Realization of an Acoustic Third-Order Topological Insulator

The recent discovery of higher-order topological insulators (TIs) has opened new possibilities in the search for novel topological materials and metamaterials. Second-order TIs have been implemented in two-dimensional (2D) systems exhibiting topological “corner states,” as well as three-dimensional (3D) systems having one-dimensional (1D) topological “hinge states.” Third-order TIs, which have topological states three dimensions lower than the bulk (which must thus be 3D or higher), have not yet been reported. Here, we describe the realization of a third-order TI in an anisotropic diamond-lattice acoustic metamaterial. The bulk acoustic band structure has nontrivial topology characterized by quantized Wannier centers. By direct acoustic measurement, we observe corner states at two corners of a rhombohedronlike structure, as predicted by the quantized Wannier centers. This work extends topological corner states from 2D to 3D, and may find applications in novel acoustic devices.

Figure 1

Structure and bulk dispersion of acoustic anisotropic diamond lattice. (a) Schematic of the cubic cell of an anisotropic diamond lattice. Red (blue) bonds indicate couplings with strength $t_{2(1)}$. (b) The unit cell of the acoustic anisotropic diamond lattice. All parameter values are given in the main text. (c) Acoustic pressure profile for the resonator eigenmode of interest in this work. (d) The first Brillouin zone of the anisotropic diamond lattice. (e) Simulated bulk bands of acoustic anisotropic diamond lattice shown in (b). Red curves show the band structure for $d{r}_c=0\mathrm{mm}$ ($r_{c1}=r_{c2}=4.9\mathrm{mm}$), and blue curves show the band structure for $d{r}_c=5.8\mathrm{mm}$ ($r_{c1}=2$ and $r_{c2}=7.8\mathrm{mm}$), where $d{r}_c=r_{c2}-r_{c1}$.

Figure 2

Topological corner states on a rhombohedronlike sample. (a) Schematic of the rhombohedronlike structure. (b) ($\overline{1}\overline{1}1)$ surface atoms of the structure shown in (a). (c) Simulated eigenfrequencies of the finite acoustic lattice. Black and red dots represent bulk and corner states, respectively. (d),(e) Simulated eigenmode profiles of the two corner states. (f) Photograph of the fabricated rhombohedronlike sample containing 52 resonators. The inset shows an enlarged view of a resonator with one stopper removed.

Figure 3

Experimental observation of corner states in a rhombohedron-like acoustic structure. (a) Measured bulk transmission spectrum. (b) Measured spectra at the two corner sites indicated in (c) and (d). (c),(d) Measured pressure maps at 2900 Hz, viewed from the front (c) and back (d). The illustration of the structure is simplified for clarity; for the real structure see Figs. 2. The balls correspond to the cylindrical resonators and the gray bonds indicate the couplings. The color of each ball represents the measured acoustic pressure at that site, as measured at the top of the cylindrical resonator. The color map does not apply to the bonds.

Figure 4

Experimental demonstration of surface states on a tetrahedronlike acoustic structure. (a) Photograph of the fabricated sample. Black dashed line denotes the (111) surface. (b) Measured spectra at four surface resonators on four surfaces of the sample. The source and probe are located at the same resonator for each curve. (c),(d) Measured pressure maps at 2900 Hz, viewed from the front (c) and back (d).