Observation of higher-order topological acoustic states protected by generalized chiral symmetry


Topological systems are inherently robust to disorder and continuous perturbations, resulting in dissipation-free edge transport of electrons in quantum solids, or reflectionless guiding of photons and phonons in classical wave systems characterized by topological invariants. Recently, a new class of topological materials characterized by bulk polarization has been introduced, and was shown to host higher-order topological corner states. Here, we demonstrate theoretically and experimentally that 3D-printed two-dimensional acoustic meta-structures can possess nontrivial bulk topological polarization and host one-dimensional edge and Wannier-type second-order zero-dimensional corner states with unique acoustic properties. We observe second-order topological states protected by a generalized chiral symmetry of the meta-structure, which are localized at the corners and are pinned to ‘zero energy’. Interestingly, unlike the ‘zero energy’ states protected by conventional chiral symmetry, the generalized chiral symmetry of our three-atom sublattice enables their spectral overlap with the continuum of bulk states without leakage. Our findings offer possibilities for advanced control of the propagation and manipulation of sound, including within the radiative continuum.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Concept and measurement of the bulk polarization transition in the deformed kagome lattice.
Fig. 2: Theoretical prediction and experimental demonstration of acoustic edge and corner states.
Fig. 3: Effect of disorder and robustness of the corner states.
Fig. 4: Experimental demonstration of corner states coexisting with continuum of bulk modes.

Data availability

Data that are not already included in the paper and/or in the Supplementary Information are available on request from the authors.


  1. 1.

    Haldane, F. D. M. Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the ‘parity anomaly’. Phys. Rev. Lett. 61, 2015–2018 (1988).

    CAS  Article  Google Scholar 

  2. 2.

    Kane, C. L. & Mele, E. J. Quantum spin Hall effect in graphene. Phys. Rev. Lett. 95, 226801 (2005).

    CAS  Article  Google Scholar 

  3. 3.

    Kane, C. L. & Mele, E. J. Z 2 topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005).

  4. 4.

    Xiao, D., Yao, W. & Niu, Q. Valley-contrasting physics in graphene: Magnetic moment and topological transport. Phys. Rev. Lett. 99, 236809 (2007).

  5. 5.

    Hasan, M. Z. & Kane, C. L. Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    CAS  Article  Google Scholar 

  6. 6.

    Qi, X. L. & Zhang, S. C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).

    CAS  Article  Google Scholar 

  7. 7.

    Bernevig, B. A. & Hughes, T. L. Topological Insulators and Topological Superconductors (Princeton Univ. Press, Princeton, 2013).

  8. 8.

    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).

  9. 9.

    Raghu, S. & Haldane, F. D. M. Analogs of quantum-Hall-effect edge states in photonic crystals. Phys. Rev. A 78, 033834 (2008).

  10. 10.

    Wang, Z., Chong, Y. D., Joannopoulos, J. D. & Soljacic, M. Observation of unidirectional backscattering-immune topological electromagnetic states. Nature 461, 772–775 (2009).

    CAS  Article  Google Scholar 

  11. 11.

    Feng, L. et al. Nonreciprocal light propagation in a silicon photonic circuit. Science 333, 729–733 (2011).

    CAS  Article  Google Scholar 

  12. 12.

    Hafezi, M., Demler, E. A., Lukin, M. D. & Taylor, J. M. Robust optical delay lines with topological protection. Nat. Phys. 7, 907–912 (2011).

    CAS  Article  Google Scholar 

  13. 13.

    Fang, K. J., Yu, Z. F. & Fan, S. H. Realizing effective magnetic field for photons by controlling the phase of dynamic modulation. Nat. Photon. 6, 782–787 (2012).

    CAS  Article  Google Scholar 

  14. 14.

    Fang, K. J. & Fan, S. H. Controlling the flow of light using the inhomogeneous effective gauge field that emerges from dynamic modulation. Phys. Rev. Lett. 111, 203901 (2013).

  15. 15.

    Khanikaev, A. B. et al. Photonic topological insulators. Nat. Mater. 12, 233–239 (2013).

    CAS  Article  Google Scholar 

  16. 16.

    Rechtsman, M. C. et al. Photonic Floquet topological insulators. Nature 496, 196–200 (2013).

    CAS  Article  Google Scholar 

  17. 17.

