Hybrid Acoustic Topological Insulator in Three Dimensions

Topological insulators (TIs), featured by a symmetry-protected gapless surface Dirac cone(s) in their complete energy band gaps, have been extended from condensed-matter physics to classical bosonic systems in the last decade. However, acoustic TIs in three dimensions remain elusive because of a lack of a spin or polarization degree of freedom for longitudinal airborne sound. Here, we experimentally demonstrate a feasible way to hybridize an acoustic TI in three dimensions based on band inversion through a three-dimensional (3D) hybrid Dirac point (HDP). Such a 3D HDP, with linear dispersion in the layer plane while quadratic out of the layer, is distinct from a general point with linear dispersion in all directions. Interestingly, a single nearly gapless conical-like dispersion for acoustic surface states can be achieved at both zigzag and armchair interfaces, supporting robust sound transport. Our findings can serve as a tabletop platform for exploring unique acoustic applications based on the two-dimensional topological interfaces.

Figure 1

(a) Schematic of a 3D hybrid Dirac point for bulk bands, where the color scale represents frequency. (b) Projection hybrid Dirac cone to the $k_{xy}$ and $k_{zy}$ planes, where the red or black line shows linear or nonlinear dispersion along different directions. (C) Hybrid TI case after symmetry breaking, where the red surface represents the directionally linear surface states.

Figure 2

(a) Schematic of the atomic structure of acoustic crystal constructed by double-layer honeycomb lattice. The $A$ layer (blue) is formed by identical atoms, and the $B$ layer (green) is formed by two different atoms. (b) One-half of the bulk BZ and surface BZ in the $k_{yz}$ plane (100-surface BZ) and $k_{xz}$ plane (010-surface BZ). (c) The triangular prism cavities and connecting tubes represent the acoustic atoms and their couplings. (d) Bands inversion by stretching and compressing in the $xy$ plane. (e)–(g), Bulk band structures at (e), $b/a=0.57$, (f), $b/a=2/3$, and (g), $b/a=0.73$. Insets show top-down views of the acoustic states of the $A$ layer. (h) Schematic of the zigzag interface with topological trivial and nontrivial acoustic crystals. (i) The projected band structures, where the blue regions represent the overlapping bulk pass band and the red lines represent the topological surface states. (j) 3D view of acoustic surface states near the $\tilde{Z}$ point with a tiny surface band gap of less than 0.2%.

Figure 3

(a) Images of the experimental sample. Lower left: magnified top-down view. Lower right: magnified lateral view. Blue lines indicate the interface. (b) Experimental configuration to measure directional transmission spectra of surface states, where $\theta$ represents the angle. (c) Three 2D-$k$ slices on conical-like surface dispersion with $\theta =0$, arctan($0.5a/8h$) and arctan($a/8h$), corresponding to $\mathrm{\Delta }=0$, 0.5, and 1 cm in the experiments, respectively. (d) Experimentally measured transmission spectra. The black and gray lines correspond to the bulk band transmission of two acoustic crystals along the $\mathrm{\Gamma }A$ direction, where the shadow regions indicate the overlap of the bulk pass band. The surface transmission spectra for slices 1–3 are represented by red, cyan, and blue lines, respectively. (e) Experimental configuration to map the surface dispersion along the $\tilde{\mathrm{\Gamma }}\tilde{Z}$ direction. (f) A slice at $k_y=0$. (g) The measured surface dispersion. The color scale represents the energy density. The dashed lines are drawn to guide the eye.

Figure 4

(a) Schematic of armchair interface. (b) The projected band structures, where the red lines represent the surface states. (c) 3D view of surface states near the $\tilde{Z}$ point. (d) The measured surface dispersion at the slice of $k_x=0$. The color scale represents the energy density. The dashed lines are drawn to guide the eye.