Acoustic Hierarchical Topological Insulators

Acoustic Hierarchical Topological Insulators Xiujuan Zhang1,#, Zhi-Kang Lin2,#, Hai-Xiao Wang2,3,#, Yuan Tian1, Ming-Hui Lu1,4,†, Yan-Feng Chen1,4,†, Jian-Hua Jiang2,†,*, 1

National Laboratory of Solid State Microstructures and Department of Materials Science and Engineering, Nanjing University, Nanjing, 210093, China

2

School of Physical Science and Technology, and Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, 1 Shizi Street, Suzhou, 215006, China 3

School of Physical Science and Technology, Guangxi Normal University, Guilin 541004, China

4

Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing, 210093, China

#

These authors contributed equally to this work.

†

Correspondence and requests for materials should be addressed to [email protected] (Jian-Hua

Jiang) or [email protected] (Ming-Hui Lu) or [email protected] (Yan-Feng Chen) *

On leave at Department of Physics, University of Toronto, Toronto, Ontario, M5S 1A7, Canada

Topological band theory has stimulated tremendous interest in quantum materials with unprecedented bulk-boundary physics that may have applications in spintronics and quantum computation [1,2]. Recently, higher-order topological insulators, featuring gapped edge states and in-gap corner/hinge states, are predicted [3-7] and observed in a few systems [8-12], establishing a new paradigm beyond the conventional bulk-boundary physics. It is still unclear, however, whether conventional and higher-order topologies can coexist in a single material, since conventional topology originates from dipole polarizations while higher-order topology originates from quadrupole (or octupole) polarizations. Here, we show that although they are incompatible in electronic systems, since a quadrupole polarization demands vanishing dipole polarization, they can be realized in acoustic systems where topological phenomena are not limited by the Fermi surface. We propose theoretically and observe experimentally a new topological state, dubbed as the hierarchical topological 1

insulator, where topological Zak phases [1,2] (first-order topology) and quantized quadrupole polarization [3,4] (second-order topology) coexist in a hierarchical manner in a two-dimensional (2D) sonic crystal. Such hierarchical band topology is induced and protected by a pair of noncommutative, orthogonal glide symmetries, revealing a new class of symmetry-enriched topology. We observe the topological corner and edge states and find that the quadrupole topological polarization yields pseudospin-momentum coupled edge states, as similar to the edge states of time-reversal breaking quantum spin Hall insulators. It is further shown that both the conventional and higher-order band topology can be controlled by tuning the geometry of the sonic crystal. Our study demonstrates an instance that classical systems with configurable structures can serve as effective, directly measurable, and elegant simulators to predict and study novel topological quantum states and their phase transitions. Despite their limited availability in nature, analog computation and simulation, which use simple and tractable systems to simulate complex systems or equations, constitute a powerful and efficient approach for the quantitative understanding of complex physical systems and processes [13-14]. The generalization of analog simulation to the quantum regime, as envisioned by R.P. Feynman, has led to the remarkable discipline of quantum simulation [15-16]. However, it is rarely demonstrated that classical systems can be effective analog simulators for quantum systems and quantum states, since they are governed by distinct wave equations. Recent studies have shown that topological energy bands, prominent ingredients of many exotic quantum states and excitations, can exist not only in quantum systems but also in classical systems, such as electromagnetic [17-26], acoustic [27-34] and mechanical [35-36] waves. These classical systems faithfully simulate the energy bands, eigenstates, dynamics and edge states of the 2

quantum systems of the same space-time symmetry, therefore, illustrating that classical systems can be effective analog simulators for quantum systems. Here, taking a step forward, we demonstrate that classical systems can even be powerful and effective tools to predict new topological quantum states and study their phase transitions. This goal is achieved by utilizing tunable 2D airborne sonic crystals which have several advantages. First, exploiting the state-of-art 3D printing technology, sonic crystals can be fabricated for any desired structure, geometry and lattice. High-quality samples can be fabricated with all kinds of space-group symmetries and tailorable acoustic energy bands. With versatile techniques for acoustic wave excitations and detections, coherence lengths much longer than the lattice constants, and no limitation imposed by the Fermi energy, acoustic systems have become a remarkable regime for the study of topological phenomena [17-36]. The new topological phase, denoted as the hierarchical topological insulator, exhibits 2D Zak phases in the first acoustic band gap and a novel quadrupole topological insulator (QTI) for the second acoustic band gap. The latter requires at least four bands below the topological band gap, which differs crucially from the QTI proposed in Ref. [3]. At the heart of the hierarchical topological insulator, there is a pair of noncommutative, orthogonal glide symmetries which induce and protect the topological order. The glide symmetries double the space group representations and lead to concurrent topological dipole (for the first band gap) and quadrupole (for the second band gap) polarizations, providing an outstanding example of symmetry-enriched topology. Furthermore, facilitated by the glide symmetries, the hierarchical topological insulator and its phase transitions can be realized in simple and tunable airborne sonic crystals, without relying on 𝜋 -flux lattices [3,4], making QTIs attainable through conventional acoustic metamaterials with very large band gaps. 3