    Lu, L., Joannopoulos, J. D. & Soljaclc, M. Topological photonics. Nat. Photon. 8, 821–829 (2014).

    CAS  Article  Google Scholar 

  18. 18.

    Plotnik, Y. et al. Observation of unconventional edge states in ‘photonic graphene’. Nat. Mater. 13, 57–62 (2014).

    CAS  Article  Google Scholar 

  19. 19.

    Skirlo, S. A., Lu, L. & Soljacic, M. Multimode one-way waveguides of large Chern numbers. Phys. Rev. Lett. 113, 113904 (2014).

  20. 20.

    Skirlo, S. A. et al. Experimental observation of large Chern numbers in photonic crystals. Phys. Rev. Lett. 115, 253901 (2015).

  21. 21.

    Wu, L. H. & Hu, X. Scheme for achieving a topological photonic crystal by using dielectric material. Phys. Rev. Lett. 114, 223901 (2015).

  22. 22.

    Cheng, X. J. et al. Robust reconfigurable electromagnetic pathways within a photonic topological insulator. Nat. Mater. 15, 542–548 (2016).

    Article  Google Scholar 

  23. 23.

    Leykam, D., Rechtsman, M. C. & Chong, Y. D. Anomalous topological phases and unpaired Dirac cones in photonic Floquet topological insulators. Phys. Rev. Lett. 117, 013902 (2016).

  24. 24.

    Khanikaev, A. B. & Shvets, G. Two-dimensional topological photonics. Nat. Photon. 11, 763–773 (2017).

    CAS  Article  Google Scholar 

  25. 25.

    Ozawa, T. et al. Topological photonics. Preprint at https://arXiv.org/abs/1802.04173v1 (2018).

  26. 26.

    Peano, V., Brendel, C., Schmidt, M. & Marquardt, F. Topological phases of sound and light. Phys. Rev. X 5, 031011 (2015).

  27. 27.

    Yang, Z. J. et al. Topological acoustics. Phys. Rev. Lett. 114, 114301 (2015).

  28. 28.

    Huber, S. D. Topological mechanics. Nat. Phys. 12, 621–623 (2016).

    CAS  Article  Google Scholar 

  29. 29.

    Susstrunk, R. & Huber, S. D. Classification of topological phonons in linear mechanical metamaterials. Proc. Natl Acad. Sci. USA 113, E4767–E4775 (2016).

    CAS  Article  Google Scholar 

  30. 30.

    Xiao, M. et al. Geometric phase and band inversion in periodic acoustic systems. Nat. Phys. 11, 240–244 (2015).

    CAS  Article  Google Scholar 

  31. 31.

    Kitaev, A. Yu. Unpaired Majorana fermions in quantum wires. Phys.-Uspekhi 44, 131 (2001).

    Article  Google Scholar 

  32. 32.

    Barik, S. et al. A topological quantum optics interface. Science 359, 666–668 (2018).

    CAS  Article  Google Scholar 

  33. 33.

    Bahari, B. et al. Nonreciprocal lasing in topological cavities of arbitrary geometries. Science 358, 636–639 (2017).

    CAS  Article  Google Scholar 

  34. 34.

    Bandres, M. A. et al. Topological insulator laser: Experiments. Science 359, eear4005 (2018).

  35. 35.

    Harari, G. et al. Topological insulator laser: Theory. Science 359, eear4003 (2018).

  36. 36.

    Ni, X. et al. Spin- and valley-polarized one-way Klein tunneling in photonic topological insulators. Sci. Adv. 4, eeap8802 (2018).

  37. 37.

    Fang, C., Gilbert, M. J. & Bernevig, B. A. Bulk topological invariants in noninteracting point group symmetric insulators. Phys. Rev. B 86, 115112 (2012).

  38. 38.

    Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators. Science 357, 61–66 (2017).

    CAS  Article  Google Scholar 

  39. 39.

    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).

  40. 40.

    Serra-Garcia, M. et al. Observation of a phononic quadrupole topological insulator. Nature 555, 342–345 (2018).

    Article  Google Scholar 

  41. 41.

    Peterson, C. W., Benalcazar, W. A., Hughes, T. L. & Bahl, G. A quantized microwave quadrupole insulator with topologically protected corner states. Nature 555, 346–350 (2018).

