Abstract
Crystalline materials can host topological lattice defects that are robust against local deformations, and such defects can interact in interesting ways with the topological features of the underlying band structure. We design and implement a three dimensional acoustic Weyl metamaterial hosting robust modes bound to a onedimensional topological lattice defect. The modes are related to topological features of the bulk bands, and carry nonzero orbital angular momentum locked to the direction of propagation. They span a range of axial wavenumbers defined by the projections of two bulk Weyl points to a onedimensional subspace, in a manner analogous to the formation of Fermi arc surface states. We use acoustic experiments to probe their dispersion relation, orbital angular momentum locked waveguiding, and ability to emit acoustic vortices into free space. These results point to new possibilities for creating and exploiting topological modes in threedimensional structures through the interplay between band topology in momentum space and topological lattice defects in real space.
Introduction
Topological lattice defects (TLDs) are crystallinitybreaking defects in lattices that cannot be eliminated by local changes to the lattice morphology, due to their nontrivial realspace topology^{1}. Although they give rise to numerous important physical effects in their own right^{2}, TLDs can have especially interesting consequences in materials with topologically nontrivial bandstructures^{3,4,5,6,7,8}. For instance, Ran et al. have shown theoretically that introducing a screw dislocation into a three dimensional (3D) topological band insulator induces the formation of onedimensional (1D) helical defect modes, which are protected by the interplay between the Burgers vector of the defect and the topology of the bulk bandstructure^{5}. Aside from topological band insulators^{5,9,10,11,12}, other topological phases are predicted to have their own unique interactions with TLDs, including Weyl semimetals, topological crystalline insulators, and higherorder topological insulators^{13,14,15,16,17,18}. TLDinduced modes provide a way to probe bandstructure topology independent of standard bulkboundary correspondences^{6,7,8,11,15,17}, and may give rise to exotic material properties such as anomalous torsional effects^{13}. Experimental confirmations have, however, been hampered by the difficulty of accessing TLDs in real topological materials^{19,20,21}.
Recently, various groups have turned to classical wave metamaterials^{22,23,24,25,26} to perform the experimental studies of the interplay between TLDs and topological bandstructures, including the demonstration of topologicallyaided trapping of light on a dislocation^{23}, robust valley Halllike waveguiding along disclination lines^{24}, and defectinduced fractional modes^{25,26}. The preceding studies have all been based on two dimensional (2D) lattices; 3D lattices with TLDinduced topological modes have thus far only been investigated theoretically.
Here, we design and experimentally demonstrate a 3D acoustic metamaterial that hosts topological modes induced by the presence of a TLD. Without the TLD, the bulk metamaterial forms a Weyl crystal, whose 3D bandstructure contains topologically nontrivial degeneracies called Weyl points^{27,28,29,30,31,32,33,34,35,36,37,38,39,40,41}. Weyl crystals are known to exhibit, along their 2D external surfaces, Fermi arc states that are protected by the topology of the Weyl points^{32,33}. The introduction of the defect generates a family of modes localised to the line of the TLD (in real space). Moreover, in a manner analogous to the formation of regular Fermi arcs, the modes span the projections of two Weyl points of opposite topological charge in the axial momentum space k_{z}. The TLDbound modes for each k_{z} can be interpreted as a 2D bound state generated by a strongly localised pseudomagnetic flux associated with the TLD, in accordance with earlier theoretical predictions about disclinations in 2D topological materials^{8}. Hence, these modes arise from the interplay between the TLD and the 3D Weyl bandstructure. The TLDbound modes carry nonzero orbital angular momentum (OAM), locked to their propagation direction. For each k_{z}, the sign of the OAM depends on the Chern number of the 2D projected band structure, and matches the chirality of the robust localised state that appears in a Chern insulator on a 2D surface with singular curvature^{42,43,44,45}—a prediction that has never previously been verified in an experiment^{8,46,47}. To our knowledge, this is also the first demonstration of a 3D topologyinduced mode carrying nonzero OAM. Classical waves with nonzero OAM have a variety of emerging applications including vortex traps and rotors^{48,49} and OAMencoded communications^{50}. Although chiral structures have previously been studied for the purposes of OAM waveguiding, those waveguides support multiple OAM modes with different propagation constants^{51}; by contrast, the present topological waveguide supports, for each k_{z}, a single robust bound mode with nonzero OAM.
