Skip to main content

Thank you for visiting nature.com. 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.

  • Review Article
  • Published:

Topological acoustics

Nature Reviews Materials volume 7, pages 974–990 (2022)Cite this article

Subjects

Abstract

Topological acoustics is an emerging field that lies at the intersection of condensed matter physics, mechanical structural design and acoustics engineering. It explores the design and construction of novel artificial structures, such as acoustic metamaterials and phononic crystals, to manipulate sound waves robustly, taking advantage of topological protection. Early work on topological acoustics was limited to duplicating topological phases that have been understood in condensed matter systems, but recent advances have shifted to exploring new topological concepts that are difficult to realize in other physical systems, such as various topological semimetal phases, and topological phases associated with Floquet engineering, fragile topology, non-Hermiticity and synthetic dimensions. These developments demonstrate the unique advantages of topological acoustic systems and their role in developing topological physics. In this Review, we survey the fundamental mechanisms, basic designs and practical realizations of topological phases in acoustic systems and provide an overview of future directions and potential applications.

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

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

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

Fig. 1: Summary of different types of topological phase.
Fig. 2: Acoustic analogues of 2D topological insulating phases.
Fig. 3: Acoustic topological phases with quantized dipole and multipole moments.
Fig. 4: Acoustic Weyl crystals.
Fig. 5: Acoustic topological semimetals beyond conventional Weyl phases.
Fig. 6: Novel acoustic topological phases.

Similar content being viewed by others

References

  1. Kurokawa, K. Fukushima nuclear accident independent investigation commission by the National Diet of Japan. Nippon Genshiryoku Gakkaishi 55, 146–151 (2013).

    Google Scholar 

  2. Leventhall, H. G. Low frequency noise and annoyance. Noise Health 6, 59 (2004).

    CAS  Google Scholar 

  3. Ozcelik, A. et al. Acoustic tweezers for the life sciences. Nat. Methods 15, 1021–1028 (2018).

    Article  CAS  Google Scholar 

  4. Balram, K. C., Davanço, M. I., Song, J. D. & Srinivasan, K. Coherent coupling between radiofrequency, optical and acoustic waves in piezo-optomechanical circuits. Nat. Photon 10, 346–352 (2016).

    Article  CAS  Google Scholar 

  5. Chu, Y. et al. Quantum acoustics with superconducting qubits. Science 358, 199–202 (2017).

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  9. Maldovan, M. Sound and heat revolutions in phononics. Nature 503, 209–217 (2013).

    Article  CAS  Google Scholar 

  10. Kushwaha, M. S., Halevi, P., Dobrzynski, L. & Djafari-Rouhani, B. Acoustic band structure of periodic elastic composites. Phys. Rev. Lett. 71, 2022 (1993).

    Article  CAS  Google Scholar 

  11. Sigalas, M. & Economou, E. N. Band structure of elastic waves in two dimensional systems. Solid State Commun. 86, 141–143 (1993).

    Article  CAS  Google Scholar 

  12. Cummer, S. A., Christensen, J. & Alù, A. Controlling sound with acoustic metamaterials. Nat. Rev. Mater. 1, 16001 (2016).

    Article  Google Scholar 

  13. Ma, G., Xiao, M. & Chan, C. T. Topological phases in acoustic and mechanical systems. Nat. Rev. Phys. 1, 281–294 (2019).

    Article  Google Scholar 

  14. Zhang, X., Xiao, M., Cheng, Y., Lu, M.-H. & Christensen, J. Topological sound. Commun. Phys. 1, 97 (2018).

    Article  Google Scholar 

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

    Article  CAS  Google Scholar 

  16. Klitzing, K. V., Dorda, G. & Pepper, M. New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Phys. Rev. Lett. 45, 494 (1980).

    Article  Google Scholar 

  17. Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405 (1982).

    Article  CAS  Google Scholar 

  18. Roux, P., de Rosny, J., Tanter, M. & Fink, M. The Aharonov–Bohm effect revisited by an acoustic time-reversal mirror. Phys. Rev. Lett. 79, 3170 (1997).

    Article  CAS  Google Scholar 

  19. Fleury, R., Sounas, D. L., Sieck, C. F., Haberman, M. R. & Alù, A. Sound isolation and giant linear nonreciprocity in a compact acoustic circulator. Science 343, 516–519 (2014).

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

  21. Ni, X. et al. Topologically protected one-way edge mode in networks of acoustic resonators with circulating air flow. New J. Phys. 17, 053016 (2015).

    Article  Google Scholar 

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

    Article  CAS  Google Scholar 

  23. Brekhovskikh, L. M. & Godin, O. A. Acoustics of Layered Media II: Point Sources and Bounded Beams (Springer, 1999).

  24. 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 (1988).

    Article  CAS  Google Scholar 

  25. Chen, Z.-G. & Wu, Y. Tunable topological phononic crystals. Phys. Rev. Appl. 5, 054021 (2016).

    Article  Google Scholar 

  26. Souslov, A., Van Zuiden, B. C., Bartolo, D. & Vitelli, V. Topological sound in active-liquid metamaterials. Nat. Phys. 13, 1091–1094 (2017).

    Article  CAS  Google Scholar 

  27. Ding, Y. et al. Experimental demonstration of acoustic Chern insulators. Phys. Rev. Lett. 122, 014302 (2019).

    Article  Google Scholar 

  28. 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 

  29. Wen, X. et al. Acoustic Landau quantization and quantum-Hall-like edge states. Nat. Phys. 15, 352–356 (2019).

    Article  CAS  Google Scholar 

  30. Guinea, F., Katsnelson, M. & Geim, A. Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering. Nat. Phys. 6, 30–33 (2010).

