constrained by the band filling, acoustic systems allow access to multiple topological band gaps by changing the frequency. In our sonic crystals, the first band gap has dipolar topology which can realize acoustic quantum spin Hall effect and topological helical edge states, while the second band gap realizes a novel anomalous quadrupole topological insulator protected by the nonsymmorphic crystalline symmetry. We observe the experimental signatures of the dipolar and quadrupolar band topologies through measurements of the corner and edge states. Our study establishes a bridge between multipole topology and subwavelength metamaterials beyond tight-binding pictures, and demonstrates an instance that classical systems can serve as powerful and elegant simulators for the discovery of novel topological quantum states.