We design our square-lattice sonic crystals such that in each unit-cell, there are four archshaped objects made of photosensitive resin (modulus 2765 MPa, density 1.3 g/cm3) serving as acoustic scatterers (see Fig. 1a). The geometries of the four scatterers are identical and are characterized by three parameters: the arch height ℎ, the arm length 𝑙 and the arm width 𝑤, which can be tuned in order to engineer the acoustic bands (see Supplementary Note 1 for examples). Those scatterers are arranged in such a way that the sonic crystal has two glide symmetries, 𝐺𝑥 : = 𝑎

𝑎

𝑎

𝑎

(𝑥, 𝑦) → ( − 𝑥, + 𝑦) and 𝐺𝑦 : = (𝑥, 𝑦) → ( + 𝑥, − 𝑦) with 𝑎 being the lattice constant. 2 2 2 2 Including the inversion and 𝐶4 rotation symmetries, the system has a nonsymmorphic space group, namely, a wallpaper group p4g. Those glide symmetries have profound impacts on the acoustic bands (see Fig. 1b). For the Bloch states (denoted as 𝜓𝑛,𝒌 for the 𝑛-th band with wavevector 𝒌), the eigenvalues of the glide operator, 𝐺𝑦 𝜓𝑛,𝒌 = 𝑔𝑦 𝜓𝑛,𝒌 with 𝑔𝑦 = ±𝑒 𝑖𝑘𝑥 𝑎/2 , have periodicity of

4𝜋 𝑎

along the ΓX direction.

The same property holds for 𝐺𝑥 along the ΓY direction. For instance, the first two acoustic bands are associated with the same eigenvalue of 𝐺𝑦 as 𝑔𝑦 = 𝑒 𝑖𝑘𝑥 𝑎/2 (see Fig. 1c). The evolution of 𝑔𝑦 with 𝑘𝑥 connects the two Bloch bands together, to fulfill the Brillouin zone, i.e., after an interval of

2𝜋 𝑎

4𝜋 𝑎

periodicity within the first

the first acoustic band evolves into the second acoustic

band. In addition, the glide symmetries result in band-sticking effects [37], leading to pseudoKramers double degeneracy on the Brillouin zone boundary (see Fig. 1b). Explicitly, when combined with the time-reversal operator 𝑇, the anti-unitary symmetry operators 𝛩𝑗 = 𝐺𝑗 ∗ 𝑇 (𝑗 = 𝑥, 𝑦) enable double degeneracy on the Brillouin zone boundary. For instance, 𝛩𝑥2 𝜓𝑛,𝒌 = 𝑒 𝑖𝑘𝑥 𝑎 𝜓𝑛,𝒌. At the Brillouin zone boundary line MX, we have 𝑘𝑥 𝑎 = 𝜋 and 𝛩𝑥2 = −1 for all acoustic bands, inducing a pseudo-Kramers double degeneracy. Similar double degeneracy emerges on the MY 4

line, due to the fact that 𝛩𝑦2 = −1. Recent studies show that the same space group leads to exotic first-order topological insulators and edge states in the compound Sr2Pb3 [38]. However, the emergence of quadrupole topological order has never been discovered in wallpaper groups. In our sonic crystals, the wallpaper space group dictates that the double degeneracy at the X (Y) point comprises two bands of opposite parities. In contrast, at the Γ point, any pair of bands connected by the glide symmetries are of the same parity (see Supplementary Note 2 for proof). The parity-switch between the Γ and X (Y) points yields a topological dipole polarization 𝒑 = 1 1

(2 , 2) of the Wannier orbitals [39]. Equivalently, this topological dipole is described by the 2D Zak phase 𝝑 = (𝜋, 𝜋) [39]. This property holds for any two bands that are related by the glide operations. Note that although acoustic waves do not have dipole or quadrupole polarizations related to electromagnetic responses as is the case for charged particles such as electrons, the dipole and quadrupole polarizations here are defined on the Wannier orbitals of acoustic bands, i.e., on the shift and deformation of the acoustic Wannier orbitals. To construct a topological quadrupole moment, one needs two cancelling dipoles. The second acoustic band gap, which has four acoustic bands below, fulfills such a requirement. However, the quantization of quadrupole polarization is a highly nontrivial consequence of the noncommutative, orthogonal glide symmetries, as proven in the Supplementary Note 3. The dipole and quadrupole polarizations can be characterized by the Wannier bands and the nested Wannier bands [3,4]. A Wannier center 𝜈𝑥 is the Berry phase of an energy band associated with a Wilson-loop in the Brillouin zone (e.g., with fixed 𝑘𝑦 while 𝑘𝑥 goes from 0 to