    Article  CAS  Google Scholar 

  31. Peri, V., Serra-Garcia, M., Ilan, R. & Huber, S. D. Axial-field-induced chiral channels in an acoustic Weyl system. Nat. Phys. 15, 357–361 (2019).

    Article  CAS  Google Scholar 

  32. Kane, C. L. & Mele, E. J. Z2 topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005).

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  34. Bernevig, B. A., Hughes, T. L. & Zhang, S.-C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314, 1757–1761 (2006).

    Article  CAS  Google Scholar 

  35. Wu, C., Bernevig, B. A. & Zhang, S.-C. Helical liquid and the edge of quantum spin Hall systems. Phys. Rev. Lett. 96, 106401 (2006).

    Article  Google Scholar 

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

    Article  CAS  Google Scholar 

  37. Mei, J., Chen, Z. & Wu, Y. Pseudo-time-reversal symmetry and topological edge states in two-dimensional acoustic crystals. Sci. Rep. 6, 32752 (2016).

    Article  CAS  Google Scholar 

  38. Zhang, Z. et al. Topological creation of acoustic pseudospin multipoles in a flow-free symmetry-broken metamaterial lattice. Phys. Rev. Lett. 118, 084303 (2017).

    Article  Google Scholar 

  39. Yves, S., Fleury, R., Lemoult, F., Fink, M. & Lerosey, G. Topological acoustic polaritons: robust sound manipulation at the subwavelength scale. New J. Phys. 19, 075003 (2017).

    Article  Google Scholar 

  40. Xia, B.-Z. et al. Topological phononic insulator with robust pseudospin-dependent transport. Phys. Rev. B 96, 094106 (2017).

    Article  Google Scholar 

  41. Deng, Y., Ge, H., Tian, Y., Lu, M. & Jing, Y. Observation of zone folding induced acoustic topological insulators and the role of spin-mixing defects. Phys. Rev. B 96, 184305 (2017).

    Article  Google Scholar 

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

    Article  Google Scholar 

  43. Deng, W. et al. Acoustic spin-Chern insulator induced by synthetic spin–orbit coupling with spin conservation breaking. Nat. Commun. 11, 3227 (2020).

    Article  CAS  Google Scholar 

  44. Christiansen, R. E., Wang, F. & Sigmund, O. Topological insulators by topology optimization. Phys. Rev. Lett. 122, 234502 (2019).

    Article  CAS  Google Scholar 

  45. Xia, J.-P. et al. Programmable coding acoustic topological insulator. Adv. Mater. 30, 1805002 (2018).

    Article  Google Scholar 

  46. Fu, L., Kane, C. L. & Mele, E. J. Topological insulators in three dimensions. Phys. Rev. Lett. 98, 106803 (2007).

    Article  Google Scholar 

  47. He, C. et al. Acoustic analogues of three-dimensional topological insulators. Nat. Commun. 11, 2318 (2020).

    Article  CAS  Google Scholar 

  48. Schaibley, J. R. et al. Valleytronics in 2D materials. Nat. Rev. Mater. 1, 16055 (2016).

    Article  CAS  Google Scholar 

  49. Lu, J., Qiu, C., Ke, M. & Liu, Z. Valley vortex states in sonic crystals. Phys. Rev. Lett. 116, 093901 (2016).

    Article  Google Scholar 

  50. Lu, J. et al. Observation of topological valley transport of sound in sonic crystals. Nat. Phys. 13, 369–374 (2017).

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

  52. Lu, J. et al. Dirac cones in two-dimensional artificial crystals for classical waves. Phys. Rev. B 89, 134302 (2014).

    Article  Google Scholar 

  53. Ye, L. et al. Observation of acoustic valley vortex states and valley-chirality locked beam splitting. Phys. Rev. B 95, 174106 (2017).

    Article  Google Scholar 

  54. Wu, X. et al. Direct observation of valley-polarized topological edge states in designer surface plasmon crystals. Nat. Commun. 8, 1304 (2017).

    Article  Google Scholar 

  55. Zhang, L. et al. Valley kink states and topological channel intersections in substrate-integrated photonic circuitry. Laser Photon. Rev. 13, 1900159 (2019).

    Article  Google Scholar 

  56. Orazbayev, B. & Fleury, R. Quantitative robustness analysis of topological edge modes in C6 and valley-Hall metamaterial waveguides. Nanophotonics 8, 1433–1441 (2019).

    Article  Google Scholar 

  57. Zhang, Z. et al. Topological acoustic delay line. Phys. Rev. Appl. 9, 034032 (2018).

    Article  Google Scholar 

  58. Ma, T. & Shvets, G. All-Si valley-Hall photonic topological insulator. New J. Phys. 18, 025012 (2016).

    Article  Google Scholar 

  59. Gao, F. et al. Topologically protected refraction of robust kink states in valley photonic crystals. Nat. Phys. 14, 140–144 (2018).

    Article  CAS  Google Scholar 

  60. Jia, D. et al. Topological refraction in dual-band valley sonic crystals. Phys. Rev. B 103, 144309 (2021).

    Article  CAS  Google Scholar 

  61. Zhang, Z. et al. Directional acoustic antennas based on valley-Hall topological insulators. Adv. Mater. 30, 1803229 (2018).

