{'fraction_non_alphanumeric': 0.03944788052496606, 'fraction_numerical': 0.013501282244682456, 'mean_word_length': 4.727429805615551, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Observation of Dirac hierarchy in three-dimensional acoustic topological insulators

Dirac cones (DCs) play a pivotal role in various unique phenomena ranging from massless electrons in graphene to robust surface states in topological insulators (TIs). Recent studies have theoretically revealed a full Dirac hierarchy comprising an eightfold bulk DC, a fourfold surface DC, and a twofold hinge DC, associated with a hierarchy of topological phases including first-order to third-order three-dimensional (3D) topological insulators, using the same 3D base lattice. Here, we report the first experimental observation of the Dirac hierarchy in 3D acoustic TIs. Using acoustic measurements, we unambiguously reveal that lifting of multifold DCs in each hierarchy can induce two-dimensional (2D) topological surface states with a fourfold DC in a first-order 3D TI, one-dimensional (1D) topological hinge states with a twofold DC in a second-order 3D TI, and zero-dimensional (0D) topological corner states in a third-order 3D TI. Our work not only expands the fundamental research scope of Dirac physics, but also opens up a new route for multidimensional robust wave manipulation.

In another context, three-dimensional (3D) TIs are a class of topological matters featuring 3D insulating bulk but conducting boundaries. According to the principle of bulk-boundary correspondence that was recently generalized by the newly-discovered high-order topology, an nth-order 3D TI has topologically robust states along (3-n) dimensional boundaries. For example, a first-order 3D TI has topological surface states with two-dimensional (2D) DCs in 3D bulk bandgaps; a second-order 3D TI has topological hinge states with one-dimensional (1D) DCs in 2D surface bandgaps; a third-order 3D TI has topological corner states in 1D hinge bandgaps. The 3D TIs thus provide a fertile field for studying Dirac physics. Although the 3D TI phases were initially proposed in condensed-matter systems, their realization in photonics and acoustics is currently under intense investigation, owning to their ability to wave trap and manipulation in a topologically robust manner across multiple dimensions, and their great potential in next-generation integrated photonics/acoustics. An important task in this field is to realize a full hierarchy of topological phases, i.e., firstorder to third-order 3D TIs, using the same base configuration; in this way, robust 2D interfaces with surface DCs, 1D hinge waveguides with line DCs, and zero-dimensional (0D) cavities can exist in a single 3D integrated platform. However, all existing experimental works on 3D TIs are limited to a single or two topological phases [12,[39][40][41][42][43]; a full topological phase hierarchy that consists of first-order, second-order and third-order 3D TIs based on the same configuration has remained elusive so far.
Recent theoretical advances have suggested a novel route to achieve the hierarchy of band topology of 3D TIs, which is intriguingly associated with a hierarchy of DCs [44] that corresponds to various topological phase transition points across multiple dimensions. 3D honeycomb lattices with the same base configuration can exhibit 3D bulk states with an eightfold DC, 2D surface states with a fourfold DC in a first-order 3D TI, 1D hinge states with twofold DC in a second-order 3D TI, and 0D corner states in a third-order 3D TI, which, however, have so far evaded experimental observation.
FIG. 1. Dirac hierarchy and topological phase hierarchy. The Dirac hierarchy comprises (a) an eightfold-degenerate bulk DC to (b) a fourfolddegenerate 2D surface DC in a first-order 3D TI, followed by (c) a twofold-degenerate 1D hinge DC in a second-order 3D TI and finally (d) a pair of 0D corner states in a third-order 3D TI. At each hierarchy, the degenerate DC in the higher dimension is broken by appropriate symmetry breaking, resulting in a complete bandgap that hosts a DC dispersion in a lower dimension. The topological phase hierarchy comprises of (e) a gapless phase, (f) a first-order TI, (g) a second-order TI, and (h) a third-order TI.