2𝜋 𝑎

). Physically, the Wannier center 𝜈𝑥

is also the coordinate of the center of the Wannier orbital of an energy band along the x direction as measured from the center of the unit cell. The Wannier band is the dependence of the Wannier 5

center 𝜈𝑥 on the wavevector 𝑘𝑦 . In the presence of mirror symmetries, the dipole polarization of 1

the Wannier orbitals is quantized with 𝜈𝑥 = 0 or 2 for a trivial or topological dipole polarization, respectively. The emergence of topological quadrupole polarization must further fulfill the following 1

requirements: (i) symmetric and gapped (i.e., 𝜈𝑥 cannot be 0 or 2 ) Wannier bands, and (ii) a quantized, finite topological polarization in each Wannier sector as characterized by the nested Wannier bands [3,4]. The former indicates that the vanishing dipole polarization in total has intricate internal structures that may allow a finite quadrupole polarization. We calculate numerically the Wannier bands associated with the four acoustic bands below the second band gap (see Supplementary Note 4 for the numerical methods and calculation details). It is found that the Wannier bands are indeed gapped. The four Wannier bands can form different combinations, representing different Wannier sectors. If the first and third Wannier bands are grouped together to form one sector, while the second and fourth Wannier bands form the other sector, then the Wannier bands summed within each sector, 𝜈𝑥− and 𝜈𝑥+ , are gapped (Fig. 2a-c). Furthermore, the topological polarizations in these Wannier sectors, as described by the nested Wannier bands, 𝜈−

𝜈+

𝑝𝑦𝑥 = −𝑝𝑦𝑥 , are quantized to

1 2

by the orthogonal glide symmetries, as shown by our first-

principle calculation, if the arch height ℎ is finite (see Supplementary Note 4 for the calculation 𝜈−

𝜈−

details of the nested Wannier bands). From the 𝐶4 rotation symmetry, we obtain that 𝑝𝑦𝑥 = 𝑝𝑥 𝑦 = − 𝜈− 𝜈

1

1

, which reveals a nontrivial topological quadrupole moment [3,4], 𝑞𝑥𝑦 = 2𝑝𝑦𝑥 𝑝𝑥 𝑦 = 2. 2 We note that the first and second (the third and fourth) Wannier bands form a Wannier sector with trivial topology since their summed-Wannier-bands are gapless (Fig. 2c). Because the four

6

1

Wannier bands are symmetric around 𝜈𝑥 = 0 and 2 (as constrained by the glide symmetries; see Supplementary Note 3), other combinations of the Wannier bands also form gapless Wannier sectors. They are not connected to the physical edge states which have a gapped spectrum, as found in our simulations and experiments. As we now show, the transition into a trivial phase with vanishing quadrupole polarization, 𝑞𝑥𝑦 = 0, can be realized by restoring the mirror symmetries [3,4]. This transition can be achieved by continuously deforming the scatterers in each unit-cell. Through a process illustrated in Fig. 2, the Wannier bands gradually evolve from gapped to gapless, i.e., the Wannier band in the 𝜈𝑥+ (𝜈𝑥− ) sector gradually moves up (down) to 0.5 (-0.5). This process corresponds to the destruction of the quadrupole topological order due to the restoration of the mirror symmetries, since the latter always yield gapless Wannier bands, except for 𝜋-flux lattices [4]. Three example sonic crystals and their Wannier bands are shown in Fig. 2a-h to illustrate the effect of geometry on the evolution of the Wannier bands. The topological phase diagram is shown in Fig. 2i. The topological transition takes place when the mirror symmetries are restored, i.e., when the arch height vanishes, ℎ = 0. Accompanying the topological transition, the bulk band gap closes and the topological corner states merge into the bulk bands (see Fig. 2i, and see Supplementary Note 5 for details on how the corner states merge into the bulk states). After this transition, the corner states disappear in the trivially gapped phase with vanishing quadrupole polarization. These findings manifest a hallmark feature of the QTI: the topological corner states are dictated directly by the bulk quadrupole topology [3,4], instead of the topology of the edge states. We now confirm the hierarchical band topology by experimental observations of the topological edge and corner states. The 2D sonic crystal with a lattice constant 𝑎 = 2 cm is