    Article  Google Scholar 

  62. Lu, J. et al. Valley topological phases in bilayer sonic crystals. Phys. Rev. Lett. 120, 116802 (2018).

    Article  CAS  Google Scholar 

  63. Zhu, Z. et al. Acoustic valley spin Chern insulators. Phys. Rev. Appl. 16, 014058 (2021).

    Article  CAS  Google Scholar 

  64. Wang, M. et al. Valley-locked waveguide transport in acoustic heterostructures. Nat. Commun. 11, 3000 (2020).

    Article  CAS  Google Scholar 

  65. Wang, M., Ye, L., Christensen, J. & Liu, Z. Valley physics in non-Hermitian artificial acoustic boron nitride. Phys. Rev. Lett. 120, 246601 (2018).

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  68. Berry, M. V. Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392, 45–57 (1984).

    Article  Google Scholar 

  69. Peterson, C. W., Li, T., Benalcazar, W. A., Hughes, T. L. & Bahl, G. A fractional corner anomaly reveals higher-order topology. Science 368, 1114–1118 (2020).

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  71. Zak, J. Berry’s phase for energy bands in solids. Phys. Rev. Lett. 62, 2747 (1989).

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

  73. Xiao, Y.-X., Ma, G., Zhang, Z.-Q. & Chan, C. T. Topological subspace-induced bound state in the continuum. Phys. Rev. Lett. 118, 166803 (2017).

    Article  Google Scholar 

  74. Li, X. et al. Su–Schrieffer–Heeger model inspired acoustic interface states and edge states. Appl. Phys. Lett. 113, 203501 (2018).

    Article  Google Scholar 

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

    Article  CAS  Google Scholar 

  76. Zangeneh-Nejad, F. & Fleury, R. Topological Fano resonances. Phys. Rev. Lett. 122, 014301 (2019).

    Article  CAS  Google Scholar 

  77. Zangeneh-Nejad, F. & Fleury, R. Topological analog signal processing. Nat. Commun. 10, 2058 (2019).

    Article  Google Scholar 

  78. Xie, B. et al. Higher-order band topology. Nat. Rev. Phys. 3, 520–532 (2021).

    Article  Google Scholar 

  79. 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).

    Article  CAS  Google Scholar 

  80. 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).

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  82. Xue, H. et al. Realization of an acoustic third-order topological insulator. Phys. Rev. Lett. 122, 244301 (2019).

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

  84. Zhang, Z. et al. Deep-subwavelength holey acoustic second-order topological insulators. Adv. Mater. 31, 1904682 (2019).

    Article  CAS  Google Scholar 

  85. 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).

    Article  CAS  Google Scholar 

  86. Zhang, X. et al. Symmetry-protected hierarchy of anomalous multipole topological band gaps in nonsymmorphic metacrystals. Nat. Commun. 11, 65 (2020).

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  90. Yang, Z.-Z., Peng, Y.-Y., Li, X., Zou, X.-Y. & Cheng, J.-C. Boundary-dependent corner states in topological acoustic resonator array. Appl. Phys. Lett. 117, 113501 (2020).

    Article  CAS  Google Scholar 

  91. Meng, F., Chen, Y., Li, W., Jia, B. & Huang, X. Realization of multidimensional sound propagation in 3D acoustic higher-order topological insulator. Appl. Phys. Lett. 117, 151903 (2020).

    Article  CAS  Google Scholar 

  92. Zheng, S. et al. Three-dimensional higher-order topological acoustic system with multidimensional topological states. Phys. Rev. B 102, 104113 (2020).

    Article  CAS  Google Scholar 

  93. Yan, M. et al. Acoustic square-root topological states. Phys. Rev. B 102, 180102 (2020).

    Article  CAS  Google Scholar 

  94. Huang, X. et al. Acoustic higher-order topology derived from first-order with built-in Zeeman-like fields. Sci. Bull. 67, 488–494 (2022).

    Article  CAS  Google Scholar 

  95. Zhang, X., Liu, L., Lu, M.-H. & Chen, Y.-F. Valley-selective topological corner states in sonic crystals. Phys. Rev. Lett. 126, 156401 (2021).

    Article  CAS  Google Scholar 

  96. Yang, Y. et al. Hybrid-order topological insulators in a phononic crystal. Phys. Rev. Lett. 126, 156801 (2021).

    Article  CAS  Google Scholar 

  97. Zheng, L.-Y. & Christensen, J. Dirac hierarchy in acoustic topological insulators. Phys. Rev. Lett. 127, 156401 (2021).

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  100. Xie, B.-Y. et al. Second-order photonic topological insulator with corner states. Phys. Rev. B 98, 205147 (2018).

    Article  CAS  Google Scholar 

  101. Benalcazar, W. A., Li, T. & Hughes, T. L. Quantization of fractional corner charge in Cn-symmetric higher-order topological crystalline insulators. Phys. Rev. B 99, 245151 (2019).

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  103. 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 

  104. Wieder, B. J. et al. Topological materials discovery from crystal symmetry. Nat. Rev. Mater. 7, 196–216 (2022).

    Article  Google Scholar 

  105. Kang, B., Shiozaki, K. & Cho, G. Y. Many-body order parameters for multipoles in solids. Phys. Rev. B 100, 245134 (2019).

    Article  CAS  Google Scholar 

  106. Wheeler, W. A., Wagner, L. K. & Hughes, T. L. Many-body electric multipole operators in extended systems. Phys. Rev. B 100, 245135 (2019).