Results
Design of the Weyl acoustic structure
The emergence of a TLDbound topological mode is conceptually illustrated in Fig. 1a. In a Weyl semimetal, topologicallycharged Weyl points in the 3D bulk imply the existence of Fermi arc modes on 2D external surfaces of the crystal. In the 2D surface momentum space, each Fermi arc extends between the projections of two oppositelycharged Weyl points. The introduction of a TLD into the Weyl crystal breaks translational symmetry in the x–y plane while maintaining it along z, and generates modes that are spatially localised to the 1D string formed by the TLD. Viewed from momentum space, the TLDbound modes extend between the projections of the two Weyl points into the 1D momentum space k_{z}.
Recently, the discovery of higherorder topological materials^{52} has led to the idea of higherorder Weyl and Weyllike phases^{53,54,55,56,57,58,59,60,61}, which can host “higherorder Fermi arcs”^{56,57,58,61}. Like the TLDbound modes discussed in this paper, higherorder Fermi arc modes are onedimensional, but they arise from a completely different mechanism involving higherorder topological indices^{56,57,61}. Moreover, they lie along external hinges, whereas the present TLDbound modes are localised to the line of the TLD, embedded inside a 3D bulk.
We designed and fabricated a 3D acoustic crystal formed by chirally structured layers stacked along z, as shown in Fig. 1b–e. Without any TLD, an x–y cross section of the structure would form a triangular lattice. The TLD is introduced by a “cutandglue” procedure in which a π/3 wedge is deleted (Fig. 1c inset) and the edges are reattached by deforming the rest of the lattice (see Methods). The experimental sample is formed by stacking 3Dprinted structures, with a total of 21 layers (see Methods); a photograph is shown in Fig. 1e.
The 3D Brillouin zone of the acoustic crystal, in the absence of the TLD, is depicted in the left panel of Fig. 2a. Weyl points exist at K and \(K^{\prime} \) (H and \(H^{\prime} \)), with topological charge +1 (−1)^{39,62,63}; for details, refer to Supplementary Note 1. Consider the Weyl point at K or \(K^{\prime} \) (the analysis for H and \(H^{\prime} \) is similar). In its vicinity, the wavefunctions are governed by the effective Hamiltonian
where τ_{i} (σ_{i}) denotes valley (sublattice) Pauli matrices, we have rescaled each spatial coordinate so that the group velocity is unity, and k_{z} is the wavenumber in the z direction.
Pseudomagnetic flux of the lattice defect
With the introduction of the TLD, k_{z} remains a good quantum number; in the x–y plane, the distortion introduced by the TLD can be modelled as a matrixvalued gauge field^{8} that mixes the valleys (i.e., K with \(K^{\prime} \) and H with \(H^{\prime} \)). The effective Hamiltonian can be brought back into blockdiagonal form by a unitary transformation^{8}, whereby the Hamiltonian for each block has the form of Eq. (1) but modified by
where \(\tau ^{\prime} =\pm\!1\) is the block index, r is the radial coordinate and e_{θ} is the azimuthal unit vector in the unfolded space, and the factor Ω = 5/6 is the number of undeleted wedges. Unlike previously studied straininduced pseudomagnetic fluxes in Weyl semimetals^{63,64,65}, the pseudomagnetic flux here is strongly localised^{8,66}. Moreover, unlike previous studies of pseudomagnetic fluxes generated by screw dislocations, the pseudomagnetic flux is k_{z}independent^{5,13}.
Viewed from 2D, the pseudomagnetic flux induces topologically protected chiral defect states. For each k_{z} > 0, one can show^{5,8} that there is a single bound solution (among the two Weyl Hamiltonians) localised at r = 0. This remains true even when k_{z} is nonperturbative. For fixed k_{z}, the lattice in the absence of the TLD maps to a 2D Chern insulator whose Chern numbers switch sign with k_{z} (the gap closes at 0 and ±π/L); upon introducing the TLD via the cutandglue construction, one of the two subblocks in the effective Hamiltonian (\(\tau ^{\prime} =1\) for 0 < k_{z} < π/L, and \(\tau ^{\prime} =1\) for −π/L < k_{z} < 0) exhibits a solution that is localised to the TLD^{8}. As we vary k_{z}, this family of solutions spans the projections of the Weyl points at K(\(K^{\prime} \)) and H(\(H^{\prime} \)). Note that the overall acoustic structure preserves timereversal symmetry (T), but the individual Hamiltonian subblocks effectively break T; the defect mode at − k_{z} thus serves as the timereversed counterpart of the defect mode at k_{z}, with opposite chirality. For further details, refer to Supplementary Note 2.
The upper panel of Fig. 2b shows the numerically computed acoustic band diagram for the TLDfree bulk structure, projected onto k_{z}. The relevant bands along KH (ML) are plotted in green (orange), and the gap region is shown in white. The lower panel of Fig. 2b shows the corresponding band diagram for a structure with a TLD, which is periodic along z and has the same x–y profile as the experimental sample (Fig. 1b–e). These numerical results reveal the existence of TLDbound modes, plotted in red, which occupy the gap and span almost the entire k_{z} range. (Near k_{z} = 0 and k_{z} = π/L, they are difficult to distinguish from bulk modes due to finitesize effects.)
In Fig. 2c,d, we show the mode distributions for the TLDbound modes at k_{z} = ±0.5π/L. The modes are strongly localised to the center of the TLD; their intensity profiles are identical since the two modes map to each other under time reversal. The phase distributions (inset) reveal that the k_{z} > 0 (k_{z} < 0) TLDbound mode has winding number +1 (−1). This winding number is tied to the Chern number of the 2D projected band structure for fixed k_{z}. The fact that the TLDbound modes carry nonzero OAM, locked to the propagation direction, distinguishes them from previously studied topological defect modes^{67,68,69,70} and hinge modes^{56,57,61} that have zero OAM. Moreover, we have verified numerically that the TLDbound modes’ localisation and OAM are robust to inplane disorder, consistent with their topological origin (see Supplementary Note 3).
Spectrum and field distribution measurements
We performed a variety of experiments to characterise the TLDbound modes in the fabricated structure. First, we investigated their dispersion curve by threading an acoustic source into the bottom layer of the sample, near the center of the TLD. A probe is inserted into the other 20 layers in turn, via the central air sheet in each layer, as indicated by the blue arrow in Fig. 1d. The acoustic pressure, measured close to the center of the TLD, is Fourier transformed to obtain the spectral plot shown in Fig. 3a. The overlaid red dashes are the numerically obtained TLDbound mode dispersion curve (Fig. 2b), which closely matches the intensity peaks in the experimental results. We then repositioned the source and probe away from the TLD, obtaining in the spectrum shown in Fig. 3b; this matches the bulk spectrum obtained numerically, with the spectral intensities peaking in the bulk bands. For details about the source and probe positions, see Supplementary Note 4.
The acoustic pressure intensity at k_{z} = π/2L is plotted versus frequency in Fig. 3c. A narrow peak corresponding to the TLDbound modes is clearly observable within the bulk gap, with only a small frequency shift of 80 Hz relative to the numerically predicted eigenfrequency. For excitation near the TLD, the measured intensity distribution at frequency f = 4.924 kHz is plotted in Fig. 3d, showing strong localisation around the TLD. The radial dependence of the intensity distribution is plotted in Fig. 3e (note that the apparent irregularity arises from the fact that the measurement points lie at different azimuthal angles). The measurement data is in good agreement with the numerically obtained TLDbound mode profiles. From a linear least squares fit of the semilogarithmic plot, using measurement data up to a radial distance of 12 cm, we find a localisation length of 2.38 cm, which is on the order of the mean distance between unit cells (i.e., the approximate lattice constant).
