Observation of Degenerate Zero-Energy Topological States at Disclinations in an Acoustic Lattice

Building upon the bulk-boundary correspondence in topological phases of matter, disclinations have recently been harnessed to trap fractionally quantized density of states (DOS) in classical wave systems. While these fractional DOS have associated states localized to the disclination’s core, such states are not protected from deconfinement due to the breaking of chiral symmetry, generally leading to resonances which, even in principle, have finite lifetimes and suboptimal confinement. Here, we devise and experimentally validate in acoustic lattices a paradigm by which topological states bind to disclinations without a fractional DOS but which preserve chiral symmetry. The preservation of chiral symmetry pins the states at the midgap, resulting in their protected maximal confinement. The integer DOS at the defect results in twofold degenerate states that, due to symmetry constraints, do not gap out. Our study provides a fresh perspective about the interplay between symmetry protection in topological phases and topological defects, with possible applications in classical and quantum systems alike.

Figure 1

(a) Top: the top view of a conventional straight tube coupled-cavity model and a coiled tube coupled-cavity model. Bottom: a 3D view of a coiled tube coupled-cavity model. (b) Frequency spectrum of the coupled cavity system with coiled coupling tubes shown in (a), bottom, as a function of the ratio between $a_0/h_0$. The blue markers represent the frequencies of the symmetric modes (lower frequency) and the antisymmetric modes (higher frequency).The average frequencies of the symmetric and antisymmetric modes are marked with red circles. The symmetric and antisymmetric modes are distributed symmetrically about the zero-energy level (4289 Hz) at $a_0/h_0=0.75$, indicating chiral symmetry. (c) The top view of the OAL(3c) honeycomb lattice with coiled coupling tubes, where the external coupling tubes are situated at the bottom of the cavity and the internal coupling tubes are located at $h_0/4$ above the center of the cavity. The figure on the top right shows a single unit cell with coiled coupling tubes.

Figure 2

(a) The OAL(3c) lattice with a $2\pi /3$ disclination. Each shade represents one unit cell. Two different sublattices are distinguished by red and blue circles. The inset figure shows a unit cell with its Wannier centers at half filling at Wyckoff position 3c. (b) Numerically computed eigenfrequencies for the OAL(3c) structure. The topological corner states, edge states, and trivial corner states are represented by red, green, and brown circles, respectively. (c) The four degenerate topological corner states at 4304 Hz. (d) The OAL(1a) lattice with a $2\pi /3$ disclination. The inset figure shows a unit cell with its Wannier centers at half filling at Wyckoff position 1a (threefold degenerate). (e) Numerically computed eigenfrequencies for the OAL(1a) structure. (f) The pair of degenerate disclination bound states at 4285 Hz. The dotted lines highlight the quadrants. Only the region surrounding the lattice core is shown for better visualization.

Figure 3

(a) Top panel: the acoustic OAL(1a) lattice. Only the inner three by 3 unit cells are shown here for better visualization. Bottom panel: a close-up view of three cavities in the dashed line box shows the position of the external and internal coupling tubes. The transparent cut plane indicates the interface between the two layers used to construct the experimental acoustic sample. (b) Photographs of the OAL(3c) acoustic lattice sample with its cavities (the larger holes) and coupling channels. The two blocks are stacked and then sealed to form the coupled-cavity lattice. The smaller holes without tubes are for mounting purposes. (c) Spectra of the normalized pressure amplitude $|p|$ of the disclination (purple) and bulk (gray) states. The degenerate disclination states are marked with the red star (two degenerate states at 4340 Hz). (d) The pressure distribution maps of the two disclination states at the frequency marked by the red star in (c). The area of the circle represents the amplitude of the pressure. Note that the entire lattice is measured, but the pressure amplitudes are too weak away from the disclination core.