Abstract
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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Data availability
Owing to their larger size, the data represented in Fig. 3 and Supplementary Fig. 10 are available on Zenodo at https://zenodo.org/record/4441748#.YAFkdznisuV. Source data are provided with this paper.
References
Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).
Qi, X. L. & Zhang, S. C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011).
Benalcazar, W. A., Bernevig, B. A. & Hughes, T. L. Quantized electric multipole insulators. Science 357, 61–66 (2017).
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).
Ezawa, M. Higher-order topological insulators and semimetals on the breathing kagome and pyrochlore lattices. Phys. Rev. Lett. 120, 026801 (2018).
Zhang, F., Kane, C. L. & Mele, E. J. Surface state magnetization and chiral edge states on topological insulators. Phys. Rev. Lett. 110, 046404 (2013).
Song, Z., Fang, Z. & Fang, C. (d – 2)-dimensional edge states of rotation symmetry protected topological states. Phys. Rev. Lett. 119, 246402 (2017).
Schindler, F. et al. Higher-order topological insulators. Sci. Adv. 4, eaat0346 (2018).
Roy, B. Antiunitary symmetry protected higher-order topological phases. Phys. Rev. Res. 1, 032048(R) (2019).
Schindler, F. et al. Higher-order topology in bismuth. Nat. Phys. 14, 918–924 (2018).
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).
Noh, J. et al. Topological protection of photonic mid-gap defect modes. Nat. Photon. 12, 408–415 (2018).
Mittal, S. et al. Photonic quadrupole topological phases. Nat. Photon. 13, 692–696 (2019).
Hassan, A. E. et al. Corner states of light in photonic waveguides. Nat. Photon. 13, 697–700 (2019).
Serra-Garcia, M. et al. Observation of a phononic quadrupole topological insulator. Nature 555, 342–345 (2018).
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).
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).
Zhang, X. et al. Second-order topology and multidimensional topological transitions in sonic crystals. Nat. Phys. 15, 582–588 (2019).
Zhang, X. et al. Dimensional hierarchy of higher-order topology in three-dimensional sonic crystals. Nat. Commun. 10, 5331 (2019).
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).
Qi, Y. et al. Acoustic realization of quadrupole topological insulators. Phys. Rev. Lett. 124, 206601 (2020).
Xue, H. et al. Observation of an acoustic octupole topological insulator. Nat. Commun. 11, 2442 (2020).
Ni, X., Li, M., Weiner, M., Alù, A. & Khanikaev, A. B. Demonstration of a quantized acoustic octupole topological insulator. Nat. Commun. 11, 2108 (2020).
Imhof, S. et al. Topoelectrical-circuit realization of topological corner modes. Nat. Phys. 14, 925–929 (2018).
Bao, J. et al. Topoelectrical circuit octupole insulator with topologically protected corner states. Phys. Rev. B 100, 201406 (2019).
Armitage, N. P., Mele, E. J. & Vishwanath, A. Weyl and Dirac semimetals in three-dimensional solids. Rev. Mod. Phys. 90, 015001 (2018).
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).
Wang, Z. et al. Dirac semimetal and topological phase transitions in A3Bi (A = Na, K, Rb). Phys. Rev. B 85, 195320 (2012).
Ezawa, M. Magnetic second-order topological insulators and semimetals. Phys. Rev. B 97, 155305 (2018).
Ezawa, M. Second-order topological insulators and loop-nodal semimetals in transition metal dichalcogenides XTe2 (X = Mo, W). Sci. Rep. 9, 5286 (2019).
Lin, M. & Hughes, T. L. Topological quadrupolar semimetals. Phys. Rev. B 98, 241103(R) (2018).
Wieder, B. J. et al. Strong and fragile topological Dirac semimetals with higher-order Fermi arcs. Nat. Commun. 11, 627 (2020).
Calugaru, D., Juricic, V. & Roy, B. Higher-order topological phases: a general principle of construction. Phys. Rev. B 99, 041301(R) (2019).
Ni, X., Gorlach, M. A., Alù, A. & Khanikaev, A. B. Topological edge states in acoustic kagome lattices. N. J. Phys. 19, 055002 (2017).
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).
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).
Wang, H., Lin, Z., Jiang, B., Guo, G. & Jiang, J. Higher-order Weyl semimetals. Phys. Rev. Lett. 125, 146401 (2020).
Ghorashi, S. A. A., Li, T. & Hughes, T. L. Higher-order Weyl semimetals. Phys. Rev. Lett. 125, 266804 (2020).
Acknowledgements
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.
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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.
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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.
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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). https://doi.org/10.1038/s41563-021-00933-4
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DOI: https://doi.org/10.1038/s41563-021-00933-4
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