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Higher-order topological semimetal in acoustic crystals


The notion of higher-order topological insulators has endowed materials with topological states beyond the first order. Particularly, a three-dimensional (3D) higher-order topological insulator can host topologically protected 1D hinge states, referred to as the second-order topological insulator, or 0D corner states, referred to as the third-order topological insulator. Similarly, a 3D higher-order topological semimetal can be envisaged if it hosts states on the 1D hinges. Here we report the realization of a second-order topological Weyl semimetal in a 3D-printed acoustic crystal, which possesses Weyl points in 3D momentum space, 2D Fermi arc states on surfaces and 1D gapless states on hinges. Like the arc surface states, the hinge states also connect the projections of the Weyl points. Our experimental results evidence the existence of the higher-order topological semimetal, which may pave the way towards innovative acoustic devices.

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Fig. 1: SOTSM for a 3D stacked breathing kagomé lattice.
Fig. 2: 3D acoustic crystal with Weyl points and Fermi arcs.
Fig. 3: Hinge state and acoustic pressure fields.
Fig. 4: Hinge states and response spectra for different structural parameters.

Data availability

Owing to their larger size, the data represented in Fig. 3 and Supplementary Fig. 10 are available on Zenodo at Source data are provided with this paper.


  1. 1.

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

    CAS  Article  Google Scholar 

  2. 2.

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

    CAS  Article  Google Scholar 

  3. 3.

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

    CAS  Article  Google Scholar 

  4. 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).

    Article  Google Scholar 

  5. 5.

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

    CAS  Article  Google Scholar 

  6. 6.

    Zhang, F., Kane, C. L. & Mele, E. J. Surface state magnetization and chiral edge states on topological insulators. Phys. Rev. Lett. 110, 046404 (2013).

    Article  Google Scholar 

  7. 7.

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

  8. 8.

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

    Article  Google Scholar 

  9. 9.

    Roy, B. Antiunitary symmetry protected higher-order topological phases. Phys. Rev. Res. 1, 032048(R) (2019).

    Article  Google Scholar 

  10. 10.

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

    CAS  Article  Google Scholar 

  11. 11.

    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).

    CAS  Article  Google Scholar 

  12. 12.

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

    CAS  Article  Google Scholar 

  13. 13.

    Mittal, S. et al. Photonic quadrupole topological phases. Nat. Photon. 13, 692–696 (2019).

    CAS  Article  Google Scholar 

  14. 14.

    Hassan, A. E. et al. Corner states of light in photonic waveguides. Nat. Photon. 13, 697–700 (2019).

    Article  Google Scholar 

  15. 15.

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

    CAS  Article  Google Scholar 

  16. 16.

    Xue, H., Yang, Y., Gao, F., Chong, Y. & Zhang, B. Acoustic higher-order topological insulator on a kagome lattice. Nat. Mater. 18, 108–112 (2019).

    CAS  Article  Google Scholar 

  17. 17.

    Ni, X., Weiner, M., Alù, A. & Khanikaev, A. B. Observation of higher-order topological acoustic states protected by generalized chiral symmetry. Nat. Mater. 18, 113–120 (2019).

    CAS  Article  Google Scholar 

  18. 18.

    Zhang, X. et al. Second-order topology and multidimensional topological transitions in sonic crystals. Nat. Phys. 15, 582–588 (2019).

    CAS  Article  Google Scholar 

  19. 19.

    Zhang, X. et al. Dimensional hierarchy of higher-order topology in three-dimensional sonic crystals. Nat. Commun. 10, 5331 (2019).

    Article  Google Scholar 

  20. 20.

    Weiner, M., Ni, X., Li, M., Alù, A. & Khanikaev, A. B. Demonstration of a third-order hierarchy of topological states in a three-dimensional acoustic metamaterial. Sci. Adv. 6, eaay4166 (2020).

    CAS  Article  Google Scholar 

  21. 21.

    Qi, Y. et al. Acoustic realization of quadrupole topological insulators. Phys. Rev. Lett. 124, 206601 (2020).

    CAS  Article  Google Scholar 

  22. 22.

    Xue, H. et al. Observation of an acoustic octupole topological insulator. Nat. Commun. 11, 2442 (2020).

    CAS  Article  Google Scholar 

  23. 23.