Figure 3f plots the phase of the measured acoustic pressure signal versus azimuthal angle for k_{z} = π/2L and f = 4.924 kHz. The different data series in this plot correspond to measurement points at different radial distances. The phase is observed to wind by +2π during a counterclockwise (CCW) loop encircling the TLD, consistent with the numerically obtained eigenmode (Fig. 2c), which implies that the TLDbound mode has OAM of +1.
Excitation by vortex sources
To demonstrate the physical significance of the OAM carried by the TLDbound modes, we studied their coupling to external acoustic vortices. The experimental setup is shown in Fig. 4a. The vortex wave is generated in a cylindrical waveguide of radius 1.7 cm, attached to the bottom layer of the sample at the center of the TLD. Figure 4b shows the acoustic pressure intensity measured in the top layer, on the opposite side of the sample from the source. This intensity is obtained by averaging over points closest to the TLD, and dividing by the averaged intensity in the bottom layer to normalise away the frequency dependence of the source. For a CCW vortex source, a strong peak is observed within the range of frequencies where TLDbound modes are predicted to exist. For a clockwise (CW) vortex source, the intensity is low (the nonvanishing intensity is likely due to finitesize effects).
Figure 4c–d shows the intensity and phase distributions measured in the top layer at 5.6 kHz, confirming that the TLDbound modes are preferentially excited by the CCW vortex.
After the TLDbound modes have passed through the structure, they emit an acoustic vortex into free space at the far surface. In Fig. 4e–h, we show the intensity and phase distributions measured by an external acoustic probe positioned 2 mm above the top surface of the sample. For a CCW vortex source in the bottom layer, a CCW vortex is emitted from the top layer, at the position of the TLD; for a CW vortex source, the emission is negligible due to the TLDbound modes not being excited. For frequencies outside the range of the TLDbound modes, the CW and CCW vortices both produce negligible emission from the top layer (see Supplementary Note 4).
Discussion
We have experimentally realised a 3D acoustic structure hosting localised topological modes induced by a topological lattice defect. In real space, the modes lie along a 1D line formed by the defect, embedded within the bulk; in momentum space, they connect the projections of the Weyl points in the defectfree crystal, and hence span the 1D Brillouin zone. This is, to our knowledge, the first experimental demonstration of a defectinduced topological mode in any 3D system. For each momentum space slice (k_{z}), the system maps onto a 2D Chern insulator trapped on a surface with singular curvature. Theoretical studies have previously shown that such a system hosts a robust localised defect mode tied to the Chern number of the 2D bulk bandstructure^{8,46,47}.
The TLDbound modes carry nonzero OAM, locked to their propagation direction. This is a striking feature not possessed by topological defect modes based on other similar schemes; for example, the localised topological modes of 2D Kekulé lattices carry zero winding number^{67,68,69,70}. Our sample therefore serves as an OAMlocked acoustic waveguide, one whose operating principles are very different from the chiral acoustic emitters^{71,72} and metasurfaces^{73,74} studied in previous works. This design may be useful for applications of acoustic vortices, such as acoustic traps and rotors^{48,49} and OAMencoded communications^{50}. Similar designs could be used to realise TLDbound modes in photonics, based on 3D photonic crystals^{30} or laserwritten waveguide arrays^{35}.
Finally, our work opens the door for further investigations into the numerous other effects of lattice defects in topological materials. Many interesting phenomena in this area have been proposed theoretically but have not thus far been observed, including torsional chiral magnetic effects in Weyl semimetals and 1D helical defect modes in 3D weak topological insulators^{5,13,15}.
Methods
Lattice generation
The lattice was optimised by the the molecular dynamics simulator LAMMPS^{75}, using two types of particle interactions: (i) a threebody Tersoff potential (SiC.tersoff), and (ii) a pairwise nearestneighbour harmonic potential \(U(r)=K{(r{r}_{0})}^{2}\) (bond_style harmonic) with K = 20 and r_{0} = 0. Note that these particle interactions have no physical significance; they are simply a convenient way to generate a lattice with minimal variation in intersite distances^{24}.
