Fig. 4

Topological corner states: boundary with codimension three. a Photograph of the fabricated corner sample, consisting of the topological SC (size: \(8\,\times 8\,\times 8\) unit cells) enclosed by the trivial SC (thickness \(4a\)) in all three directions. To ensure pump–probe experimental measurements, the corner sample is cut to allow open interfaces, as shown in the figure. b Measured transmission spectra for four types of pump–probe configurations: the bulk probe (gray), the surface probe (orange), the hinge probe (dark blue), and the corner probe (dark green). The inset depicts the locations of the source and the surface, hinge, and corner probes (the same colors as in the transmission spectra). The location of the bulk probe (not shown in the inset) is in the middle of the topological SC. c Acoustic pressure profile for the corner states excited at the frequency of 7.9 kHz (indicated by the black arrow in b) and presented using slice plots (the locations of the slices are indicated in the left panel).