    Article  CAS  Google Scholar 

  107. Chen, K. et al. Nonlocal topological insulators: deterministic aperiodic arrays supporting localized topological states protected by nonlocal symmetries. Proc. Natl Acad. Sci. USA 118, e2100691118 (2021).

    Article  CAS  Google Scholar 

  108. Wei, Q. et al. 3D hinge transport in acoustic higher-order topological insulators. Phys. Rev. Lett. 127, 255501 (2021).

    Article  CAS  Google Scholar 

  109. Du, J., Li, T., Fan, X., Zhang, Q. & Qiu, C. Acoustic realization of surface-obstructed topological insulators. Phys. Rev. Lett. 128, 224301 (2022).

    Article  CAS  Google Scholar 

  110. Hasan, M. Z. et al. Weyl, Dirac and high-fold chiral fermions in topological quantum matter. Nat. Rev. Mater. 6, 784–803 (2021).

    Article  CAS  Google Scholar 

  111. 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 

  112. 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).

    Article  CAS  Google Scholar 

  113. Ge, H. et al. Experimental observation of acoustic Weyl points and topological surface states. Phys. Rev. Appl. 10, 014017 (2018).

    Article  CAS  Google Scholar 

  114. Xie, B. et al. Experimental realization of type-II Weyl points and Fermi arcs in phononic crystal. Phys. Rev. Lett. 122, 104302 (2019).

    Article  CAS  Google Scholar 

  115. Huang, X., Deng, W., Li, F., Lu, J. & Liu, Z. Ideal type-II Weyl phase and topological transition in phononic crystals. Phys. Rev. Lett. 124, 206802 (2020).

    Article  CAS  Google Scholar 

  116. Zangeneh-Nejad, F. & Fleury, R. Zero-index Weyl metamaterials. Phys. Rev. Lett. 125, 054301 (2020).

    Article  CAS  Google Scholar 

  117. Nielsen, H. & Ninomiya, M. A no-go theorem for regularizing chiral fermions. Phys. Lett. B 105, 219–223 (1981).

    Article  Google Scholar 

  118. 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 

  119. Xiao, D., Chang, M.-C. & Niu, Q. Berry phase effects on electronic properties. Rev. Mod. Phys. 82, 1959 (2010).

    Article  CAS  Google Scholar 

  120. He, H. et al. Topological negative refraction of surface acoustic waves in a Weyl phononic crystal. Nature 560, 61–64 (2018).

    Article  CAS  Google Scholar 

  121. Luo, L. et al. Observation of a phononic higher-order Weyl semimetal. Nat. Mater. 20, 794–799 (2021).

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  123. Cheng, H., Sha, Y., Liu, R., Fang, C. & Lu, L. Discovering topological surface states of Dirac points. Phys. Rev. Lett. 124, 104301 (2020).

    Article  CAS  Google Scholar 

  124. Cai, X. et al. Symmetry-enforced three-dimensional Dirac phononic crystals. Light Sci. Appl. 9, 38 (2020).

    Article  CAS  Google Scholar 

  125. Xie, B. et al. Dirac points and the transition towards Weyl points in three-dimensional sonic crystals. Light Sci. Appl. 9, 201 (2020).

    Article  CAS  Google Scholar 

  126. Su, Z., Gao, W., Liu, B., Huang, L. & Wang, Y. Three-dimensional Dirac semimetal metamaterial enabled by negative couplings. New J. Phys. 24, 033025 (2022).

    Article  Google Scholar 

  127. Kargarian, M., Randeria, M. & Lu, Y.-M. Are the surface Fermi arcs in Dirac semimetals topologically protected? Proc. Natl Acad. Sci. USA 113, 8648–8652 (2016).

    Article  CAS  Google Scholar 

  128. Qiu, H., Xiao, M., Zhang, F. & Qiu, C. Higher-order Dirac sonic crystals. Phys. Rev. Lett. 127, 146601 (2021).

    Article  CAS  Google Scholar 

  129. Xia, C.-H., Lai, H.-S., Sun, X.-C., He, C. & Chen, Y.-F. Experimental demonstration of bulk-hinge correspondence in a three-dimensional topological Dirac acoustic crystal. Phys. Rev. Lett. 128, 115701 (2022).

    Article  CAS  Google Scholar 

  130. Yang, Y. et al. Topological triply degenerate point with double Fermi arcs. Nat. Phys. 15, 645–649 (2019).

    Article  CAS  Google Scholar 

  131. Deng, W. et al. Acoustic spin-1 Weyl semimetal. Sci. China Phys. Mech. Astron. 63, 287032 (2020).

    Article  Google Scholar 

  132. Fang, C., Gilbert, M. J., Dai, X. & Bernevig, B. A. Multi-Weyl topological semimetals stabilized by point group symmetry. Phys. Rev. Lett. 108, 266802 (2012).

    Article  Google Scholar 

  133. He, H. et al. Observation of quadratic Weyl points and double-helicoid arcs. Nat. Commun. 11, 1820 (2020).

    Article  CAS  Google Scholar 

  134. Deng, W. et al. Nodal rings and drumhead surface states in phononic crystals. Nat. Commun. 10, 1769 (2019).

    Article  Google Scholar 

  135. Geng, Z.-G. et al. Topological nodal line states in three-dimensional ball-and-stick sonic crystals. Phys. Rev. B 100, 224105 (2019).

