Acoustic Topological Metamaterials of Large Winding Number

Longer-range interactions, or couplings beyond the second nearest neighbors, have long been overlooked in topological models for either topological insulators or semimetals. For natural (electronic) materials, such ignorance is reasonable as near-range couplings are always dominant. However, artificially constructed metamaterials can break such a limitation, allowing the longer-range couplings comparable with the nearer ones, achieving alternative properties unattainable in natural materials. Here, we report a one-dimensional acoustic metamaterial designed in terms of an extended Su-Schrieffer-Heeger (SSH) model, containing the nearest, and in particular the third-nearest couplings. In contrast to the conventional SSH model with the topological phases of the winding number one, the extended SSH model can host topological phases of winding number two, besides one, exhibiting amazing rotonlike dispersions. The larger winding number implies more end states emerging. All these predictions are confirmed by the experiments with the fabricated acoustic metamaterials. Such longer-range couplings can also be extended to two- or three-dimensional lattices, to further achieve the topological states exceptionally in metamaterials.

Figure 1

Tight binding model for an extended SSH chain. (a) A schematic of the lattice with the nearest intracell and intercell couplings, $w_1$ and $v_1$, and the third-nearest ones, $w_3$ and $v_3$. The blue dashed box manifests one unit cell. (b) Phase diagram with respect to $w_3$ and $v_3$, obtained by fixing $w_1=-1.5$ and $v_1=-0.5$. The color regions are determined by absolute values of the bulk winding numbers $|W|$. The black solid lines denote phase boundaries corresponding to gap closure. The A, B, and O points label three representative states of different phases. (c) Bulk band dispersion of state A. (d) Winding loop for the first band of state A. (e) Eigenmodes of finite chains (with 120 sites) in states A, B, and O, respectively. (f),(g) Evolutions of the energy band and the winding number along the specific path denoted by gray dashed lines in (b).

Figure 2

Acoustic metamaterial with stronger third-range couplings. (a) Photo of the 3D printing acoustic metamaterial sample. (b) Schematic structure of the sample, containing three unit cells. (c) Bulk dispersion along the $k_x$ direction. The color maps and circles represent the experimental and numerical data, respectively. (d) Left panel: enlarged band structure of the lower band in (c). The color segments denote the fitting bands. Right panel: the ratio between the group and phase velocities corresponding to the segments in left panel. (e) Pseudospin winding distributions for the upper and lower bands.

Figure 3

Fourfold degenerate end states of the acoustic metamaterial. (a) Left panel: simulated eigenfrequencies for the sample containing 41 unit cells. Red and gray balls present end and bulk states. Right panel: measured response spectra (RS) of bulk (gray dashed curves) and end (red curve) states. The light blue bar highlights the band gap. (b) Simulated sound pressure fields of two pairs of the end states. (c) Measured sound pressure level (SPL) distribution at the frequency of $3.25\mathrm{kHz}$. (d),(e) Measured phase profiles along the z direction in the end resonators.

Figure 4

Schematic of the experimental setup.