Demonstration of a third-order hierarchy of topological states in a three-dimensional acoustic metamaterial

Topological acoustic metamaterial design

A 3D acoustic metamaterial emulator of the pyrochlore lattice (28, 30) is realized using metamolecules (unit cells) schematically shown in Fig. 1 (A and B), whose specific dimensions are given in Materials and Methods. Each molecule consists of four acoustic resonators, with acoustic pressure modes oscillating along the axial direction, as shown in the inset to Fig. 1A. We choose to work with the fundamental mode, whose only node appears in the center of the individual cavity of the resonator. To form the lattice, the resonators are coupled through narrow cylindrical channels whose position on the resonator is carefully chosen to respect the crystalline symmetries of the pyrochlore lattice. The control over coupling between resonators within and between adjacent metamolecules of the metacrystal, which is required to induce bulk topological polarization, is achieved by varying the diameter of the cylindrical coupling channels. Because the acoustic modes are strongly bound to the resonant cavities and the design of the structure only allows nearest neighbor coupling, the tight-binding model (TBM) can be an effective modeling tool for our system, with nearest-neighbor hopping described by intra-cell κ and inter-cell γ coupling parameters. An analytical construction of the Hamiltonian is presented in section S1. COMSOL Multiphysics (acoustic module) has been used to verify the results with full-wave finite element method (FEM) simulations. If the diameter of the intraconnected channels is smaller than the one of the interconnected channels, κ < γ, then the structure is referred to as the “expanded” crystal, implying stronger coupling among resonators of the adjacent metamolecules. In the opposite condition, κ > γ, the structure is “shrunken,” and modes are strongly coupled within the resonators of the metamolecule within the same unit cell. The acoustic resonators in the metamaterial are aligned to show the sites and primitive vectors of the face-centered cubic (fcc) lattice in Fig. 1B and the corresponding first Brillouin zone shown in Fig. 1C. As the dimensions of the crystal exceeded the print volume of currently available stereolithography (SLA) 3D printers, the hybrid approach, combining 3D printing of the constitutive elements and subsequent snap assembly, was used. The metamolecules were 3D printed and snapped together using the interlocking features that are deliberately introduced in the design, thus allowing the assembly of rigid and stable large-scale metacrystals. The assembled finite structure consisting of 20 metamolecules (80 resonators) is shown in Fig. 1D. Note that the boundaries of the crystal are terminated by smaller resonators, which are designed to have resonant frequencies detuned from the spectral region of interest to high frequency range, yielding an effective boundary condition analogous to the terminated boundary in TBM. These are required to preserve the Γ₄ generalized chiral symmetry present in the pyrochlore structure with nearest neighbor coupling (section S3) (20).

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Nontrivial topology and bulk polarization

First, we theoretically explore the band structure of the unit cell, as shown in Fig. 2 (A and B). In the ideal pyrochlore lattice, with κ = γ (Fig. 2A), a line of degeneracy along the X-W high symmetry path can be observed. The degeneracy of this line is broken by making the inter-cell and intra-cell couplings different, κ ≠ γ (Fig. 2B). Through both semi-analytical TBM and the first-principles FEM modeling of the structure, we observe band inversion at the W point between two cases of expanded and shrunken crystals, which indicates a nontrivial topological transition. However, energy bands alone are not sufficient to determine the topological difference between expanded and shrunken crystals, and further analysis of the associated eigenstates is required.

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To explicitly confirm the topological phase transition between expanded and shrunken crystals, we calculated the bulk topological polarization using the eigenvalue problem of the Wilson loops (31). The bulk polarization of the lowest energy band, defining the emergence of Wannier-type HOT states in the first bandgap above this band for both cases of the expanded and shrunken lattices, can be obtained through the Wannier band v₃(k₁, k₂), where (k₁, k₂) vary along the reciprocal lattice vectors b₁, b₂ in momentum space, and the Wilson loop is evaluated along the direction of the reciprocal lattice vector b₃ (Fig. 1C). Then, the bulk polarization of the lowest energy band of interest is defined asp3(k1,k2)=1N1N2∑b3v3(k1,k2)(1)where N_1,2 is the total number of points in the discretized momentum space. Because of the spatial symmetries of the pyrochlore crystal, the bulk polarizations with cyclically permutated indexes are equivalent to each other, and therefore, the system is characterized by nontrivial bulk polarization in all three dimensions. As shown in Fig. 2 (C and D), the bulk polarization of the expanded crystal calculated through Eq. 1 yields ¹/₄, which indicates that the average position of the modes belonging to the lowest energy band has a a04 deviation (shift) from the center of the sites in the unit cell. According to the bulk-boundary correspondence principle, when boundaries are introduced in the expanded pyrochlore lattice with nonvanishing bulk polarization, the boundary modes such as surface, edge, and corner states should appear hanging at the dangling sites of the corresponding terminations of the crystal. Meanwhile, the bulk polarization of the shrunken crystal is zero, implying that the average position of the desired modes resides at the center of the four sites in the metamolecule, and no boundary modes are present at any boundary (section S4).