    Article  CAS  Google Scholar 

  136. Qiu, H. et al. Straight nodal lines and waterslide surface states observed in acoustic metacrystals. Phys. Rev. B 100, 041303 (2019).

    Article  CAS  Google Scholar 

  137. Xiao, M., Sun, X.-Q. & Fan, S. Nodal chain semimetal in geometrically frustrated systems. Phys. Rev. B 99, 094206 (2019).

    Article  CAS  Google Scholar 

  138. Lu, J. et al. Nodal-chain semimetal states and topological focusing in phononic crystals. Phys. Rev. Appl. 13, 054080 (2020).

    Article  CAS  Google Scholar 

  139. Yang, Y. et al. Observation of a topological nodal surface and its surface-state arcs in an artificial acoustic crystal. Nat. Commun. 10, 5185 (2019).

    Article  Google Scholar 

  140. Xiao, M. et al. Experimental demonstration of acoustic semimetal with topologically charged nodal surface. Sci. Adv. 6, eaav2360 (2020).

    Article  Google Scholar 

  141. Rudner, M. S. & Lindner, N. H. Band structure engineering and non-equilibrium dynamics in Floquet topological insulators. Nat. Rev. Phys. 2, 229–244 (2020).

    Article  CAS  Google Scholar 

  142. Fleury, R., Khanikaev, A. B. & Alù, A. Floquet topological insulators for sound. Nat. Commun. 7, 11744 (2016).

    Article  CAS  Google Scholar 

  143. Liu, X., Guo, Q. & Yang, J. Miniaturization of Floquet topological insulators for airborne acoustics by thermal control. Appl. Phys. Lett. 114, 054102 (2019).

    Article  Google Scholar 

  144. Liang, G. & Chong, Y. Optical resonator analog of a two-dimensional topological insulator. Phys. Rev. Lett. 110, 203904 (2013).

    Article  CAS  Google Scholar 

  145. Pasek, M. & Chong, Y. Network models of photonic Floquet topological insulators. Phys. Rev. B 89, 075113 (2014).

    Article  Google Scholar 

  146. Gao, F. et al. Probing topological protection using a designer surface plasmon structure. Nat. Commun. 7, 11619 (2016).

    Article  CAS  Google Scholar 

  147. Peng, Y.-G. et al. Experimental demonstration of anomalous Floquet topological insulator for sound. Nat. Commun. 7, 13368 (2016).

    Article  CAS  Google Scholar 

  148. Peng, Y.-G., Shen, Y.-X., Zhao, D.-G. & Zhu, X.-F. Low-loss and broadband anomalous Floquet topological insulator for airborne sound. Appl. Phys. Lett. 110, 173505 (2017).

    Article  Google Scholar 

  149. Wei, Q., Tian, Y., Zuo, S.-Y., Cheng, Y. & Liu, X.-J. Experimental demonstration of topologically protected efficient sound propagation in an acoustic waveguide network. Phys. Rev. B 95, 094305 (2017).

    Article  Google Scholar 

  150. Zhang, Z., Delplace, P. & Fleury, R. Superior robustness of anomalous non-reciprocal topological edge states. Nature 598, 293–297 (2021).

    Article  CAS  Google Scholar 

  151. Rudner, M. S., Lindner, N. H., Berg, E. & Levin, M. Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems. Phys. Rev. X 3, 031005 (2013).

    CAS  Google Scholar 

  152. Peng, Y.-G., Geng, Z.-G. & Zhu, X.-F. Topologically protected bound states in one-dimensional Floquet acoustic waveguide systems. J. Appl. Phys. 123, 091716 (2018).

    Article  Google Scholar 

  153. Peng, Y.-G. et al. Chirality-assisted three-dimensional acoustic Floquet lattices. Phys. Rev. Res. 1, 033149 (2019).

    Article  CAS  Google Scholar 

  154. Zhu, W., Xue, H., Gong, J., Chong, Y. & Zhang, B. Time-periodic corner states from Floquet higher-order topology. Nat. Commun. 13, 11 (2022).

    Article  CAS  Google Scholar 

  155. Po, H. C., Watanabe, H. & Vishwanath, A. Fragile topology and Wannier obstructions. Phys. Rev. Lett. 121, 126402 (2018).

    Article  CAS  Google Scholar 

  156. Song, Z. et al. All magic angles in twisted bilayer graphene are topological. Phys. Rev. Lett. 123, 036401 (2019).

    Article  CAS  Google Scholar 

  157. Wang, H.-X., Guo, G.-Y. & Jiang, J.-H. Band topology in classical waves: Wilson-loop approach to topological numbers and fragile topology. New J. Phys. 21, 093029 (2019).

    Article  CAS  Google Scholar 

  158. Alexandradinata, A., Höller, J., Wang, C., Cheng, H. & Lu, L. Crystallographic splitting theorem for band representations and fragile topological photonic crystals. Phys. Rev. B 102, 115117 (2020).

    Article  CAS  Google Scholar 

  159. Peri, V. et al. Experimental characterization of fragile topology in an acoustic metamaterial. Science 367, 797–800 (2020).

    Article  CAS  Google Scholar 

  160. Ghatak, A. & Das, T. New topological invariants in non-Hermitian systems. J. Phys. Condens. Matter 31, 263001 (2019).

    Article  CAS  Google Scholar 

  161. Bergholtz, E. J., Budich, J. C. & Kunst, F. K. Exceptional topology of non-Hermitian systems. Rev. Mod. Phys. 93, 015005 (2021).