Here, we report unambiguous experimental evidence of the full hierarchy of DCs and the corresponding hierarchy of band topology in 3D acoustic crystals composed of stacked 2D honeycomb lattices of interconnected acoustic resonators. We start from a 3D acoustic crystal with an extended honeycomb cell containing multiple primitive unit cells, which features a 3D eightfold-degenerate bulk DC [see Fig. 1 To experimentally demonstrate the Dirac and topological phases hierarchy, we fabricate a series of 3D acoustic crystals using a standard 3D printing technique. As shown in Fig. 2(a), the first experimental sample exhibits the first-order topological insulating phase, supporting 2D topological surface states with a fourfold DC. The details of the unit cell are shown in Fig. 2 (1) in which is a N-by-N identity matrix ( = 2, 6), ℋ and ℋ are the Hamiltonians for the monolayer Kekulé lattice and the out-of-plane SSH chain, respectively. Both of them feature block-off-diagonal forms that ℋ = [0, ; † , 0] and ℋ = [0, ; * , 0], in which and are the couplings between different sublattices for the Kekulé lattice and SSH chain, respectively. For the explicit form of , and the Hamiltonian (1) under a specific basis, please refer to the Supplementary Material. After some algebra operation, one can find that the eigenvalue of ℋ is = + , with the corresponding eigenstate = ⨂ , in which ( ) and ( ) are the eigenvalue and eigenstate of ℋ ( ℋ ), respectively. This fact implies that: if and are both bulk states, then is also a bulk state, while if is an edge state and is a bulk (edge or 2D corner) state, then is a surface (hinge or 3D corner) state [44].
According to the zone-folding mechanism, when both interlayer couplings equal to each other ( = = 3.2 mm) and the intercell coupling is equal to the intracell coupling ( = = 4.0 mm), an eightfold bulk DC (red sphere) emerges at Z ( = ) in the band structure with the frequency around 4.6 kHz for the lowest twelves bands, as shown in Fig. 2(c). To demonstrate the first-order topological insulating phase and the 2D fourfold surface DC in the first acoustic crystal, we carry out a series of measurements. In the experiments, an acoustic point source [marked by a cyan star in Fig. 2(a)] is placed at the top center of the acoustic sample, which can generate a broadband signal from 2 kHz to 7 kHz to excite the surface and bulk modes simultaneously. A probe (microphone) is inserted into surface (bulk) resonators to measure the surface (bulk) transmission. The measured transmission along the surface (blue region) and in bulk (red region) are shown in Fig. 3(a). We observe a wide transmission dip from approximately 3.8 kHz to 5.0 kHz in the transmission of the bulk states, corresponding to an extremely wide bulk bandgap with more than 27 % relative bandwidth. On the surface, however, we observe the high transmission spanning the whole bandgap, indicating the existence of gapless topological surface states.
To map out the dispersion band structure of the topological surface states, we perform pump-probe surface measurements at the top surface of the sample. The measured and simulated field distributions on the sample's external surface at 4.2 kHz are shown in Fig. 3(b), exhibiting good confinement of waves on the top surface. We then conduct spatial Fourier transform to the measured acoustic field distributions to obtain the surface dispersion. The measured surface dispersion along the high-symmetry line K ̅ -Γ ̅ -M ̅ -K ̅ (the projections of the high-symmetry points on the xy plane) is displayed in Fig. 3(c), revealing a family of topological surface states spanning the frequency range of the bulk bandgap with a gapless fourfold DC around 4.3 kHz. The simulated surface dispersion (cyan hollow circles highlight the surface DC and transparent grey dots represent the bulk state) is also shown for comparison, matching well with the measured counterparts. We also plot the measured isofrequency contours in 2D reciprocal space from 3.9 kHz to 4.7 kHz, as shown in Fig. 3(d), confirming the conical nature of the surface dispersion. Moreover, we present in Fig. 3  The topological nature of the first 3D acoustic crystal can be understood as follows. According to Eq. (1), the lattice configuration of the first sample indicates that ℋ has a gapless eigen-spectrum, and is a bulk state, while ℋ has a topological edge state because the winding number is nonzero [44]. Consequently, = ⨂ appears to be 2D topological surface states.
