Fig. 4: Topological corner and edge states in the second band gap due to the anomalous quadrupole.

a Calculated acoustic spectrum for a box-shaped finite structure consisting of 10 × 10 sonic crystal unit cells enclosed by hard-wall boundaries with an air channel of width 0.25a separating them (see the inset for the structure illustration). b Frequency-resolved transmission spectra for three pump–probe configurations: bulk probe, edge probe and corner probe. Inset: photo of the sample with a zoom-in photo of the unit-cell. Source (“S”) and detector (“D”) for each pump–probe configuration (including the bulk, edge, and corner probes) are marked in the inset using the same color as that of the corresponding transmission curve. c, d Measured and simulated acoustic pressure profiles of the corner states, respectively. Four degenerate corner states (related by the C4 rotation) are shown in d, whereas only one of them is measured in c. Unit-cell geometry parameters are the same as that in Fig. 1.