Numerical simulation
All bandstructure calculations were performed using COMSOL Multiphysics, with air density 1.18 kg m^{−3} and sound speed 343 ms^{−1}. All airsolid interfaces are modeled as hard acoustic boundaries. For the dispersion plot in the lower panel of Fig. 2b, we used periodic boundary conditions in the z direction, and plane wave radiation boundary conditions in x and y.
Experiments
The experimental samples were fabricated from photosensitive resin via stereolithographic 3D printing. For the dispersion measurements in Fig. 3, the bottom surface of the sample is covered by a square plexiglass plate (length 500 mm), which acts as a hard acoustic boundary. A broadband acoustic signal is launched from a balanced armature speaker of around 1 mm radius, driven by a power amplifier, and located at the center of the TLD at the interface between the plate and the sample. Each acoustic probe is a microphone (Brüel & Kjær Type 4961, of about 3.2 mm radius) in a sealed sleeve with a tube of 1 mm radius and 250 mm length. The probes can be threaded into the sample along the horizontal air regions to scan different positions within each layer of the sample (see Supplementary Note 4). The measured data was processed by a Brüel & Kjær 3160A022 module to extract the frequency spectrum, with 2 Hz resolution. Spatial Fourier transforms are applied to the complex acoustic pressure signals to obtain the dispersion relation and field distributions.
For the experiment shown in Fig. 4, the CW and CCW waves are generated in a circular waveguide of radius 1.7 cm, into which three balanced armature speakers are inserted. The signal amplitudes in the three speakers are kept the same, and the phases are controlled by two waveform generators (Agilent type 33500B). The CW and CCW waves were generated by setting the relative phases to (0^{∘}, ±120^{∘}, ±240^{∘}).
Data availability
The data supporting the findings of this study are available from the Digital Repository of Nanyang Technological University (DRNTU) at https://doi.org/10.21979/N9/THY532.
Code availability
All numerical codes are available from the corresponding authors on reasonable request.
References
Mermin, N. D. The topological theory of defects in ordered media. Rev. Mod. Phys. 51, 591–648 (1979).
Kosterlitz, J. M. Nobel lecture: topological defects and phase transitions. Rev. Mod. Phys. 89, 040501 (2017).
Jackiw, R. & Rossi, P. Zero modes of the vortexfermion system. Nucl. Phys. B 190, 681–691 (1981).
Lammert, P. E. & Crespi, V. H. Topological phases in graphitic cones. Phys. Rev. Lett. 85, 5190–5193 (2000).
Ran, Y., Zhang, Y. & Vishwanath, A. Onedimensional topologically protected modes in topological insulators with lattice dislocations. Nat. Phys. 5, 298–303 (2009).
Teo, J. C. Y. & Kane, C. L. Topological defects and gapless modes in insulators and superconductors. Phys. Rev. B 82, 115120 (2010).
Juričić, V., Mesaros, A., Slager, R.J. & Zaanen, J. Universal probes of twodimensional topological insulators: dislocation and π flux. Phys. Rev. Lett. 108, 106403 (2012).
Rüegg, A. & Lin, C. Bound states of conical singularities in graphenebased topological insulators. Phys. Rev. Lett. 110, 046401 (2013).
Slager, R.J., Mesaros, A., Juričić, V. & Zaanen, J. The space group classification of topological bandinsulators. Nat. Phys. 9, 98–102 (2013).
Slager, R.J., Mesaros, A., Juričić, V. & Zaanen, J. Interplay between electronic topology and crystal symmetry: dislocationline modes in topological band insulators. Phys. Rev. B 90, 241403 (2014).
Slager, R.J., Rademaker, L., Zaanen, J. & Balents, L. Impuritybound states and green’s function zeros as local signatures of topology. Phys. Rev. B 92, 085126 (2015).
Slager, R.J. The translational side of topological band insulators. J. Phys. Chem. Solids 128, 24–38 (2019).
