Quantized octupole acoustic topological insulator

The Berry phase associated with energy bands in crystals can lead to quantized quantities, such as the quantization of electric dipole polarization in an insulator, known as a one-dimensional (1D) topological insulator (TI) phase. Recent theories have generalized such quantization from dipole to higher multipole moments, giving rise to the discovery of multipole TIs, which exhibit a cascade hierarchy of multipole topology at boundaries of boundaries: A quantized octupole moment in the three-dimensional (3D) bulk can induce quantized quadrupole moments on its two-dimensional (2D) surfaces, which then produce quantized dipole moments along 1D hinges. The model of 2D quadrupole TI has been realized in various classical structures, exhibiting zero-dimensional (0D) in-gap corner states. Here we report on the realization of a quantized octupole TI on the platform of a 3D acoustic metamaterial. By direct acoustic measurement, we observe 0D corner states, 1D hinge states, 2D surface states, and 3D bulk states, as a consequence of the topological hierarchy from octupole moment to quadrupole and dipole moment. The critical conditions of forming a nontrivial octupole moment are further demonstrated by comparing with another two samples possessing a trivial octupole moment. Our work thus establishes the multipole topology and its full cascade hierarchy in 3D geometries.

topological phases, has remained as an open question until recent seminal works on quantized electric multipole TIs 5,6 . Take the quantized quadrupole as an example. It has been shown in various classical systems 7-12 that a 2D quantized quadrupole TI exhibits topological states at "boundaries of boundaries": it lacks gapless 1D topological edge states, as in conventional 2D topological insulators, but instead hosts topologically protected 0D corner states. Such a generalized bulk-boundary correspondence has opened the door to the pursuit of higher-order TIs 5-12, 15-18 that can arise from both quantized dipole and multipole moments.
The most striking feature of multipole TIs is their cascade hierarchy of topology. Fig. 1a schematically illustrates such a three-level hierarchy. At the lowest level, the corner states (Q) are a direct result of quantized dipoles (px, py, pz), similar to the well-known 1D TIs 4 . At the intermediate level, the quantized dipoles are induced along 1D edges by the nontrivial quantized quadrupole (qxy, qyz, qzx) in a 2D bulk, as demonstrated in previous works [7][8][9][10][11][12] . At the highest level, the quantized quadrupoles arise on 2D surfaces of a 3D bulk which exhibit a nontrivial quantized octupole (oxyz). From the perspective of the cascade hierarchy of multipole topology, the previously observed quadrupole TI phases [7][8][9][10][11][12] can be treated as the 2D projection of a 3D octupole TI, and the 1D TI phases appear as 1D projection along the hinges. To our knowledge, the 3D octupole TI as well as its cascade hierarchy of multipole topology has never been observed.
Here we report on the observation of an octupole TI in an acoustic metamaterial. This metamaterial consists of coupled acoustic resonators, whose local resonances serve as artificial atomic orbitals and thus fulfil the description of tight-binding models 5,6 . The positive and negative couplings 7,8 between resonators are achieved by connecting the resonators with thin waveguides at different sides of resonance's nodal line. By direct local acoustic measurements sweeping over all resonators of a finite sample, we are able to observe the in-gap 0D corner states, gapped 1D 4 hinge states, gapped 2D surface states, and gapped 3D bulk states, all of which arise from the nontrivial octupole moment (oxyz = 1/2) in the bulk. Furthermore, we corroborate our results by measuring another two samples with trivial octupole moments (oxyz = 0). These trivial samples exhibit completely different boundary signatures from those in the nontrivial sample. Our results demonstrate a full hierarchy of multipole topology from octupole moment to quadruple and dipole moment, which is unique to a quantized octupole TI.
The lattice model to realize an octupole TI 5,6 is shown in Fig. 1b, where intra-cell (left panel) and inter-cell (right panel) couplings has strength of γ and λ. Solid and dashed lines denote positive and negative couplings, respectively. The unit cell can be regarded as two coupled unit cells of quadrupole TIs with opposite settings. Due to the negative couplings, a π flux exists on each facet, which not only opens a spectrum gap at half-filling, but also maintains the mirror symmetries (up to a gauge transformation). In the presence of mirror and inversion symmetries, the bulk octupole Oxyz moment can only take quantized values of 0 or 1/2, depending on the relative strength of γ and λ 5,6 .