    Ni, X., Li, M., Weiner, M., Alù, A. & Khanikaev, A. B. Demonstration of a quantized acoustic octupole topological insulator. Nat. Commun. 11, 2108 (2020).

    CAS  Article  Google Scholar 

  24. 24.

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

    CAS  Article  Google Scholar 

  25. 25.

    Bao, J. et al. Topoelectrical circuit octupole insulator with topologically protected corner states. Phys. Rev. B 100, 201406 (2019).

    CAS  Article  Google Scholar 

  26. 26.

    Armitage, N. P., Mele, E. J. & Vishwanath, A. Weyl and Dirac semimetals in three-dimensional solids. Rev. Mod. Phys. 90, 015001 (2018).

    CAS  Article  Google Scholar 

  27. 27.

    Wan, X., Turner, A. M., Vishwanath, A. & Savrasov, S. Y. Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates. Phys. Rev. B 83, 205101 (2011).

    Article  Google Scholar 

  28. 28.

    Wang, Z. et al. Dirac semimetal and topological phase transitions in A3Bi (A = Na, K, Rb). Phys. Rev. B 85, 195320 (2012).

    Article  Google Scholar 

  29. 29.

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

    CAS  Article  Google Scholar 

  30. 30.

    Ezawa, M. Second-order topological insulators and loop-nodal semimetals in transition metal dichalcogenides XTe2 (X = Mo, W). Sci. Rep. 9, 5286 (2019).

    Article  Google Scholar 

  31. 31.

    Lin, M. & Hughes, T. L. Topological quadrupolar semimetals. Phys. Rev. B 98, 241103(R) (2018).

    Article  Google Scholar 

  32. 32.

    Wieder, B. J. et al. Strong and fragile topological Dirac semimetals with higher-order Fermi arcs. Nat. Commun. 11, 627 (2020).

    CAS  Article  Google Scholar 

  33. 33.

    Calugaru, D., Juricic, V. & Roy, B. Higher-order topological phases: a general principle of construction. Phys. Rev. B 99, 041301(R) (2019).

    Article  Google Scholar 

  34. 34.

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

    Article  Google Scholar 

  35. 35.

    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).

    CAS  Article  Google Scholar 

  36. 36.

    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 (2018).

    CAS  Article  Google Scholar 

  37. 37.

    Wang, H., Lin, Z., Jiang, B., Guo, G. & Jiang, J. Higher-order Weyl semimetals. Phys. Rev. Lett. 125, 146401 (2020).

    CAS  Article  Google Scholar 

  38. 38.

    Ghorashi, S. A. A., Li, T. & Hughes, T. L. Higher-order Weyl semimetals. Phys. Rev. Lett. 125, 266804 (2020).

    CAS  Article  Google Scholar 

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This work is supported by the National Key R & D Program of China (Grant no. 2017YFA0304203), the National Natural Science Foundation of China (Grant nos 11890701, 11674200, 11704128, 11774275, 11804101, 11974120, 11974005, 12034012, 12074128, and 12074232) and the Shanxi ‘1331 Project’ Key Subjects Construction.

Author information




G.C., Z.L. and S.J. conceived the idea. Q.W. and X.Z. calculated the theoretical results, designed the experiments and carried out the numerical simulations. Q.W., X.Z. and M.Y. performed the experiments. W.D., J.L. and X.H. guided the experimental measurement and analysed the experimental data. G.C. and Z.L. supervised the project. All the authors contributed to the preparation of the manuscript.

Corresponding authors

Correspondence to Gang Chen or Zhengyou Liu.

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The authors declare no competing interests.

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Peer review information Nature Materials thanks Alexander Khanikaev and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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

Supplementary information

Supplementary Information

Supplementary Sections I–VIII and Figs. 1–13.

Source data

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Source Data Fig. 2

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Supplementary Data 1

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Supplementary Data 2

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Supplementary Data 3

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Source Data for Supplementary Fig. 7.

Supplementary Data 8

Source Data for Supplementary Fig. 9.

Supplementary Data 9

Source Data for Supplementary Fig. 12.

Supplementary Data 10

Source Data for Supplementary Fig. 13.

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Wei, Q., Zhang, X., Deng, W. et al. Higher-order topological semimetal in acoustic crystals. Nat. Mater. 20, 812–817 (2021).

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