Sumiyoshi, H. & Fujimoto, S. Torsional chiral magnetic effect in a weyl semimetal with a topological defect. Phys. Rev. Lett. 116, 166601 (2016).
Liu, J. & Balents, L. Anomalous hall effect and topological defects in antiferromagnetic weyl semimetals: Mn 3 sn/ge. Phys. Rev. Lett. 119, 087202 (2017).
SotoGarrido, R., Muñoz, E. & Juričić, V. Dislocation defect as a bulk probe of monopole charge of multiweyl semimetals. Phys. Rev. Res. 2, 012043 (2020).
van Miert, G. & Ortix, C. Dislocation charges reveal twodimensional topological crystalline invariants. Phys. Rev. B 97, 201111 (2018).
Li, T., Zhu, P., Benalcazar, W. A. & Hughes, T. L. Fractional disclination charge in twodimensional C_{n}symmetric topological crystalline insulators. Phys. Rev. B 101, 115115 (2020).
Queiroz, R., Fulga, I. C., Avraham, N., Beidenkopf, H. & Cano, J. Partial lattice defects in higherorder topological insulators. Phys. Rev. Lett. 123, 266802 (2019).
Yazyev, O. V. & Louie, S. G. Electronic transport in polycrystalline graphene. Nat. Mater. 9, 806–809 (2010).
Huang, P. Y. et al. Grains and grain boundaries in singlelayer graphene atomic patchwork quilts. Nature 469, 389–392 (2011).
Hamasaki, H., Tokumoto, Y. & Edagawa, K. Conductive and nonconductive dislocations in bisb topological insulators. J. Phys. Soc. Japan 89, 023703 (2020).
Lin, Q., Sun, X.Q., Xiao, M., Zhang, S.C. & Fan, S. A threedimensional photonic topological insulator using a twodimensional ring resonator lattice with a synthetic frequency dimension. Sci. Adv. 4, eaat2774 (2018).
Li, F.F. et al. Topological lighttrapping on a dislocation. Nat. Commun. 9, 2462 (2018a).
Wang, Q., Xue, H., Zhang, B. & Chong, Y. D. Observation of protected photonic edge states induced by realspace topological lattice defects. Phys. Rev. Lett. 124, 243602 (2020a).
Liu, Y. et al. Bulk–disclination correspondence in topological crystalline insulators. Nature 589, 381–385 (2021).
Peterson, C. W., Li, T., Jiang, W., Hughes, T. L. & Bahl, G. Trapped fractional charges at bulk defects in topological insulators. Nature 589, 376–380 (2021).
Wan, X., Turner, A. M., Vishwanath, A. & Savrasov, S. Y. Topological semimetal and fermiarc surface states in the electronic structure of pyrochlore iridates. Phys. Rev. B 83, 205101 (2011).
Yang, B.J. & Nagaosa, N. Classification of stable threedimensional dirac semimetals with nontrivial topology. Nat. Comm. 5, 1–10 (2014).
Liu, Z. K. et al. Discovery of a threedimensional topological dirac semimetal, na3bi. Science 343, 864–867 (2014).
Lu, L. et al. Experimental observation of weyl points. Science 349, 622–624 (2015).
Fang, C., Chen, Y., Kee, H.Y. & Fu, L. Topological nodal line semimetals with and without spinorbital coupling. Phys. Rev. B 92, 081201 (2015).
Xu, S.Y. et al. Discovery of a weyl fermion semimetal and topological fermi arcs. Science 349, 613–617 (2015).
Lv, B. Q. et al. Experimental discovery of weyl semimetal taas. Phys. Rev. X 5, 031013 (2015).
Bian, G. et al. Topological nodalline fermions in spinorbit metal pbtase 2. Nat. Comm. 7, 1–8 (2016).
Noh, J. et al. Experimental observation of optical weyl points and fermi arclike surface states. Nat. Phys. 13, 611–617 (2017).
Wang, Q., Xiao, M., Liu, H., Zhu, S. & Chan, C. T. Optical interface states protected by synthetic weyl points. Phys. Rev. X 7, 031032 (2017).