7

fabricated using the 3D printing technology (see Materials and Methods). Cladding boards above and below the printed structure lead to a quasi-2D acoustic system, with acoustic band-structures that are nearly identical to the 2D limit for the low-lying acoustic bands (see Supplementary Note 6 for supporting data). We first measure the edge states in the second acoustic band gap, as induced by the topological quadrupole moment. The edge channels are formed between the sonic crystal and a hard-wall boundary made of photosensitive resin, separated by an air layer of thickness 0.4𝑎. The measured and calculated dispersions of the edge states, shown in Fig. 3a, agree well with each other. Here, the measured edge-states dispersion is derived from the Fourier transformations of the frequency-dependent acoustic pressure profiles along the edge when they are excited by an acoustic point-like source at the center of the edge (see Supplementary Note 7 for more details). Limited by the fabrication precision (0.1 mm) and finite-size effect, as well as by the fact that there might be a discrepancy between the measured quasi-2D acoustic systems in the laboratory and the 2D acoustic systems in our simulation, the measured dispersion of the edge states is slightly blueshifted from the simulated dispersion. Nonetheless, the measured dispersion agrees fairly well with the simulated dispersion, demonstrating the emergence of gapped edge states within the bulk band gap (see Fig. 3a). The acoustic pressure profile for a particular edge state (marked by the blue dot in Fig. 3a) is shown in Fig. 3b. The simulated acoustic pressure profile for the edge state, shown in Fig. 3c, is consistent with the measured acoustic pressure profile. Interestingly, we find that the edge states carry finite orbital angular momentum (OAM), which is observed from the acoustic pressure fields in Fig. 3d in two complementary ways: the phase and amplitude distributions of the acoustic pressure fields. First, the phase distributions exhibit phase singularities and phase vortices, indicating finite OAM. The two edge states (indicated by red and blue dots in Fig. 3a), which are related to each other by time-reversal operation, have opposite 8

phase winding numbers around the vortex cores. At the vortex cores, the acoustic pressure amplitudes vanish and the phases become singular. In addition, the distributions of the energy flow (i.e., the time-averaged Poynting vector, indicated by the blue arrows; see Supplementary Note 8 for calculation details) of the acoustic fields also manifest the acoustic OAM for the edge states. The edge states thus exhibit pseudospin-momentum locking, where the pseudospins are emulated by the OAM. This result is quite surprising, since such pseudospin-momentum locked edge states were understood to belong to the QSHE systems [23,31]. Here the gapped edge states resemble the edge states in time-reversal symmetry broken QSHE systems [1,2]. We remark that in our sonic crystals, the bulk and edge band gaps are very large, as seen in Fig. 3a. From our calculation, the second bulk band gap ranges from 10.9 kHz to 15.7 kHz, reaching a very large band gap ratio of 37%. Inside the bulk band gap, the edge band gap ranges from 11.5 kHz to 13.5 kHz, with a large band gap ratio of 16%. The large bulk band gap leads to highly confined edge states, while the large edge band gap stabilizes the in-gap corner states at the subwavelength scale. These phenomena allow very strong enhancement of the acoustic wave intensity for the topological edge and corner modes. Furthermore, the large bulk and edge gaps makes it much easier to distinguish the spectra and responses of the bulk, edge and corner states. We further measure the corner states in a box-shaped finite-size structure surrounded by hardwall boundaries (see the inset of Fig. 4a). The distance between the sonic crystal and the hard-wall boundary is fixed to 0.4𝑎 (a similar structure is used to calculate the corner modes in Fig. 2i). The calculated acoustic spectrum near the edge band gap is shown in Fig. 4a. Four degenerate acoustic modes emerge in the edge band gap, which are localized at the four corners of the box-shaped structure. The emergence of a single topologically localized mode at each corner is a smoking-gun feature of the quadrupole topological order [3,4,8-11]. 9

To confirm the coexistence of the bulk, edge and corner modes in a single acoustic chip, we measure the frequency-resolved responses of three types of pump-probe configurations. We separately name these pump-probe configurations the bulk-probe, edge-probe and corner-probe. The bulk-probe corresponds to the measurements of the acoustic pressure at the detecting location (marked by the dark-blue “D” in the inset of Fig. 4b) under the excitation of a separated source (marked by the dark-blue “S” in the inset of Fig. 4b), where both the detector and the source are placed in the bulk region. The edge-probe represents the case with separated source and detector on an edge. The corner-probe stands for the case with separated source and detector near a corner. The measured transmission spectra for those pump-probe configurations are shown in Fig. 4b. The transmission spectra clearly indicate that the peak of the corner-probe lies in the spectral gap of the edge-probe. Besides, the peaks of the edge-probe lie in the spectral gap of the bulk-probe. These features clearly demonstrate the coexistence of the bulk, edge and corner states in a single acoustic chip, where the corner states emerge in the edge band gap and the edge states emerge in the bulk band gap. We also measured the acoustic pressure profile at the peak frequency of the corner-probe, 13.0 kHz, which is presented in Fig. 4c. The measured acoustic pressure profile agrees well with the theoretical acoustic pressure profile obtained from the eigen-mode calculation (see Fig. 4d). It is worth mentioning that the measured transmission spectra in Fig. 4b exhibit a small frequency blue-shift, about 3%, when compared with the theoretical spectrum from the 2D eigen-mode calculation. This blue-shift should again be associated with the geometry deviation of the fabricated sonic crystal and its quasi-2D nature from our ideal 2D simulation. The first acoustic band gap carries quantized 2D Zak phases 𝝑 = (𝜋, 𝜋) which lead to nontrivial edge states similar to those found in Ref. [39]. The theoretical and experimental study of such edge states are presented in the Supplementary Note 8. We now illustrate that the first 10