Experimental demonstration of a hierarchy of Wannier-type HOT states

We numerically calculated the energy spectrum of the metamaterial sample in Fig. 1D as a function of the ratio κ/γ (Fig. 3A) by using TBM of the finite crystal. Based on the prediction from nontrivial bulk polarization outlined in the previous section, for the case of expanded lattice (when κγ<1), we expect surface (purple-colored), edge (blue-colored), and corner modes (red-colored) to emerge at the boundaries. Both TBM and numerical simulations based on FEM fully confirm these predictions. When κγ<13, fourfold degeneracy corner modes split from the bulk continuum (yellow-colored bands) and, as can be proven analytically (see the Supplementary Materials for details), the transition point occurs at κγ=13. In the 3D printed structure fabricated for our experimental investigation, we choose to work in the κ/γ = 0.111 regime of the energy spectrum of the lattice (indicated by dashed lines in Fig. 3A). In this case, the zero-energy corner states, as well as the edge and surface states, appear fully immersed into the bulk bandgap and show no overlap with the bulk continuum. In agreement with the TBM energy spectrum, the frequency response spectra of different sites extracted from acoustic measurements on the structure (Fig. 3B) reveal a wide bulk bandgap with lower and upper bulk bands, as well as a hierarchy of topological boundary modes, including surface states, edge states, and (fourfold degenerate) corner states (localized at the four corners of the structure). In contrast to the lower bulk band, the upper bulk band also shows up the spectra for surface and edge sites. This observation is related to the fact that the modes of the upper bulk band penetrate into the surface and edge resonators, as opposed to the lower band, where the topological polarization removes the corresponding surface and edge degrees of freedom from the bulk eigenmodes. Materials and Methods provides details of the procedure to extract the frequency response spectra for different regions from experimental measurements. The corner states appear at “zero-energy,” i.e., their frequency corresponds to the frequency of a separate individual resonator; however, the collective nature of these modes is evidenced by the nonvanishing field strength in other sites of the lattice. All the topological states appear within the bulk bandgap, with some overlap due to the finite bandwidth of the corresponding spectral lines caused by their finite lifetime. Some overlap among the bulk, surface, edge, and corner spectra can be seen in Fig. 3B and is attributed to inhomogeneous broadening due to energy dissipation. Two major loss mechanisms in our experiment are (i) the absorption in the resin used in 3D printing and (ii) the leakage through the probe holes, which are deliberately introduced in the design of the individual resonators to allow excitation and probing of the acoustic field. The resultant experimentally measured quality factors of the individual resonators are within the range of 50 to 60, and therefore, the corresponding finite lifetimes of the modes of the lattice are long enough not to alter their topological nature, making them clearly observable in our system. To validate our findings and highlight the topological nature of the observed states, we also fabricated a trivial shrunken structure, which was found to host bulk modes only, further evidencing the topological origin of the observed surface, edge, and corner states. The frequency response spectra for the shrunken structure are shown in section S4. In this scenario, the spectra for surface, edge, and corner sites all appear in the lower bulk band as well, thus confirming that the topological polarization is trivial and the wavefunction of the bulk modes penetrates into boundary sites (section S4 and fig. S2). This is an important distinction from the nontrivial case, when these sites are excluded from the wavefunction of the lower bulk band due to the topological bulk polarization of the lower bulk band as in Fig. 3B.

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The experimentally measured field profiles of all the modes, shown in Fig. 3C, are in excellent agreement with the numerically calculated profiles of Wannier-type higher order states. From Fig. 3C, one can easily ascertain the difference between field profiles of the exponentially decaying bulk modes (excited at the frequency within the bulk band indexed by label I in Fig. 3B) and of the extremely localized zero-energy corner state excited at the corresponding frequency (indexed by label II in Fig. 3B). Note that the field decay of the bulk modes is less rapid, following a 1/r² dependence with an exponential correction due to loss. It is worth mentioning that the generalized chiral symmetry in the pyrochlore lattice provides inherent robustness of the corner modes, as the corner states are always pinned to zero-energy states, and their field profiles are ideally localized at one of the sublattices, which is evident in the corner mode spectral profile in Fig. 3C. The generalized chiral symmetry for the present case of 3D pyrochlore lattice is described by the operator Γ̂4, with the property Γ̂44=I, connecting a sequence of inequivalent Hamiltonians Ĥn with the same spectrum Γ̂nĤ0Γ̂−n=Ĥn, where n = 1,2,3, and H₀ + H₁ + H₂ + H₃ = 0 (section S3). As can be seen from the logarithmic field maps in Fig. 3C (right), the corner state predominantly localizes on the same sublattice. Thus, the acoustic pressure amplitude at the atoms of different sublattices within the same unit cell appears to be at least 500 times smaller compared to the amplitude in the same sublattice of the adjacent cells (only 40 times weaker due to the exponential localization of the states). As expected, the sublattice localization is not observed for the bulk modes plotted in the left panel of Fig. 3C.

In addition to corner states localized in three dimensions, the nonzero value of bulk topological polarization gives rise to the formation of another class of Wannier-type higher-order states–second-order edge states confined to the edges and first-order surface states confined to the surface of the fabricated finite metacrystal. These modes are also shown in Fig. 3C, where we can see that they exhibit extreme localization to a particular edge or surface, respectively. For the edge mode, the mode is excited halfway between the terminating corners, and it forms a standing wave-like profile with two nodes located at the respective corners. For the surface mode, the mode is excited in the inner part of the surface at the starred site, where it also forms a membrane-like standing wave profile with fixed constraint boundary conditions located along the edges.