    Article  Google Scholar 

  42. 42.

    Imhof, S. et al. Topolectrical circuit realization of topological corner modes. Nat. Phys. 14, 925–929 (2018).

    CAS  Article  Google Scholar 

  43. 43.

    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).

  44. 44.

    Ezawa, M. Higher-order topological insulators and semimetals on the breathing kagome and pyrochlore lattices. Phys. Rev. Lett. 120, 026801 (2018).

  45. 45.

    Geier, M., Trifunovic, L., Hoskam, M. & Brouwer, P. W. Second-order topological insulators and superconductors with an order-two crystalline symmetry. Phys. Rev. B 97, 205135 (2018).

  46. 46.

    Ezawa, M. Minimal models for Wannier-type higher-order topological insulators and phosphorene. Phys. Rev. B 98, 045125 (2018).

  47. 47.

    Khalaf, E. Higher-order topological insulators and superconductors protected by inversion symmetry. Phys. Rev. B 97, 205136 (2018).

  48. 48.

    Kunst, F. K., van Miert, G. & Bergholtz, E. J. Lattice models with exactly solvable topological hinge and corner states. Phys. Rev. B 97, 241405(R) (2018).

  49. 49.

    Ezawa, M. Strong and weak second-order topological insulators with hexagonal symmetry and Z3 index. Phys. Rev. B 97, 241402(R) (2018).

  50. 50.

    Song, Z. D., Fang, Z. & Fang, C. (d−2)-dimensional edge states of rotation symmetry protected topological states. Phys. Rev. Lett. 119, 246402 (2017).

  51. 51.

    Schindler, F. et al. Higher-order topological insulators. Sci. Adv. 4, eaat0346 (2018).

  52. 52.

    Ni, X., Gorlach, M. A., Alù, A. & Khanikaev, A. B. Topological edge states in acoustic kagome lattices. New J. Phys. 19, 055002 (2017).

  53. 53.

    Noh, J. et al. Topological protection of photonic mid-gap defect modes. Nat. Photon. 12, 408–415 (2018).

    CAS  Article  Google Scholar 

  54. 54.

    Marinica, D. C., Borisov, A. G. & Shabanov, S. V. Bound states in the continuum in photonics. Phys. Rev. Lett. 100, 183902 (2008).

  55. 55.

    Hsu, C. W. et al. Observation of trapped light within the radiation continuum. Nature 499, 188–191 (2013).

    CAS  Article  Google Scholar 

  56. 56.

    Zhen, B., Hsu, C. W., Lu, L., Stone, A. D. & Soljacic, M. Topological nature of optical bound states in the continuum. Phys. Rev. Lett. 113, 257401 (2014).

  57. 57.

    Hsu, C. W., Zhen, B., Stone, A. D., Joannopoulos, J. D. & Soljacic, M. Bound states in the continuum. Nat. Rev. Mater. 1, 16048 (2016).

  58. 58.

    Doeleman, H. M., Monticone, F., den Hollander, W., Alù, A. & Koenderink, A. F. Experimental observation of a polarization vortex at an optical bound state in the continuum. Nat. Photon. 12, 397–401(2018).

  59. 59.

    Xue, H., Yang, Y., Gao, F., Chong, Y. D. & Zhang, B. Acoustic higher-order topological insulator on a kagome lattice. Nat. Mater. https://doi.org/10.1038/s41563-018-0251-x (2018).

Download references


The work was supported by the Defense Advanced Research Projects Agency under the Nascent programme with grant number HR00111820040, and by the National Science Foundation with grant numbers CMMI-1537294, EFRI-1641069 and DMR-1809915. Research was carried out in part at the Center for Functional Nanomaterials, Brookhaven National Laboratory, which is supported by the US Department of Energy, Office of Basic Energy Sciences, under contract number DE-SC0012704.

Author information




All authors contributed extensively to the work presented in this paper.

Corresponding authors

Correspondence to Andrea Alù or Alexander B. Khanikaev.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Sections 1–11, Supplementary Figures 1–9, Supplementary References 1–8

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ni, X., Weiner, M., Alù, A. et al. Observation of higher-order topological acoustic states protected by generalized chiral symmetry. Nature Mater 18, 113–120 (2019). https://doi.org/10.1038/s41563-018-0252-9

Download citation

Further reading