acoustic band gap can also mimic the quantum spin Hall effect (QSHE). The pseudo-Kramers degeneracy at the Brillouin zone boundary as induced by the glide symmetries provides an instrumental element for the simulation of pseudo-spin degeneracy and the QSHE in acoustic systems. The emergence of the QSHE relies on the band inversion at the M point, which can be controlled by rotating the arch-shaped scatterers. The parity order at the M point for the first four acoustic bands is shown in Fig. 5a, while acoustic band structures are shown in Fig. 5b for three different rotation angles. Since all acoustic bands at the M point are doubly degenerate, the first four acoustic bands consist of two doublets: one doublet with even parity and the other doublet with odd parity. The dependence of the parity order on the rotation angle has 180ºperiodicity. Parity order switch takes place when the rotation angle is equal to an integer of 90º,where a Dirac point with four-fold degeneracy emerge at the M point (see Fig. 5b). For rotation angles between 90ºand 180º, we associate the acoustic Hamiltonian near the M point as a negative Dirac mass equation, i.e., the QSHE phase, whereas the rotation angles between 0ºand 90ºrepresent the normal phase with positive Dirac masses. The band structure calculation and Hamiltonian analysis indicate that those phases are similar to the negative- and positive-mass Dirac equations in the Bernevig-Hughes-Zhang model for the QSHE [2] (see Supplementary Note 9 for details). Helical edge states emerge in the first acoustic band gap at the boundary between sonic crystals with opposite Dirac masses, as shown in Fig. 5c. The calculated and measured dispersions of the acoustic edge states are consistent with each other. The measurements of the edge states and their dispersions are performed following the same experimental procedure as the measurements of the edge states in the second band gap in Fig. 3. It is noticed that the dispersions of the edge states are not so well captured for the low-frequency part. The underlying reason is mainly due to the small group velocity of the edge states in the low-frequency section (see the green curves in Fig. 5c). 11

The decreased group velocity leads to longer propagation time and severer propagation loss and thus yields reduced fidelity of the recorded real-space acoustic pressure profiles and the extracted dispersions. Overall, the measured edge states dispersions are consistent with our theoretical calculation, since the blue-shift of the measured dispersions can be associated as the geometry deviation and quasi-2D nature of the fabricated sonic crystals. The pseudospin-momentum-locking feature of the edge states is illustrated in Fig. 5d, where both the existence of phase vortices in the acoustic pressure profile and the winding Poynting vectors indicate the finite OAM of the edge states. The acoustic OAM emulate the pseudospins of the edge states. The pseudospin-momentumlocking is manifested by the fact that the pseudospin-up and pseudospin-down edge states have opposite wavevector and group velocities. In conclusion, we discovered a new topological state, the hierarchical topological insulator, where the conventional and higher-order band topologies coexist in a single physical system. Our results demonstrate that classical systems with remarkable controllability and measurability can be effective and powerful analog simulators for the study and discovery of quantum states with nontrivial topology. Furthermore, our study provides the first example of realizing quadrupole topological order using conventional subwavelength acoustic metamaterials, thereby opening a new pathway toward higher-order topological materials in other classical (e.g., photonic) systems as well as in electronic systems.

References [1] Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 30453067 (2010) 12

[2] Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057-1110 (2011). [3] Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators. Science 357, 61-66 (2017). [4] Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators. Phys. Rev. B 96, 245115 (2017). [5] Schindler, F. et al. Higher-order topological insulators. Sci. Adv. 4, eaat0346 (2018). [6] Langbehn, J., Peng, Y., Trifunovic, L., von Oppen, F. & Brouwer, P. W. Reflection-symmetric second-order topological insulators and superconductors. Phys. Rev. Lett. 119, 246401 (2017). [7] Song, Z. D., Fnag, Z. & Fang, C. (d-2)-dimensional edge states of rotation symmetry protected topological states. Phys. Rev. Lett. 119, 246402 (2017). [8] Serra-Garcia, M. et al. Observation of a phononic quadrupole topological insulator. Nature 555, 342-345 (2018). [9] Peterson, C. W., Benalcazar, W. A., Hughes, T. L. & Bahl, G. A quantized mircowave quadrupole insulator with topological protected corner states. Nature 555, 346-350 (2018). [10] Imhof, S. et al. Topolectrical circuit realization of topological corner modes. Nat. Phys. 14, 925-929 (2018). [11] Noh, J. et al. Topological protection of photonic mid-gap defect modes. Nat. Photon. 12, 408415 (2018).