We adopt coupled acoustic resonators [19][20][21][22][23][24] to implement this model. The unit cell design is illustrated in Fig. 1c. Here each lattice site is a hard-wall cuboid resonator filled with air of size 80 mm × 40 mm × 10 mm. The dimensions are carefully designed so that a single resonator supports a dipole resonance mode (Fig. 1d, left panel) at f0 = 2162.5 Hz, which is far away from other modes (Fig. 5a). The nearest neighbor coupling is realized by connecting resonators with thin waveguides.
As shown in Fig. 1d, it is well known that when two resonators are coupled together with a positive coupling coefficient γ, the resulted eigenmodes will exhibit symmetric and antisymmetric phase relations with split eigenfrequencies of f0+ γ and f0-γ. If the sign of coupling is flipped, then the eigenmodes also exchange 25 . To achieve that, we simply relocate the connecting waveguide to the 5 other side of the dipole mode's nodal line. As plotted in the right panel of Fig. 1d, the two eigenmodes switch their eigenfrequencies. In addition, the amplitude of coupling is controlled by tuning the width of connecting waveguide. Following an optimization approach which takes into account effects such as the resonance frequency shift and the coupling between the dipole resonance mode and other resonance modes (Methods and Fig. 5), we design the acoustic octupole TI structure as in In contrast, the second sample possesses the intra-cell couplings larger than the inter-cell couplings (γ = 53.3 Hz > λ = 9.8 Hz). In such a case, { , , } = {0, 0, 0}, meaning that only bulk states emerge and no boundary signatures can be found (Figs. 2b, e). The third sample has a more subtle feature. The intra-cell couplings are smaller than the inter-cell ones along x and y directions (γx,y = 9.8 Hz < λx,y = 53.3 Hz). In the z direction, however, intra-cell couplings are larger than the inter-cell ones (γz = 53.3 Hz > λz = 9.8 Hz). In such a case, the Wilson loops yield the topological indices of { , , }={0.5, 0.5, 0}, leading to hinge states along z and surface states on surfaces normal to x and y directions, apart from the bulk states (Figs. 2d, f).
We then probe above signatures of the octupole TI experimentally. We measure the acoustic response upon a local excitation at each site of the first sample ( Fig. 3a) consisting of 1000 sites.
Left panel of Fig. 3b shows measured results at an arbitrary frequency (2000 Hz). We divide the whole sample into four regions, as schematically shown in the right panel of Fig. 3b. The "corner" region consists of 8 sites at corners. The "hinge" region refers to 12 hinges, each of which covers 8 sites. The "surface" region has 6 surfaces, each of which is composed of 64 sites. The other 512 sites constitute the "bulk" region. The resulted average intensity spectra (see Methods for details) for different regions are shown in Fig. 3c. Peak frequencies of the spectra are in agreement with the calculation in Fig. 2a. We further plot the integrated intensity spectra over frequencies (integration regions are indicated in Fig. 3c) around peaks of corresponding spectra. As shown in Figs. 3d-g, the corner, hinge, surface and bulk states are successfully identified. Note that the measured intensities on surfaces are larger than those in the bulk in Fig. 3f, but are smaller than in the bulk in Fig. 3g. The most pronounced feature here is the in-gap corner states, which is away 7 from other states and thus can be clearly observed (Fig. 3d). These corner states are consequences of nontrivial bulk and feature unique robustness (Fig. 8), which have great potential for applications. The hinge, surface and bulk states are closer in frequency, leading to overlaps in the integrated intensity spectra (Figs. 3e-g).
Similar measurements are also conducted on the second and third samples. For the second sample, since there are no boundary states, the measured average intensity spectra for different regions have identical peaks (Fig. 4a). For the third sample, because of the anisotropy, we further treat hinges and surfaces along different directions separately. Because there is no corner state, the measured spectra in the corner region shows overlapped peaks as those along z direction hinges

Competing interests
The authors declare no competing interests.