    Article  Google Scholar 

  162. Zhu, W. et al. Simultaneous observation of a topological edge state and exceptional point in an open and non-Hermitian acoustic system. Phys. Rev. Lett. 121, 124501 (2018).

    Article  CAS  Google Scholar 

  163. Zhang, Z., López, M. R., Cheng, Y., Liu, X. & Christensen, J. Non-Hermitian sonic second-order topological insulator. Phys. Rev. Lett. 122, 195501 (2019).

    Article  CAS  Google Scholar 

  164. López, M. R., Zhang, Z., Torrent, D. & Christensen, J. Multiple scattering theory of non-Hermitian sonic second-order topological insulators. Commun. Phys. 2, 132 (2019).

    Article  Google Scholar 

  165. Hu, B. et al. Non-Hermitian topological whispering gallery. Nature 597, 655–659 (2021).

    Article  CAS  Google Scholar 

  166. Gao, H. et al. Observation of topological edge states induced solely by non-Hermiticity in an acoustic crystal. Phys. Rev. B 101, 180303 (2020).

    Article  CAS  Google Scholar 

  167. Gao, H. et al. Non-Hermitian route to higher-order topology in an acoustic crystal. Nat. Commun. 12, 1888 (2021).

    Article  CAS  Google Scholar 

  168. Zhang, X., Tian, Y., Jiang, J.-H., Lu, M.-H. & Chen, Y.-F. Observation of higher-order non-Hermitian skin effect. Nat. Commun. 12, 5377 (2021).

    Article  CAS  Google Scholar 

  169. Zhang, L. et al. Acoustic non-Hermitian skin effect from twisted winding topology. Nat. Commun. 12, 6297 (2021).

    Article  CAS  Google Scholar 

  170. Zhang, K. et al. Observation of topological properties of non-Hermitian crystal systems with diversified coupled resonators chains. J. Appl. Phys. 130, 064502 (2021).

    Article  CAS  Google Scholar 

  171. Kawabata, K., Shiozaki, K., Ueda, M. & Sato, M. Symmetry and topology in non-Hermitian physics. Phys. Rev. X 9, 041015 (2019).

    CAS  Google Scholar 

  172. Brody, D. C. Biorthogonal quantum mechanics. J. Phys. A 47, 035305 (2013).

    Article  Google Scholar 

  173. Gong, Z. et al. Topological phases of non-Hermitian systems. Phys. Rev. X 8, 031079 (2018).

    CAS  Google Scholar 

  174. Yao, S. & Wang, Z. Edge states and topological invariants of non-Hermitian systems. Phys. Rev. Lett. 121, 086803 (2018).

    Article  CAS  Google Scholar 

  175. Ding, K., Ma, G., Xiao, M., Zhang, Z. & Chan, C. T. Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization. Phys. Rev. X 6, 021007 (2016).

    Google Scholar 

  176. Tang, W. et al. Exceptional nexus with a hybrid topological invariant. Science 370, 1077–1080 (2020).

    Article  CAS  Google Scholar 

  177. Tang, W., Ding, K. & Ma, G. Direct measurement of topological properties of an exceptional parabola. Phys. Rev. Lett. 127, 034301 (2021).

    Article  CAS  Google Scholar 

  178. Ozawa, T. & Price, H. M. Topological quantum matter in synthetic dimensions. Nat. Rev. Phys. 1, 349–357 (2019).

    Article  Google Scholar 

  179. Fan, X. et al. Probing Weyl physics with one-dimensional sonic crystals. Phys. Rev. Lett. 122, 136802 (2019).

    Article  CAS  Google Scholar 

  180. Zangeneh-Nejad, F. & Fleury, R. Experimental observation of the acoustic Z2 Weyl semimetallic phase in synthetic dimensions. Phys. Rev. B 102, 064309 (2020).

    Article  CAS  Google Scholar 

  181. Fan, X., Xia, T., Qiu, H., Zhang, Q. & Qiu, C. Tracking valley topology with synthetic Weyl paths. Phys. Rev. Lett. 128, 216403 (2022).

    Article  CAS  Google Scholar 

  182. Ni, X. et al. Observation of Hofstadter butterfly and topological edge states in reconfigurable quasi-periodic acoustic crystals. Commun. Phys. 2, 55 (2019).

    Article  Google Scholar 

  183. Long, Y. & Ren, J. Floquet topological acoustic resonators and acoustic Thouless pumping. J. Acoust. Soc. Am. 146, 742–747 (2019).

    Article  CAS  Google Scholar 

  184. Apigo, D. J., Cheng, W., Dobiszewski, K. F., Prodan, E. & Prodan, C. Observation of topological edge modes in a quasiperiodic acoustic waveguide. Phys. Rev. Lett. 122, 095501 (2019).

    Article  CAS  Google Scholar 

  185. Xu, X. et al. Physical observation of a robust acoustic pumping in waveguides with dynamic boundary. Phys. Rev. Lett. 125, 253901 (2020).

    Article  CAS  Google Scholar 

  186. Chen, Z.-G., Tang, W., Zhang, R.-Y., Chen, Z. & Ma, G. Landau–Zener transition in the dynamic transfer of acoustic topological states. Phys. Rev. Lett. 126, 054301 (2021).

    Article  CAS  Google Scholar 

  187. Chen, Z.-G., Zhu, W., Tan, Y., Wang, L. & Ma, G. Acoustic realization of a four-dimensional higher-order Chern insulator and boundary-modes engineering. Phys. Rev. X 11, 011016 (2021).