Wu, W. et al. Nodal surface semimetals: theory and material realization. Phys. Rev. B 97, 115125 (2018).
Armitage, N. P., Mele, E. J. & Vishwanath, A. Weyl and dirac semimetals in threedimensional solids. Rev. Mod. Phys. 90, 015001 (2018).
Li, F., Huang, X., Lu, J., Ma, J. & Liu, Z. Weyl points and fermi arcs in a chiral phononic crystal. Nat. Phys. 14, 30–34 (2018b).
Yan, Q. et al. Experimental discovery of nodal chains. Nat. Phys. 14, 461–464 (2018).
Xiao, M. et al. Experimental demonstration of acoustic semimetal with topologically charged nodal surface. Sci. Adv. 6, eaav2360 (2020).
Wen, X. G. & Zee, A. Shift and spin vector: new topological quantum numbers for the hall fluids. Phys. Rev. Lett. 69, 953–956 (1992).
Parrikar, O., Hughes, T. L. & Leigh, R. G. Torsion, parityodd response, and anomalies in topological states. Phys. Rev. D 90, 105004 (2014).
Rüegg, A., Coh, S. & Moore, J. E. Corner states of topological fullerenes. Phys. Rev. B 88, 155127 (2013).
Schine, N., Ryou, A., Gromov, A., Sommer, A. & Simon, J. Synthetic landau levels for photons. Nature 534, 671–675 (2016).
Can, T., Chiu, Y. H., Laskin, M. & Wiegmann, P. Emergent conformal symmetry and geometric transport properties of quantum hall states on singular surfaces. Phys. Rev. Lett. 117, 266803 (2016).
Biswas, R. R. & Son, D. T. Fractional charge and interlandau–level states at points of singular curvature. Proc. Nat. Acad. Sci. (USA) 113, 8636–8641 (2016).
Skeldon, K. D., Wilson, C., Edgar, M. & Padgett, M. J. An acoustic spanner and its associated rotational doppler shift. New J. Phys. 10, 013018 (2008).
Baresch, D., Thomas, J.L. & Marchiano, R. Observation of a singlebeam gradient force acoustical trap for elastic particles: acoustical tweezers. Phys. Rev. Lett. 116, 024301 (2016).
Shi, C., Dubois, M., Wang, Y. & Zhang, X. Highspeed acoustic communication by multiplexing orbital angular momentum. Proc. Natl Acad. Sci. 114, 7250–7253 (2017).
Wong, G. et al. Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber. Science 337, 446–449 (2012).
Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators. Science 357, 61–66 (2017).
Lin, M. & Hughes, T. L. Topological quadrupolar semimetals. Phys. Rev. B 98, 241103 (2018).
Roy, B. Antiunitary symmetry protected higherorder topological phases. Phys. Rev. Res. 1, 032048 (2019).
Călugăru, D., Juričić, V. & Roy, B. Higherorder topological phases: a general principle of construction. Phys. Rev. B 99, 041301 (2019).
Wang, H.X., Lin, Z.K., Jiang, B., Guo, G.Y. & Jiang, J.H. Higherorder weyl semimetals. Phys. Rev. Lett. 125, 146401 (2020b).
Ghorashi, S. A. A., Li, T. & Hughes, T. L. Higherorder weyl semimetals. Phys. Rev. Lett. 125, 266804 (2020).
Luo, L. et al. “Observation of a phononic higherorder weyl semimetal,” arXiv preprint arXiv:2011.01351 (2020).
Wieder, B. J. et al. Strong and fragile topological dirac semimetals with higherorder fermi arcs. Nat. Comm. 11, 1–13 (2020).
Wu, W., Yu, Z.M., Zhou, X., Zhao, Y. X. & Yang, S. A. Higherorder dirac fermions in three dimensions. Phys. Rev. B 101, 205134 (2020).
Wei, Q. et al. Higherorder topological semimetal in acoustic crystals. Nat. Mater. https://doi.org/10.1038/s41563021009334 (2021).
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).
