Fig. 1. Symmetry-protected hierarchy of anomalous topological multipoles in nonsymmorphic sonic crystals.

a A bird’s-eye view of the 2D nonsymmorphic sonic crystal cladded by plastic boards from below and above (not shown). The inset illustrates the unit-cell structure with four arch-shaped scatterers made of photosensitive resin (an acoustically rigid material). b Acoustic bands and the hierarchy of multipole topological band gaps. The dipole and (anomalous) quadrupole are quantized and protected by the glide symmetries in the p4g group. Inset: Brillouin zone. Symbols +/− represent even/odd parity, respectively, at the Γ and X points in the Brillouin zone. c Gapped and nondegenerate Wannier bands for the anomalous quadrupole topological gap. d Wannier bands for the sum (“1 + 2” and “4 + 3”) and difference (“1 − 2” and “4 − 3”) sectors. e Schematic illustration of the linear combinations of the Wannier orbits in different sectors (“1 + 2” and “1 − 2”), which respectively yield the dipole topology and the anomalous quadrupole topology. Unit-cell geometry parameters are h = 0.21a, l = 0.42a, w = 0.1a, and a = 2 cm.