    CAS  Google Scholar 

  188. Cheng, W., Prodan, E. & Prodan, C. Revealing the boundary Weyl physics of the four-dimensional Hall effect via phason engineering in metamaterials. Phys. Rev. Appl. 16, 044032 (2021).

    Article  CAS  Google Scholar 

  189. Chen, H. et al. Creating synthetic spaces for higher-order topological sound transport. Nat. Commun. 12, 5028 (2021).

    Article  CAS  Google Scholar 

  190. Mermin, N. D. The topological theory of defects in ordered media. Rev. Mod. Phys. 51, 591 (1979).

    Article  CAS  Google Scholar 

  191. Kosterlitz, J. M. Nobel lecture: topological defects and phase transitions. Rev. Mod. Phys. 89, 040501 (2017).

    Article  Google Scholar 

  192. Ran, Y., Zhang, Y. & Vishwanath, A. One-dimensional topologically protected modes in topological insulators with lattice dislocations. Nat. Phys. 5, 298–303 (2009).

    Article  CAS  Google Scholar 

  193. Teo, J. C. & Kane, C. L. Topological defects and gapless modes in insulators and superconductors. Phys. Rev. B 82, 115120 (2010).

    Article  Google Scholar 

  194. Wang, Q. et al. Vortex states in an acoustic Weyl crystal with a topological lattice defect. Nat. Commun. 12, 3654 (2021).

    Article  CAS  Google Scholar 

  195. Ye, L. et al. Topological dislocation modes in three-dimensional acoustic topological insulators. Nat. Commun. 13, 508 (2022).

    Article  CAS  Google Scholar 

  196. Xue, H. et al. Observation of dislocation-induced topological modes in a three-dimensional acoustic topological insulator. Phys. Rev. Lett. 127, 214301 (2021).

    Article  CAS  Google Scholar 

  197. Lin, Z.-K. et al. Topological Wannier cycles induced by sub-unit-cell artificial gauge flux in a sonic crystal. Nat. Mater. 21, 430–437 (2022).

    Article  CAS  Google Scholar 

  198. Smirnova, D., Leykam, D., Chong, Y. & Kivshar, Y. Nonlinear topological photonics. Appl. Phys. Rev. 7, 021306 (2020).

    Article  CAS  Google Scholar 

  199. Chen, B. G.-g, Upadhyaya, N. & Vitelli, V. Nonlinear conduction via solitons in a topological mechanical insulator. Proc. Natl Acad. Sci. USA 111, 13004–13009 (2014).

    Article  Google Scholar 

  200. Pal, R. K., Vila, J., Leamy, M. & Ruzzene, M. Amplitude-dependent topological edge states in nonlinear phononic lattices. Phys. Rev. E 97, 032209 (2018).

    Article  CAS  Google Scholar 

  201. Snee, D. D. & Ma, Y.-P. Edge solitons in a nonlinear mechanical topological insulator. Extrem. Mech. Lett. 30, 100487 (2019).

    Article  Google Scholar 

  202. Chaunsali, R. & Theocharis, G. Self-induced topological transition in phononic crystals by nonlinearity management. Phys. Rev. B 100, 014302 (2019).

    Article  CAS  Google Scholar 

  203. Darabi, A. & Leamy, M. J. Tunable nonlinear topological insulator for acoustic waves. Phys. Rev. Appl. 12, 044030 (2019).

    Article  CAS  Google Scholar 

  204. Kinsler, L. E., Frey, A. R., Coppens, A. B. & Sanders, J. V. Fundamentals of Acoustics (Wiley, 2000).

  205. Lumer, Y., Plotnik, Y., Rechtsman, M. C. & Segev, M. Self-localized states in photonic topological insulators. Phys. Rev. Lett. 111, 243905 (2013).

    Article  Google Scholar 

  206. Leykam, D. & Chong, Y. D. Edge solitons in nonlinear-photonic topological insulators. Phys. Rev. Lett. 117, 143901 (2016).

    Article  Google Scholar 

  207. Hadad, Y., Khanikaev, A. B. & Alu, A. Self-induced topological transitions and edge states supported by nonlinear staggered potentials. Phys. Rev. B 93, 155112 (2016).

    Article  Google Scholar 

  208. Chiu, C.-K., Teo, J. C., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2016).

    Article  Google Scholar 

  209. Zhao, Y., Huang, Y.-X. & Yang, S. A. Z2-projective translational symmetry protected topological phases. Phys. Rev. B 102, 161117 (2020).

    Article  CAS  Google Scholar 

  210. Zhao, Y. X., Chen, C., Sheng, X.-L. & Yang, S. A. Switching spinless and spinful topological phases with projective PT symmetry. Phys. Rev. Lett. 126, 196402 (2021).

    Article  CAS  Google Scholar 

  211. Shao, L. B., Liu, Q., Xiao, R., Yang, S. A. & Zhao, Y. X. Gauge-field extended k · p method and novel topological phases. Phys. Rev. Lett. 127, 076401 (2021).

    Article  CAS  Google Scholar 

  212. Xue, H. et al. Projectively enriched symmetry and topology in acoustic crystals. Phys. Rev. Lett. 128, 116802 (2022).

    Article  CAS  Google Scholar 

  213. Li, T. et al. Acoustic Möbius insulators from projective symmetry. Phys. Rev. Lett. 128, 116803 (2022).

    Article  CAS  Google Scholar 

  214. Yang, Y. et al. Demonstration of negative refraction induced by synthetic gauge fields. Sci. Adv. 7, eabj2062 (2021).

