Three-dimensional higher-order topological acoustic system with multidimensional topological states

Topologically protected gapless edge/surface states are phases of quantum matter which behave as massless Dirac fermions and are immune to disorders and continuous perturbations. Recently, a new class of topological insulators (TIs) with gapped edge states and in-gap corner states has been theoretically predicted in electronic systems and experimentally realized in two-dimensional (2D) mechanical and electromagnetic systems, electrical circuits, optical and sonic crystals, and elastic phononic plates. Here we elaborately design a three-dimensional topological acoustic system by arranging acoustic meta-atoms in a simple cubic lattice. Under the direct field measurements, besides the 2D surface propagations on all of the six surfaces, the one-dimensional hinge propagations behaving as acoustic fibers along the 12 hinges, and the zero-dimensional corner modes working as localized resonances at the eight corners are experimentally confirmed. As these multidimensional topological states are activated in different frequencies and independent spaces, our works may provide a new pathway for designing high-performance integrated acoustic devices that can manipulate the propagation of sound waves in multiple dimensions.

Figure 1

(a) Schematic presentation of the 3D SSH model. The unit cell contains eight nodes, highlighted by a gray box. The red and blue bonds represent intracell ($\gamma$) and intercell ($\lambda$) couplings. (b) A realistic design of the 3D acoustic system on the basis of the 3D SSH model. The insert is a unit cell including eight meta-atoms, each of which consists of a cavity and six channels. The widths of channels within and between unit cells are $d_1$ and $d_2$, representing intracell and intercell couplings. (c), (d) Dispersion relations of the unit cell with $d_1>d_2$ and $d_1<d_2$, respectively.

Figure 2

(a) A sketch of the supercell composed of $3\times 3$ unit cells aligned in the $xy$ plane with periodic boundary condition along the $z$ direction. (b), (c) Simulated acoustic pressure field profiles of the surface and hinge states, respectively. (d) Simulated dispersions of the supercell composed of $3\times 3$ trivial unit cells with $d_1>d_2$. In the band gap of bulk states, there are no surface and hinge states. (e) Simulated dispersions of the supercell composed of $3\times 3$ nontrivial unit cells with $d_1<d_2$. The surface (green) and hinge (blue) states emerge in the band gap of bulk states. (f), (g) Simulated dispersions of the surface and hinge states, respectively.

Figure 3

(a), (b) Numerical models of the higher-order acoustic networks with $3\times 3\times 3$ trivial and nontrivial unit cells, respectively. (c), (d) Numerically evaluated eigenfrequencies of acoustic networks with trivial and nontrivial unit cells. Black, green, blue, and red dots denote the bulk, surface, hinge, and corner modes, respectively. (e)–(h) Simulated acoustic pressure field profiles of bulk (4466.1 Hz), surface (5135.3 Hz), hinge (5011.4 Hz), and corner (4883.1 Hz) modes.

Figure 4

(a) The complete acoustic network system with $N\times N\times N$ nontrivial unit cells composed of $2N\times 2N\times 2N$ cubic air cavities. By removing one side of the cubic air cavities from the complete acoustic network system in the $xy$, $xz$, and $yz$ planes, respectively, as shown in the blue area, the incomplete acoustic network system composed of $\left(2N-1\right)\times \left(2N-1\right)\times \left(2N-1\right)$ cubic air cavities can be obtained. Here, $N=3$. (b), (c) The eigenvalues of topological corner states of the complete and incomplete acoustic network system as a function of $N$, respectively. (d) The simulated acoustic pressure field profiles of topological corner modes in the incomplete acoustic network system with $N=3$.

Figure 5

The simulated acoustic pressure field profiles of topological corner modes in the complete acoustic network system with $N=6$. Only topological corner modes of one of the topologically protected corners are shown here.

Figure 6

(a) Photograph of the fabricated acoustic system consisting of $3\times 3\times 3$ nontrivial unit cells. (b) Measured bulk (black), surface (green), hinge (blue), and corner (red) transmission spectra for a topological hexagon-shaped sample. (c), (d), (e) Measured pressure field profiles for the surface (5136 Hz), hinge (5010 Hz), and corner (4888 Hz) states, respectively. Full wavefield numerical simulations of the topological acoustic system are constructed in Fig. S3 (Supplemental Material [36]).