Fig. 3.

Topological hinge states: boundary with codimension two. a Calculated acoustic dispersions of the bulk (gray), surface (orange), and hinge (dark blue) states for the supercell with both surfaces and hinges (see the inset). b Photograph of the fabricated hinge sample, consisting of the topological SC (with $8\times 8\times 10$ unit cells) surrounded by the trivial SC in both the $x$- and $y$-directions (of thickness $4a$). c Measured transmission spectra for four types of pump–probe configurations: the bulk probe (gray), two surface probes (orange), and the hinge probe (dark blue). The inset depicts the positions of the source and the probes for the four pump–probe configurations (the color of each probe is the same as in the transmission spectra). d Acoustic pressure profile for the hinge states measured at the excitation frequency of 5.7 kHz (indicated by the black arrow in c) and presented using slice plots (the locations of the slices are indicated in b).