Data availability
The data that support the findings of this study are available from the corresponding authors on reasonable request.

Design of the acoustic octupole topological insulator.
To build an acoustic metamaterial that can be mapped to the desired tight-binding model, we make use of two kinds of building blocks: resonators and coupling waveguides. Here a resonator is a hard-wall cavity filled with air, which supports various modes at different frequencies. The mode of our interest is a dipole mode which has a nodal line that can be used to achieve negative couplings. The cavity is designed to have a size of 80 mm × 40 mm × 10 mm to achieve a large separation between the dipole mode of our interest and other modes (Fig. 5a). Then we couple the resonators with small coupling waveguides, which is also hard-wall and filled with air. The strategy to realize altering sign of the coupling is just to tune the position of waveguide so that different parts of the nodal line are connected (see Fig. 1d in the main text). The coupling strength is controlled by the width of the waveguide. With the building blocks and construction strategies mentioned above, we proceed to implement the octupole TI model. Noticing the tight-binding model for a quantized octupole TI (Fig. 1b in the main text) is constructed from two quadrupole TIs with opposite settings coupled along z direction, we firstly design the two layers as shown in Figs. 5b, c. Then these two layers are coupled along the third direction with dimerized coupling strength to build the acoustic octupole TI. Note along z direction we use two coupling waveguides to connect adjacent resonators in order to firmly hold the structure.
With the preliminary design in hand, we now proceed to the optimization process. There are several issues we consider. Firstly, the design is generally anisotropic. Especially the coupling strengths along z do not equal to those in the xy plane. Secondly, the coupling waveguides will introduce resonance frequency shifts, which may be different at different sites. Thirdly, the coupling waveguides will not only couple the dipole modes of different sites, but also couple all other modes. Although the first issue will not destroy the quantization of octupole moment, the last two will and may even lead to topological phase transition if their effects are sufficiently strong.
Fortunately, recent studies 7,26 have introduced the method of SW transformation 32,33 to design metamaterials from tight-binding models. We extract the effective tight-binding model of our acoustic design by following steps. Firstly, we solve the lowest 5 eigenmodes of a single resonator using COMSOL Multiphysics, eigenmode solver. Then we also solve a sample containing eight unit cells (64 sites) for the lowest 320 modes (5 modes × 64 sites, excluding modes introduced by coupling waveguides). Next, we grid the data from simulations, resulting in matrices Uj whose ith column contain ith eigenmode at site j. Subsequently, the prepared data is projected onto the basis of single resonator eigenmodes, given by Pj=(A T A) -1 A T Uj where columns of matrix A contain the 13 eigenmodes of a single resonator. The coupling matrix V then can be calculated through V=PDP -1 -H0. Here P = ( P 1 ⋮ P 64 ), D is a square matrix with eigenfrequencies of the 8-unit-cell sample contained in its diagonal elements, and H0 is also a square matrix whose diagonal elements contain eigenfrequencies of single resonator and repeated 64 times. To get an effective description within the space of the mode of our interest, we perform SW transformation perturbatively 25,33 to account for the couplings between the mode of our interest and other modes as long range couplings among the mode of our interest. With the effective tight-binding model in hand, we then tune the widths of coupling waveguides and distances between resonators so that the nearest coupling strengths along z is almost the same as those in the xy plane, and the effects of resonate frequency shifts and long ranges couplings are small while the frequency gap is still large enough for experiments.
Besides, small holes are introduced to resonators located at the boundaries to shift their resonance frequencies to be the same as those in the bulk. In the final design, the lattice constants along x, y, z directions are 200 mm, 200 mm and 100 mm, respectively, and the widths for intra-cell (intercell) coupling waveguides are 1.6 mm (4 mm) in the xy plane and 1.34 mm (3.08 mm) along z direction. The final design has a ratio of γ/λ ≈ 0.18 with λ = 9.8×(1±4%) Hz and γ = 53.3×(1±4%) Hz, and the resonance frequencies of eight resonators in the unit cell are f0 = 2145.1×(1±0.05%) Hz. The dispersion calculated from effective tight-binding model matches well with the simulated one (Fig. 5d). Due to the large separation between the target mode and other modes, the nextnearest couplings in the extracted tight-binding model are quite small (Figs. 5e, f). Thus, the designed acoustic lattice deviates almost negligibly from the ideal tight-binding model.