Peri, V., SerraGarcia, M., Ilan, R. & Huber, S. D. Axialfieldinduced chiral channels in an acoustic weyl system. Nat. Phys. 15, 357–361 (2019).
Jia, H. et al. Observation of chiral zero mode in inhomogeneous threedimensional weyl metamaterials. Science 363, 148–151 (2019).
Ilan, R., Grushin, A. G. & Pikulin, D. I. Pseudoelectromagnetic fields in 3d topological semimetals. Nat. Rev. Phys. 2, 29 (2019).
González, J., Guinea, F. & Vozmediano, M. A. H. The electronic spectrum of fullerenes from the dirac equation. Nucl. Phys. B 406, 771–794 (1993).
Gao, P. et al. Majoranalike zero modes in kekulé distorted sonic lattices. Phys. Rev. Lett. 123, 196601 (2019).
Menssen, A. J., Guan, J., Felce, D., Booth, M. J. & Walmsley, I. A. Photonic topological mode bound to a vortex. Phys. Rev. Lett. 125, 117401 (2020).
Gao, X. et al. Diracvortex topological cavities. Nat. Nanotech. 15, 1012–1018 (2020).
Noh, J. et al. Braiding photonic topological zero modes. Nat. Phys. 16, 989–993 (2020).
Ealo, J. L., Prieto, J. C. & Seco, F. Airborne ultrasonic vortex generation using flexible ferroelectrets. IEEE Transact. Ultrason. Ferroelectr. Freq. Control 58, 1651–1657 (2011).
Jiang, X. et al. Broadband and stable acoustic vortex emitter with multiarm coiling slits. Appl. Phys. Lett. 108, 203501 (2016a).
Jiang, X., Li, Y., Liang, B., Cheng, J.c & Zhang, L. Convert acoustic resonances to orbital angular momentum. Phys. Rev. Lett. 117, 034301 (2016b).
Fu, Y. et al. Sound vortex diffraction via topological charge in phase gradient metagratings, Sci. Adv. 6 https://doi.org/10.1126/sciadv.aba9876 (2020).
Plimpton, S. Fast parallel algorithms for shortrange molecular dynamics. J. Comp. Phys. 117, 1–19 (1995).
Acknowledgements
Q.W., H.X., B.Z. and Y.C. acknowledge support from Singapore MOE Academic Research Fund Tier 3 Grant MOE2016T31006, Tier 1 Grant RG187/18, and Tier 2 Grant MOE2019T22085. Y.G., H.X. S., D.J., Y.J.G. and S.Q.Y. acknowledge support from the National Natural Science Foundation of China under Grants No. 11774137 and 51779107, National Key R&D Program Project (No. 2020YFC1512403 and 2020YFC1512400) and the State Key Laboratory of Acoustics, Chinese Academy of Science under Grant No. SKLA202016.
Author information
Authors and Affiliations
Contributions
Q.W. and Y.G. contributed equally to this work. Q.W., B.Z. and Y.C. conceived the idea. Q.W. designed the acoustic structures and performed the numerical simulations. Q.W., H.X., and H.X.S. designed the experiments and fabricated the sample. Y.G., H.X.S., D.J. and Y.J.G. conducted the measurements. S.Q.Y., B.Z. and Y.C. supervised the project. All authors contributed extensively to the interpretation of the results and the writing of the paper.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Peer review information Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work. Peer reviewer reports are available.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Wang, Q., Ge, Y., Sun, Hx. et al. Vortex states in an acoustic Weyl crystal with a topological lattice defect. Nat Commun 12, 3654 (2021). https://doi.org/10.1038/s41467021239637
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41467021239637
This article is cited by

Vortex nanolaser based on a photonic disclination cavity
Nature Photonics (2023)

Topological phenomena at defects in acoustic, photonic and solidstate lattices
Nature Reviews Physics (2023)

Topological acoustics
Nature Reviews Materials (2022)

Topological dislocation modes in threedimensional acoustic topological insulators
Nature Communications (2022)

Bound vortex light in an emulated topological defect in photonic lattices
Light: Science & Applications (2022)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.