13

[12] Schindler, F. et al. Higher-order topology in bismuth. Nat. Phys. 14, 918-924 (2018). [13] H. T. Siegelmann and S. Fishman, Analog computation with dynamical systems. Physica D 120, 214 (1998). [14] Siegelmann, H. T. & Sontag, E. D. Analog computation via neural networks. Theor. Comput. Sci. 131, 331-360 (1994). [15] Feynman, R. Simulating physics with computers. Int. J. Theor. Phys. 21, 467 (1982). [16] Georgescu, I. M., Ashhab, S. & Nori, F. Quantum simulation. Rev. Mod. Phys. 86, 153185 (2014). [17] Haldane, F. D. M. & Raghu, S. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. Phys. Rev. Lett. 100, 013904 (2008). [18] Wang, Z., Chong, Y., Joannopoulos, J. D. & Soljacic, M. Observation of unidirectional backscattering immune topological electromagnetic states. Nature 461, 772-775 (2009). [19] Lu, L., Fu, L., Joannopoulos, J. D. & Soljačić, M. Weyl points and line nodes in gyroid photonic crystals. Nat. Photon. 7, 294-299 (2013). [20] Hafezi, M., Mittal, S., Fan, J., Migdall, A. & Taylor, J. M. Imaging topological edge states in silicon photonics. Nat. Photon. 7, 1001-1005 (2013). [21] Rechtsman, M. C. et al. Photonic Floquet topological insulators. Nature 496, 196-200 (2013). [22] Khanikaev, A. B. et al. Photonic topological insulators. Nat. Mater. 12, 233-239 (2013).

14

[23] Wu, L.-H. & Hu, X. Scheme for achieving a topological photonic crystal by using dielectric material. Phys. Rev. Lett. 114, 223901 (2015). [24] Zilberberg, O. et al. Photonic topological boundary pumping as a probe of 4D quantum Hall physics. Nature 553, 59-62 (2018). [25] Kraus, Y. E., Lahini, Y., Ringel, Z., Verbin, M. & Zilberberg, O. Topological states and adiabatic pumping in quasicrystals. Phys. Rev. Lett. 109, 106402 (2012). [26] Lu, L., Joannopoulos, J. D. & Soljacic, M. Topological photonics. Nat. Photon. 8, 821-829 (2014). [27] Chen, Z. G. et al. Accidental degeneracy of double Dirac cones in a phononic crystal. Sci. Rep. 4, 4613 (2014). [28] Yang, Z. J., Gao, F., Shi, X. H., Lin, X., Gao, Z., Chong, Y. D. & Zhang, B. L. Topological acoustics. Phys. Rev. Lett. 114, 114301 (2015). [29] Ni, X. et al. Topologically protected one-way edge mode in networks of acoustic resonators with circulating air flow. New J. Phys. 17, 053016 (2015). [30] Xiao, M. et al. Geometric phase and band inversion in periodic acoustic systems. Nat. Phys. 11, 240-244 (2015). [31] He, C., Ni, X. Ge, H., Sun, X. C., Chen, Y.-B., Lu, M.-H., Liu, X.-P. & Chen, Y.-F. Acoustic topological insulator and robust one-way sound transport. Nat. Phys. 12, 1124-1129 (2016).

15

[32] Lu, J., Qiu, C., Ye, L., Fan, X., Ke, M., Zhang, F., Liu, Z. Observation of topological valley transport of sound in sonic crystals. Nat. Phys. 13, 369-374 (2017). [33] Xiao, M., Chen, W. J., He, W. Y. & Chan, C. T. Synthetic gauge flux and Weyl points in acoustic systems. Nat. Phys. 11, 920-924 (2015). [34] Li, F. et al. Weyl points and Fermi arcs in a chiral phononic crystal. Nat. Phys. 14, 30-34 (2017). [35] Süsstrunk, R. & Huber, S. D. Observation of phononic helical edge states in a mechanical topological insulator. Science 349, 47-50 (2015). [36] Nash, L. M. et al. Topological mechanics of gyroscopic metamaterials. Proc. Natl Acad. Sci. USA 112, 14495-14500 (2015). [37] Parameswaran, S. A., Turner, A. M., Arovas, D. P. & Vishwanath, A. Topological order and absence of band insulators at integer filling in non-symmorphic crystals. Nat. Phys. 9, 299-303 (2013). [38] Wieder, B. J. et al. Wallpaper fermions and the nonsymmorphic Dirac insulator. Science 361, 246-251 (2018). [39] Liu, F. & Wakabayashi, K. Novel topological phase with a zero Berry curvature. Phys. Rev. Lett. 118, 076803 (2017).