Next, we experimentally demonstrate the second-order acoustic TI with twofold Dirac hinge states, by inducing inplane Kekulé distortion to the first acoustic TIs to lift the fourfold surface DC. The second acoustic crystal sample with unequal intracell and intercell couplings ( = 3.0 mm, = 3.6 mm, see the left inset of Fig. 4(a)) is shown in Fig. 4(a), which has a molecule-zigzag type hinge parallel to the axis [see the right inset of Fig. 4(a)]. Because of the in-plane Kekulé distortion, the fourfold degeneracy of the surface DC is lifted, and a complete second-order topological bandgap emerges approximately from 4.0 kHz to 4.6 kHz, wherein a pair of helical topological hinge states with twofold DC appear. As shown in Fig. 4(b), the measured (background colour) and simulated (cyan hollow circles) dispersion diagrams of the hinge state indeed reveal a gapless hinge Dirac point appears near the frequency of 4.2 kHz. We also measure the transmission spectra of the surface states (blue region) with a bandgap from 4.0 kHz to 4.6 kHz, but the transmission of the hinge state (green region) remains high with an enhancement of about 30 dB to that of the surface states, as shown in Fig. 4(c). To verify the hinge states are tightly confined and propagate on the hinge of the second-order acoustic TIs, we present in Fig. 4(d) the measured and simulated acoustic pressure distributions at 4.2 kHz. Both unambiguously show the sound is confined and guided along the hinge. Note that the discrepancy between the measured and simulated acoustic field distributions originates from the intrinsic absorption loss of the materials.
The emergence of these hinge states can be ascribed to a net-winding number when considering a specific lattice termination [44]. An alternative explanation can also be established by considering the mirror symmetry regarding the perpendicular direction of the moleculezigzag hinge and calculating the mirror winding numbers. Since commutes with ℋ at Γ point, the off-diagonal block can be block-diagonalized into two sectors ± , associated with the +1 and −1 eigenvalues of [45]. This further allows us to calculate the winding numbers in each sector separately as  Finally, we experimentally demonstrate the third-order 3D acoustic TI with 0D corner states. The third acoustic crystal sample has rhomb-shaped surface terminations with molecule-zigzag boundaries, as shown in Fig. 5 (a). The widths of the air tubes of the unit cell on the left side ( < 0) and on the right side ( > 0) are − = 2.8 mm, − = 3.4 mm and + = 3.1 mm, + = 3.7mm, respectively [see inset of Fig. 5(a)], which breaks the mirror symmetry . We measure the transmission spectra of the surface (blue region), hinge (green region), and corner (yellow region) states under the excitation of a point source placed near a sample's corner. As shown in Fig. 5 (b), the corner measurement exhibits a transmission peak within the surface and hinge bandgap, which agrees well with the respective eigenfrequency ranges of the numerically calculated corner, hinge and surface eigenstates [ Fig. 5(c)]. We also plot the measured and simulated acoustic pressure distributions at 4.18 kHz in Fig. 5(d). The resulting acoustic field distributions are indeed concentrated at the corner, revealing the localized characteristic of the topological corner state.
The appearance of the 0D topological corner states in the third acoustic crystal arises from the breaking of mirror symmetry . This is because the twofold hinge DC in the second-order 3D acoustic TI is protected by the mirror symmetry , breaking the mirror symmetry can lift the degeneracy of the hinge DC, resulting to the third-order topological phase.
We have thus experimentally demonstrated the full Dirac hierarchy across multiple dimensions with an eightfolddegenerate bulk DC, a fourfold-degenerate surface DC, and a twofold-degenerate hinge DC in 3D acoustic TIs using the same 3D base configuration. Moreover, a hierarchy of topological phases encomprassing first-order TIs with 2D DC surface states, second-order TIs with 1D DC hinge states, and third-order TIs with 0D corner states are experimentally realized via step-by-step breaking different symmetries in a single acoustic 3D honeycomb lattice. Our work not only establishes a versatile platform for exploring exotic phenomena related to the hierarchy of Dirac physics and topological phases, but also opens up new avenues to achieve robust wave manipulation and trap devices across multiple dimensions, including 2D surfaces, 1D waveguides, and 0D cavities, in a single integrated photonic/acoustic circuit.