Nested Wilson loops and topological invariants.
From the effective tight-binding model obtained by the method described in previous section, we can calculate the topological numbers from the nested Wilson loop method 5,6 . Firstly, we calculate a Wilson loop within the four bands 14 below the bandgap along z direction, yielding 2D Wannier bands (Fig. 6a). This procedure splits the original four bands which are almost degenerate in frequency into two Wannier sectors (labelled as + and − in Fig. 6b) that are spatially separated along z. Being gapped, the two Wannier sectors can carry their topological invariants. Next, we calculate a nested Wilson loop within one of the Wannier sectors (here we choose − ) along y, which again yields two separated Wannier bands, denoted as + and − in Fig. 6c In the experiments, the acoustic signal is launched from a balanced armature speaker, guided into the samples through a narrow tube (r = 1.5 mm) and collected by a microphone (Brüel&Kjaer Type 4182). Then the measured data is processed by Brüel&Kjaer 3160-A-022 module to get the frequency domain sprectrum.
In the measurements of site-resolved local response (Fig. 3 and Fig. 4 in the main text), the source and probe are always located at the same site. At each site i, we obtain the intensity of acoustic field normalized by the intensity of the source over a range of frequencies (1900 Hz -2400 Hz), Verification of π flux. Although the most remarkable feature of an octupole TI is the topological corner states located at corners of a finite sample, such corner states in general can also be found in systems without an octupole moment [34][35][36] . Thus, it is important to verify the bulk topology to ensure the topological corner states come from the bulk octupole moment. Here we verify the π flux in the limit of λ→0 (Fig. 7b) and γ→0 (Fig. 7c) to ensure our design can be mapped to the tight-binding lattice shown in Fig. 1b. As shown in Figs. 7d and f, the measured spectra when the source and microphone are placed at the same site (R1 or R7) features two peaks, corresponding to the two branches of eigenmodes (denoted by black circle on the horizontal axis). Furthermore, when we fix the source at R1 (or R7) and measure the acoustic field over all eight sites, features of π flux on each facet of the cubic arise 8,25 . As plotted in Figs. 7, e and g, intensity on the resonators located at diagonal positions of the excitation is almost zero and phases on the resonators adjacent to the excitation are 0 (π) if the coupling is positive (negative).

Robustness of corner state.
Here we present tight-binding calculations as well as experimental investigations on the robustness of the corner states. Two types of perturbation are considered (see inset of Fig. 8c): resonance frequency shifts on the three sites next to the corner (perturbation 1) and resonance frequency shift on the corner site (perturbation 2). For perturbation 1, we tune the resonant frequencies of the three sites next to one of the corners to be 0 (1 + ), where 0 is the unperturbed resonant frequency and are random numbers uniformly distributed from -δ to δ. Fig. 8a shows calculated eigenfrequencies with different perturbation strength δ. As can be seen, the corner state stays almost unaffected. For perturbation 2, the perturbation is added on one of the corner sites. This perturbation will lead to the frequency shift of one of the corner states (Fig. 8b).
To test the robustness of the corner states experimentally, we deliberately introduce on-site perturbations by putting metal balls into resonators which will shift the resonant frequency of a single resonator about 10 Hz. As shown in Fig. 8c, the spectra measured at the corner features a single peak both for the unperturbed case (grey curve) and the two perturbed cases (blue and orange curves   c-e, Integrated intensity maps over frequencies around peaks of spectra in b, which reveal that the peaks in b correspond to hinge, surface and bulk states, respectively.