16

Acknowledgements J.H.J thanks Prof. Arun Paramekanti and Prof. Hae-Young Kee for helpful discussions. He also thanks Prof. Sajeev John and the University of Toronto for hospitality where this work is finalized. Fundings: X.J.Z, Y.T, M.H.L and Y.F.C are supported by the National Key R&D Program of China (2017YFA0303702，2018YFA0306200）and the National Natural Science Foundation of China (Grant No. 11625418 and No. 51732006). Z.K.L, H.X.W and J.H.J are supported by the National Natural Science Foundation of China (Grant No. 11675116) and the Jiangsu Province Distinguished Professor Funding. Author contributions: J.H.J conceived the idea. J.H.J, Y.F.C and M.H.L guided the research. J.H.J, Z.K.L and H.X.W established the theory. Z.K.L, X.J.Z and H.X.W performed the numerical calculations and simulations. X.J.Z and Y.T achieved the experimental set-up and measurements. X.J.Z, J.H.J and M.H.L performed data-analysis. All the authors contributed to the discussions of the results and the manuscript preparation. J.H.J, X.J.Z, Z.K.L and M.H.L wrote the manuscript. Competing interests: The authors declare no competing interests. Data and materials availability: All data are available in the manuscript or the Supplementary materials.

Materials and Methods: Our sonic crystals consist of arch-shaped scatterers made of photosensitive resin (modulus 2765 MPa, density 1.3 g/cm3). A stereo lithography apparatus (with a fabrication tolerance of roughly 0.1 mm) is utilized to fabricate the samples, which include a ribbon-like sample for the edge-state measurements, and a box-shaped sample for the corner-state measurements, with ℎ = 0.212𝑎 , 𝑙 = 0.424𝑎 and 𝑤 = 0.1𝑎, where 𝑎 = 2 cm is the lattice constant. The vertical height of the sample is 1 cm. Two acoustic hard boards are used for cladding 17

from the top and the bottom of the sample to form quasi-2D acoustic systems for the frequency range of interest (i.e., less than 20 kHz). The measured edge-state dispersions are obtained by the following procedure. We first scan the acoustic pressure field distribution along the edge for monofrequency excitations. An acoustic transducer is placed under the sample to generate acoustic waves, which are further guided into the sample through an open channel (with a diameter of ~4 mm) at the bottom of the waveguide. The channel is located at the center of the edge (marked by the red star in Fig. 3B). An acoustic detector (B&K-4939 ¼-inch microphone), whose position can be controlled by an automatic stage, is used to probe the spatial dependence of the acoustic pressure from a circular open window (with a diameter slightly larger than the detector) on the top of the cladding layer. The data are collected and analyzed by a DAQ card (NI PCI-6251). The measured acoustic pressure profiles at different frequencies is then Fourier-transformed to obtain the edgestate dispersions. The Fourier transformation is implemented by using the Matlab built-in function fft. The transmission measurements are performed using a similar set-up, but with fixed positions of the source and the detector when the frequency is varied. In the experimental measurements, the upper board of the waveguide (which is attached to an automatic stage) is required to be able to move freely, but without affecting the stabled samples, in order to record the acoustic pressure filed data. To accomplish this goal, we leave a tiny air gap (about 1 mm) between the upper board and the samples below it. This treatment might affect our measurements and could be another reason (additional to the fabrication imperfection) that the measurements are slightly deviated from the simulations on frequencies. Additionally, the condition for the environment atmosphere that varies upon weather change might also affect the sound speed and the air mass density and is the third reason to the frequency shift between the experiments and the simulations. 18

Numerical simulations are performed using a commercial finite-element simulation software (COMSOL MULTIPHYSICS) via the acoustic module. The resin objects are treated as hard boundaries. In the eigen-value calculations, the Floquet periodic boundaries are implemented. The projected band structures of the ribbon-like supercell and the band spectrum of the box-shaped supercell are calculated by setting the truncation boundaries as hard boundaries. For the simulated acoustic-pressure distributions of the edge and corner states, the frequency-domain study is performed. A point source, located at the center of the edge (near the corner), is utilized to excite the edge (corner) states. The energy flow is calculated through the time-averaged Poynting vector of the acoustic fields, 𝑺 = (4𝜋𝜌𝑓)−1 |𝑝|2 ∇𝜙, where 𝜌 is the density of air, 𝑓 is the eigen-frequency, and |𝑝| and 𝜙 are the amplitude and the phase of the acoustic pressure, respectively. The transmission spectra presented in Fig. 4b are normalized by the maximum of each measurement (i.e., the bulk-probe, edge-probe and corner-probe, respectively), so that they can be plotted at the same quantitative scale. The original transmission spectra are presented separately in the Supplementary Figures S7a-S7c, for the bulk-, edge- and corner-probes, respectively. We find that the corner-probe yields a much stronger signal, more than 60 times stronger than the bulk-probe, indicating very strong enhancement of the local acoustic wave intensity due to the strongly localized, subwavelength corner mode.