    Article  Google Scholar 

  215. Jiang, B. et al. Experimental observation of non-Abelian topological acoustic semimetals and their phase transitions. Nat. Phys. 17, 1239–1246 (2021).

    Article  CAS  Google Scholar 

  216. Wang, M. et al. Experimental observation of non-Abelian earring nodal links in phononic crystals. Phys. Rev. Lett. 128, 246601 (2022).

    Article  CAS  Google Scholar 

  217. Zangeneh-Nejad, F. & Fleury, R. Disorder-induced signal filtering with topological metamaterials. Adv. Mater. 32, 2001034 (2020).

    Article  CAS  Google Scholar 

  218. König, M. et al. Quantum spin Hall insulator state in HgTe quantum wells. Science 318, 766–770 (2007).

    Article  Google Scholar 

  219. Hsieh, D. et al. A topological Dirac insulator in a quantum spin Hall phase. Nature 452, 970–974 (2008).

    Article  CAS  Google Scholar 

  220. Liu, Z. et al. Discovery of a three-dimensional topological Dirac semimetal, Na3Bi. Science 343, 864–867 (2014).

    Article  CAS  Google Scholar 

  221. Ju, L. et al. Topological valley transport at bilayer graphene domain walls. Nature 520, 650–655 (2015).

    Article  CAS  Google Scholar 

  222. Xu, S.-Y. et al. Discovery of a Weyl fermion semimetal and topological Fermi arcs. Science 349, 613–617 (2015).

    Article  CAS  Google Scholar 

  223. Lv, B. et al. Experimental discovery of Weyl semimetal TaAs. Phys. Rev. X 5, 031013 (2015).

    Google Scholar 

  224. Bian, G. et al. Topological nodal-line fermions in spin-orbit metal PbTaSe2. Nat. Commun. 7, 10556 (2016).

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  226. Kempkes, S. et al. Robust zero-energy modes in an electronic higher-order topological insulator. Nat. Mater. 18, 1292–1297 (2019).

    Article  CAS  Google Scholar 

  227. Rao, Z. et al. Observation of unconventional chiral fermions with long Fermi arcs in CoSi. Nature 567, 496–499 (2019).

    Article  CAS  Google Scholar 

  228. Sanchez, D. S. et al. Topological chiral crystals with helicoid-arc quantum states. Nature 567, 500–505 (2019).

    Article  CAS  Google Scholar 

  229. Takane, D. et al. Observation of chiral fermions with a large topological charge and associated Fermi-arc surface states in CoSi. Phys. Rev. Lett. 122, 076402 (2019).

    Article  CAS  Google Scholar 

  230. Fu, B.-B. et al. Dirac nodal surfaces and nodal lines in ZrSiS. Sci. Adv. 5, eaau6459 (2019).

    Article  CAS  Google Scholar 

  231. Nassar, H. et al. Nonreciprocity in acoustic and elastic materials. Nat. Rev. Mater. 5, 667–685 (2020).

    Article  CAS  Google Scholar 

  232. Aharonov, Y. & Bohm, D. Significance of electromagnetic potentials in the quantum theory. Phys. Rev. 115, 485 (1959).

    Article  Google Scholar 

  233. Berry, M., Chambers, R., Large, M., Upstill, C. & Walmsley, J. Wavefront dislocations in the Aharonov–Bohm effect and its water wave analogue. Eur. J. Phys. 1, 154 (1980).

    Article  Google Scholar 

  234. Matlack, K. H., Serra-Garcia, M., Palermo, A., Huber, S. D. & Daraio, C. Designing perturbative metamaterials from discrete models. Nat. Mater. 17, 323–328 (2018).

    Article  CAS  Google Scholar 

Download references

Acknowledgements

H.X. and B.Z. acknowledge support from National Research Foundation Singapore Competitive Research Program no. NRF-CRP23-2019-0007, and Singapore Ministry of Education Academic Research Fund Tier 3 under grant no. MOE2016-T3-1-006 and Tier 2 under grant no. MOE2019-T2-2-085. Y.Y. acknowledges support from the National Natural Science Foundation of China (NNSFC) under grant no. 62175215.

Author information

Authors and Affiliations

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

    Haoran Xue & Baile Zhang

  2. Interdisciplinary Center for Quantum Information, State Key Laboratory of Modern Optical Instrumentation, College of Information Science and Electronic Engineering, ZJU-UIUC Institute, Zhejiang University, Hangzhou, China

    Yihao Yang

  3. ZJU-Hangzhou Global Science and Technology Innovation Center, Key Laboratory of Advanced Micro/Nano Electronic Devices & Smart Systems of Zhejiang, ZJU-UIUC Institute, Zhejiang University, Hangzhou, China

    Yihao Yang

  4. Centre for Disruptive Photonic Technologies, Nanyang Technological University, Singapore, Singapore

    Baile Zhang

Authors
  1. Haoran Xue
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yihao Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Baile Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed significantly to all aspects of the article.

Corresponding authors

Correspondence to Haoran Xue, Yihao Yang or Baile Zhang.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Reviews Materials thanks Romain Fleury, Zhengyou Liu, Guancong Ma and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

Publisher’s note

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

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xue, H., Yang, Y. & Zhang, B. Topological acoustics. Nat Rev Mater 7, 974–990 (2022). https://doi.org/10.1038/s41578-022-00465-6

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41578-022-00465-6

This article is cited by

Search

Advanced search

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