Letter | Published:

Acoustic higher-order topological insulator on a kagome lattice

Nature Materialsvolume 18pages108112 (2019) | Download Citation


Higher-order topological insulators1,2,3,4,5 are a family of recently predicted topological phases of matter that obey an extended topological bulk–boundary correspondence principle. For example, a two-dimensional (2D) second-order topological insulator does not exhibit gapless one-dimensional (1D) topological edge states, like a standard 2D topological insulator, but instead has topologically protected zero-dimensional (0D) corner states. The first prediction of a second-order topological insulator1, based on quantized quadrupole polarization, was demonstrated in classical mechanical6 and electromagnetic7,8 metamaterials. Here we experimentally realize a second-order topological insulator in an acoustic metamaterial, based on a ‘breathing’ kagome lattice9 that has zero quadrupole polarization but a non-trivial bulk topology characterized by quantized Wannier centres2,9,10. Unlike previous higher-order topological insulator realizations, the corner states depend not only on the bulk topology but also on the corner shape; we show experimentally that they exist at acute-angled corners of the kagome lattice, but not at obtuse-angled corners. This shape dependence allows corner states to act as topologically protected but reconfigurable local resonances.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Additional information

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


  1. 1.

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

  2. 2.

    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).

  3. 3.

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

  4. 4.

    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).

  5. 5.

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

  6. 6.

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

  7. 7.

    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).

  8. 8.

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

  9. 9.

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

  10. 10.

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

  11. 11.

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

  12. 12.

    Lin, M. & Hughes, T. L. Topological quadrupolar semimetals. Preprint at https://arxiv.org/abs/1708.08457 (2017).

  13. 13.

    Ezawa, M. Magnetic second-order topological insulators and semimetals. Phys. Rev. B 97, 155305 (2018).

  14. 14.

    Geier, M. et al. Second-order topological insulators and superconductors with an order-two crystalline symmetry. Phys. Rev. B 97, 205135 (2018).

  15. 15.

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

  16. 16.

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

  17. 17.

    Su, W. P., Schrieffer, J. R. & Heeger, A. J. Solitons in polyacetylene. Phys. Rev. Lett. 42, 1698–1701 (1979).

  18. 18.

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

  19. 19.

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

  20. 20.

    Khanikaev, A. B., Fleury, R., Mousavi, S. H. & Alu, A. Topologically robust sound propagation in an angular-momentum-biased graphene-like resonator lattice. Nat. Commun. 6, 8260 (2015).

  21. 21.

    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).

  22. 22.

    Yang, Z. & Zhang, B. Acoustic type-II Weyl nodes from stacking dimerized chains. Phys. Rev. Lett. 117, 224301 (2016).

  23. 23.

    He, C. et al. Acoustic topological insulator and robust one-way sound transport. Nat. Phys. 12, 1124–1129 (2016).

  24. 24.

    Lu, J. et al. Observation of topological valley transport of sound in sonic crystals. Nat. Phys. 13, 369–374 (2016).

  25. 25.

    Yang, Z., Gao, F., Yang, Y. & Zhang, B. Strain-induced gauge field and Landau levels in acoustic structures. Phys. Rev. Lett. 118, 194301 (2017).

  26. 26.

    Li, F. et al. Weyl points and Fermi arcs in a chiral phononic crystal. Nat. Phys. 14, 30–34 (2017).

  27. 27.

    Vanderbilt, D. & King-Smith, R. D. Electric polarization as a bulk quantity and its relation to surface charge. Phys. Rev. B 48, 4442–4455 (1993).

  28. 28.

    King-Smith, R. D. & Vanderbilt, D. Theory of polarization of crystalline solids. Phys. Rev. B 47, 1651(R)–1654(R) (1993).

  29. 29.

    Ni, X. et al. Observation of bulk polarization transitions and higher-order embedded topological eigenstates for sound. Nat. Mater. https://dx.doi.org/10.1038/s41563-018-0252-9 (2018).

  30. 30.

    Zhang, X. et al. Observation of second-order topological insulators in sonic crystals. Preprint at https://arxiv.org/abs/1806.10028 (2018).

  31. 31.

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

  32. 32.

    Schindler, F. et al. Higher-order topology in bismuth. Nat. Phys. 14, 918–924 (2018).

Download references


This work was sponsored by the Singapore Ministry of Education under grant nos MOE2015-T2-1-070, MOE2015-T2-2-008, MOE2016-T3-1-006 and Tier 1 RG174/16 (S), and the Young Thousand Talent Plan, China, National Natural Science Foundation of China under grant no. 61801426.

Author information


  1. Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore

    • Haoran Xue
    • , Yahui Yang
    • , Yidong Chong
    •  & Baile Zhang
  2. State Key Laboratory of Modern Optical Instrumentation, and College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou, China

    • Fei Gao
  3. Centre for Disruptive Photonic Technologies, Nanyang Technological University, Singapore, Singapore

    • Yidong Chong
    •  & Baile Zhang


  1. Search for Haoran Xue in:

  2. Search for Yahui Yang in:

  3. Search for Fei Gao in:

  4. Search for Yidong Chong in:

  5. Search for Baile Zhang in:


All the authors contributed extensively to this work. H.X. and Y.Y. fabricated the structures and performed measurements. H.X., Y.Y. and F. G. performed the simulations. Y.C. and B.Z. supervised the project.

Competing interests

The authors declare no competing interests.

Corresponding authors

Correspondence to Fei Gao or Yidong Chong or Baile Zhang.

Supplementary information

  1. Supplementary Information

    Supplementary Sections A–E, Supplementary Figures 1–5

About this article

Publication history




Issue Date



Further reading