19

Figures

Figure 1 | Hierarchical topological insulator in an airborne sonic crystal. a, A bird’s-eye view of the square-lattice sonic crystal comprising arch-shaped scatterers made of photosensitive resin in a background of air. The inset illustrates the geometry of a unit-cell. b, Acoustic bands and hierarchical topology. The first acoustic band gap (gray region) exhibits the first-order topological gap, while the second band gap (cyan region) exhibits the secondorder topological gap. Such a hierarchical band topology defines the hierarchical topological insulator. The inset depicts the Brillouin zone. Geometric parameters: ℎ = 0.18𝑎, 𝑙 = 0.36𝑎, 𝑤 = 0.15𝑎 and 𝑎 = 2 cm. c, Evolution of the glide eigenvalue 𝑔𝑦 (shown by color; the color map is given at the upper-right inset) of the first two acoustic bands along the ΓX direction. The symbols +/− represent even/odd parity of the acoustic bands, respectively. The black arrow indicates folding the dispersion and the glide eigenvalue into the first Brillouin zone.

20

Figure 2 | Quadrupole topology and topological phase transition in the second band gap. a, b and c, Respectively, the unit-cell structure, Wannier bands and summed Wannier bands (with different combinations) for the sonic crystal with ℎ = 0.25𝑎, 𝑙 = 0.50𝑎 and 𝑤 = 0.05𝑎. d, e and f, Similar quantities as in a, b and c, for the sonic crystal with ℎ = 0.07𝑎, 𝑙 = 0.50𝑎 and 𝑤 = 0.05𝑎. g and h, Respectively, the unit-cell structure and Wannier bands for the sonic crystal with ℎ = 0, 𝑙 = 0.35𝑎 and 𝑤 = 0.2𝑎 . i, Topological phase transition induced by continuously tuning the geometry, i.e., first reduce the arch height ℎ, then reduce the length 𝑙, and finally increase the width 𝑤. The band diagram shows the bulk band gap closing and reopening, while the corner states disappear in the trivial region. The corner states are 21

calculated using a box-shaped finite structure, as illustrated in Fig. 4a. The topological transition takes place when the mirror symmetry is restored (i.e., when ℎ = 0).

Figure 3 | Characterization and measurements of topological edge states in the second acoustic band gap. a, Calculated (green curves) and measured (hot color) dispersions of the acoustic edge states in the second band gap (see Materials and Methods for details). Gray regions denote the bulk acoustic bands. b and c, Measured and simulated acoustic pressure profiles of the

22

edge states launched by a point source (indicated by the red star), respectively. Cut-line plots perpendicular to the edge are also shown. The inset in b depicts a part of the fabricated sample with a scale bar. d, Acoustic pressure profiles for the two edge states marked by the red and blue dots in a, with the phase, amplitude and Poynting vectors (the blue arrows) shown around the edge. Vortices in the phase distributions of the acoustic pressure are shown, indicating finite orbital angular momentum. The vortex centers and the winding direction are, respectively, marked by the black dots and arrows.

Figure 4 | Characterization and measurements of topological corner states in the second acoustic band gap. a, Calculated acoustic spectrum for the box-shaped structure (schematically illustrated in the inset) near the edge band gap. The sonic crystal structure has an area of 8 × 8𝑎2 , which is enclosed by hard-wall boundaries at a distance of 0.4𝑎. There are four degenerate corner 23

states in the edge band gap, as highlighted by the orange dots. b, Frequency-resolved transmission spectrum for the three pump-probe configurations denoted as the bulk-probe, edge-probe and corner-probe. The source “S” and detector “D” are marked in the inset with the same colors as used for the three curves in the main figure. The transmission spectra are normalized to set the peak of the three curves to unity. A zoom-in structure of the sonic crystal with a scale bar is shown in the inset. c and d, The measured and the simulated acoustic pressure profiles of the corner mode, respectively.

24

Figure 5 | Characterization of helical edge states in the first acoustic band gap. a, Parity evolution of the first four acoustic bands at the M point by tuning the angle 𝜃 (defined in the inset) when ℎ = 0.18𝑎, 𝑙 = 0.35𝑎, 𝑤 = 0.1𝑎. The acoustic pressure profiles of the doublets at the four marked points. b, Acoustic band structures for 𝜃 = 90°, 𝜃 = 45° and 𝜃 = 135° . c, Calculated (green curves) and measured (hot color) dispersions of the acoustic edge states in the first band gap at the boundary between the two sonic crystals with different 𝜃 (45°and 135°) when ℎ = 0.21𝑎, 𝑙 = 0.42𝑎 and 𝑤 = 0.1𝑎. The gray regions denote the bulk bands. d, Acoustic pressure profiles for the two edge states marked by the red and blue dots in c, with the amplitude, the Poynting vectors (blue arrows) and phase shown near the boundary.

25