Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Acoustic higher-order topological insulator on a kagome lattice


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.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Prices vary by article type



Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Kagome lattice and its acoustic implementation.
Fig. 2: Eigenmode simulations of a triangular acoustic structure.
Fig. 3: Observation of topological corner states in a triangle-shaped finite acoustic structure.
Fig. 4: Observation of topological corner states in a parallelogram-shaped finite acoustic structure.

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.


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

    Article  CAS  Google Scholar 

  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).

    Article  Google Scholar 

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

    Article  Google Scholar 

  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).

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  CAS  Google Scholar 

  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).

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  CAS  Google Scholar 

  12. Lin, M. & Hughes, T. L. Topological quadrupolar semimetals. Preprint at (2017).

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

  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).

    Article  CAS  Google Scholar 

  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).

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

  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).

    Article  Google Scholar 

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

    Article  Google Scholar 

  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).

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

  29. Ni, X. et al. Observation of bulk polarization transitions and higher-order embedded topological eigenstates for sound. Nat. Mater. (2018).

  30. Zhang, X. et al. Observation of second-order topological insulators in sonic crystals. Preprint at (2018).

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

Authors and Affiliations



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.

Corresponding authors

Correspondence to Fei Gao, Yidong Chong or Baile Zhang.

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 A–E, Supplementary Figures 1–5

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xue, H., Yang, Y., Gao, F. et al. Acoustic higher-order topological insulator on a kagome lattice. Nature Mater 18, 108–112 (2019).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing