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Topological phases in acoustic and mechanical systems

Nature Reviews Physics volume 1, pages 281–294 (2019)Cite this article

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A Publisher Correction to this article was published on 15 April 2019

This article has been updated

Abstract

The study of classical wave physics has been reinvigorated by incorporating the concept of the geometric phase, which has its roots in optics, and topological notions that were previously explored in condensed matter physics. Recently, sound waves and a variety of mechanical systems have emerged as excellent platforms that exemplify the universality and diversity of topological phases. In this Review, we introduce the essential physical concepts that underpin various classes of topological phenomena realized in acoustic and mechanical systems: Dirac points, the quantum Hall, quantum spin Hall and valley Hall effects, Floquet topological phases, 3D gapless states and Weyl crystals.

Key points

  • Acoustic and mechanical systems are versatile platforms to study a wide range of topological phases that were first investigated in condensed matter physics.

  • Topological phenomena that can be observed include Dirac points and analogues of the quantum Hall effect, quantum spin Hall effect, valley Hall effect, Floquet topological phases, gapless states and Weyl systems.

  • Because classical acoustic systems are different from condensed matter systems (for example, they lack Kramers degeneracy), new approaches are needed to realize topological phases.

  • Schemes of symmetry breaking in phononic crystals play a key role in the realization of these topological phases, and their consequences and limitations are discussed.

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Fig. 1: Geometric phase in a 1D phononic crystal.
Fig. 2: Dirac cone dispersion.
Fig. 3: Breaking time-reversal symmetry.
Fig. 4: Acoustic and mechanical analogues of the quantum spin Hall effect and valley Hall effect.
Fig. 5: Acoustic Weyl semimetal.

Change history

  • 15 April 2019

    This article has been corrected to add a missing image credit to the caption of Fig. 4. The credit line of Fig. 4 now reads “Panel d is adapted from ref.78, CC-BY-4.0 and an image courtesy of Dr Miniaci, Swiss Federal Laboratories for Materials Science and Technology.”

References

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  3. Martinez-Sala, R. et al. Sound attenuation by sculpture. Nature 378, 241–241 (1995).

    ADS  Google Scholar 

  4. Montero de Espinosa, F. R., Jiménez, E. & Torres, M. Ultrasonic band gap in a periodic two-dimensional composite. Phys. Rev. Lett. 80, 1208–1211 (1998).

    ADS  Google Scholar 

  5. Lu, L., Joannopoulos, J. D. & Soljačć, M. Topological photonics. Nat. Photon. 8, 821–829 (2014).

    ADS  Google Scholar 

  6. Pancharatnam, S. Generalized theory of interference and its applications. Proc. Natl. Acad. Sci. India A 44, 398–417 (1956).

    MathSciNet  Google Scholar 

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

    ADS  MathSciNet  MATH  Google Scholar 

  8. von Bergmann, J. & von Bergmann, H. Foucault pendulum through basic geometry. Am. J. Phys. 75, 888–892 (2007).

    ADS  Google Scholar 

  9. Tomita, A. & Chiao, R. Y. Observation of Berry’s topological phase by use of an optical fiber. Phys. Rev. Lett. 57, 937–940 (1986).

    ADS  Google Scholar 

  10. Boulanger, J., Le Bihan, N., Catheline, S. & Rossetto, V. Observation of a non-adiabatic geometric phase for elastic waves. Ann. Phys. 327, 952–958 (2012).

    ADS  MATH  Google Scholar 

  11. Wang, S., Ma, G. & Chan, C. T. Topological transport of sound mediated by spin-redirection geometric phase. Sci. Adv. 4, eaaq1475 (2018).

    ADS  Google Scholar 

  12. Heeger, A. J., Kivelson, S., Schrieffer, J. R. & Su, W. P. Solitons in conducting polymers. Rev. Mod. Phys. 60, 781–850 (1988).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  14. Zak, J. Berry’s phase for energy bands in solids. Phys. Rev. Lett. 62, 2747–2750 (1989).

    ADS  Google Scholar 

  15. Xiao, M., Zhang, Z. Q. & Chan, C. T. Surface impedance and bulk band geometric phases in one-dimensional systems. Phys. Rev. X 4, 021017 (2014).

    Google Scholar 

  16. Gao, W. et al. Controlling interface states in 1D photonic crystals by tuning bulk geometric phases. Opt. Lett. 42, 1500–1503 (2017).

    ADS  Google Scholar 

  17. Choi, K. H., Ling, C. W., Lee, K. F., Tsang, Y. H. & Fung, K. H. Simultaneous multi-frequency topological edge modes between one-dimensional photonic crystals. Opt. Lett. 41, 1644–1647 (2016).

    ADS  Google Scholar 

  18. Yan, M. et al. Designing topological interface states in phononic crystals based on the full phase diagrams. New J. Phys. 20, 073032 (2018).

    Google Scholar 

  19. Xiao, M. et al. Geometric phase and band inversion in periodic acoustic systems. Nat. Phys. 11, 240–244 (2015). This paper reports the first observation of the Zak phase and topological transition in a phononic crystal.

    Google Scholar 

  20. Chaunsali, R., Kim, E., Thakkar, A., Kevrekidis, P. G. & Yang, J. Demonstrating an In Situ topological band transition in cylindrical granular chains. Phys. Rev. Lett. 119, 024301 (2017).

    ADS  Google Scholar 

  21. Yin, J. et al. Band transition and topological interface modes in 1D elastic phononic crystals. Sci. Rep. 8, 6806 (2018).

    ADS  Google Scholar 

  22. Esmann, M. et al. Topological nanophononic states by band inversion. Phys. Rev. B 97, 155422 (2018).

    ADS  Google Scholar 

  23. Huang, X., Xiao, M., Zhang, Z.-Q. & Chan, C. T. Sufficient condition for the existence of interface states in some two-dimensional photonic crystals. Phys. Rev. B 90, 075423 (2014).

    ADS  Google Scholar 

  24. Li, S., Zhao, D., Niu, H., Zhu, X. & Zang, J. Observation of elastic topological states in soft materials. Nat. Commun. 9, 1370 (2018).

    ADS  Google Scholar 

  25. Delplace, P., Ullmo, D. & Montambaux, G. Zak phase and the existence of edge states in graphene. Phys. Rev. B 84, 195452 (2011).

    ADS  Google Scholar 

  26. Castro Neto, A. H., Guinea, F., Peres, N. M. R., Novoselov, K. S. & Geim, A. K. The electronic properties of graphene. Rev. Mod. Phys. 81, 109–162 (2009).

    ADS  Google Scholar 

  27. Hikami, S., Larkin, A. I. & Nagaoka, Y. Spin-Orbit interaction and magnetoresistance in the two dimensional random system. Prog. Theor. Phys. 63, 707–710 (1980).

    ADS  Google Scholar 

  28. Liu, F., Ming, P. & Li, J. Ab initio calculation of ideal strength and phonon instability of graphene under tension. Phys. Rev. B 76, 064120 (2007).

    ADS  Google Scholar 

  29. Guinea, F., Katsnelson, M. I. & Geim, A. K. Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering. Nat. Phys. 6, 30 (2009).

    Google Scholar 

  30. Wen, X. et al. Observation of acoustic Landau quantization and quantum-Hall-like edge states. Preprint at arXiv https://arxiv.org/abs/1807.08454v1 (2018).

  31. Rechtsman, M. C. et al. Strain-induced pseudomagnetic field and photonic Landau levels in dielectric structures. Nat. Photon. 7, 153–158 (2013).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  33. Sepkhanov, R. A., Bazaliy, Y. B. & Beenakker, C. W. J. Extremal transmission at the Dirac point of a photonic band structure. Phys. Rev. A 75, 063813 (2007).

    ADS  Google Scholar 

  34. Zhang, X. & Liu, Z. Extremal transmission and beating effect of acoustic waves in two-dimensional sonic crystals. Phys. Rev. Lett. 101, 264303 (2008).

    ADS  Google Scholar 

  35. Torrent, D. & Sánchez-Dehesa, J. Acoustic analogue of graphene: observation of Dirac cones in acoustic surface waves. Phys. Rev. Lett. 108, 174301 (2012).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  37. Yu, S.-Y. et al. Surface phononic graphene. Nat. Mater. 15, 1243 (2016).

    ADS  Google Scholar 

  38. Zhong, W. & Zhang, X. Acoustic analog of monolayer graphene and edge states. Phys. Lett. A 375, 3533–3536 (2011).

    ADS  Google Scholar 

  39. Dai, H., Xia, B. & Yu, D. Dirac cones in two-dimensional acoustic metamaterials. J. Appl. Phys. 122, 065103 (2017).

    ADS  Google Scholar 

  40. Torrent, D., Mayou, D. & Sánchez-Dehesa, J. Elastic analog of graphene: Dirac cones and edge states for flexural waves in thin plates. Phys. Rev. B 87, 115143 (2013).

    ADS  Google Scholar 

  41. Joshua, E. S. S., Tom, C. L. & Charles, L. K. Mechanical graphene. New J. Phys. 19, 025003 (2017).

    Google Scholar 

  42. Chong, C., Kevrekidis, P. G., Ablowitz, M. J. & Ma, Y.-P. Conical wave propagation and diffraction in two-dimensional hexagonally packed granular lattices. Phys. Rev. E 93, 012909 (2016).

    ADS  MathSciNet  Google Scholar 

  43. Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982).

    ADS  Google Scholar 

  44. Zhang, Y., Tan, Y.-W., Stormer, H. L. & Kim, P. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438, 201 (2005).

    ADS  Google Scholar 

  45. Novoselov, K. S. et al. Room-temperature quantum Hall effect in graphene. Science 315, 1379–1379 (2007).

    ADS  Google Scholar 

  46. Haldane, F. D. M. & Raghu, S. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. Phys. Rev. Lett. 100, 013904 (2008).

    ADS  Google Scholar 

  47. Klitzing, Kv, Dorda, G. & Pepper, M. New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Phys. Rev. Lett. 45, 494–497 (1980).

    ADS  Google Scholar 

  48. Wang, Z., Chong, Y., Joannopoulos, J. D. & Soljačć, M. Observation of unidirectional backscattering-immune topological electromagnetic states. Nature 461, 772–775 (2009).

    ADS  Google Scholar 

  49. 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).

    ADS  Google Scholar 

  50. 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).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  52. 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).

    ADS  Google Scholar 

  53. 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). Together with Yang et al. (2015) and Ni et al. (2015), this paper proposes using circulation of a fluid to break time-reversal symmetry to realize the quantum Hall effect in sound.

    ADS  Google Scholar 

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

    Google Scholar 

  55. Hatsugai, Y. Chern number and edge states in the integer quantum Hall effect. Phys. Rev. Lett. 71, 3697–3700 (1993).

    ADS  MathSciNet  MATH  Google Scholar 

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

    ADS  Google Scholar 

  57. Chen, Z.-G. & Wu, Y. Tunable topological phononiccrystals. Phys. Rev. Appl. 5, 054021 (2016).

    ADS  Google Scholar 

  58. Zhu, Y. et al. Experimental realization of acoustic chern insulator. Phys. Rev. Lett. 122, 014302 (2019).

    ADS  Google Scholar 

  59. Wang, P., Lu, L. & Bertoldi, K. Topological phononic crystals with one-way elastic edge waves. Phys. Rev. Lett. 115, 104302 (2015).

    ADS  Google Scholar 

  60. Nash, L. M. et al. Topological mechanics of gyroscopic metamaterials. Proc. Natl. Acad. Sci. 112, 14495–14500 (2015). This work uses electric motors to break time-reversal symmetry to realize a mechanical Chern insulator.

    ADS  Google Scholar 

  61. Yao-Ting, W., Pi-Gang, L. & Shuang, Z. Coriolis force induced topological order for classical mechanical vibrations. New J. Phys. 17, 073031 (2015).

    Google Scholar 

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

    ADS  Google Scholar 

  63. 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).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  65. Süsstrunk, R. & Huber, S. D. Observation of phononic helical edge states in a mechanical topological insulator. Science 349, 47–50 (2015). This is the first realization of the analogy of the QSH effect in a mechanical system.

    ADS  Google Scholar 

  66. Hofstadter, D. R. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, 2239–2249 (1976).

    ADS  Google Scholar 

  67. He, C. et al. Acoustic topological insulator and robust one-way sound transport. Nat. Phys. 12, 1124–1129 (2016). This is the first experimental realization of a 2D topological insulator for sound.

    Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  70. Simon, Y., Romain, F., Fabrice, L., Mathias, F. & Geoffroy, L. Topological acoustic polaritons: robust sound manipulation at the subwavelength scale. New J. Phys. 19, 075003 (2017).

    Google Scholar 

  71. Hafezi, M., Demler, E. A., Lukin, M. D. & Taylor, J. M. Robust optical delay lines with topological protection. Nat. Phys. 7, 907–912 (2011).

    Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  74. Li, J., Wang, J., Wu, S. & Mei, J. Pseudospins and topological edge states in elastic shear waves. AIP Adv. 7, 125030 (2017).

    ADS  Google Scholar 

  75. Shiqiao, W., Ying, W. & Jun, M. Topological helical edge states in water waves over a topographical bottom. New J. Phys. 20, 023051 (2018).

    Google Scholar 

  76. Yu, S.-Y. et al. Elastic pseudospin transport for integratable topological phononic circuits. Nat. Commun. 9, 3072 (2018).

    ADS  Google Scholar 

  77. Mousavi, S. H., Khanikaev, A. B. & Wang, Z. Topologically protected elastic waves in phononic metamaterials. Nat. Commun. 6, 8682 (2015).

    ADS  Google Scholar 

  78. Miniaci, M., Pal, R. K., Morvan, B. & Ruzzene, M. Experimental observation of topologically protected helical edge modes in patterned elastic plates. Phys. Rev. X 8, 031074 (2018).

    Google Scholar 

  79. Zheng, L.-Y., Theocharis, G., Tournat, V. & Gusev, V. Quasitopological rotational waves in mechanical granular graphene. Phys. Rev. B 97, 060101 (2018).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  81. Lu, J. et al. Observation of topological valley transport of sound in sonic crystals. Nat. Phys. 13, 364–374 (2016). This paper reports the realization of the valley Hall effect in phononic crystals.

    Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  84. Xia, B.-Z. et al. Observation of valleylike edge states of sound at a momentum away from the high-symmetry points. Phys. Rev. B 97, 155124 (2018).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  86. Yang, Y., Yang, Z. & Zhang, B. Acoustic valley edge states in a graphene-like resonator system. J. Appl. Phys. 123, 091713 (2018).

    ADS  Google Scholar 

  87. Yan, M. et al. On-chip valley topological materials for elastic wave manipulation. Nat. Mater. 17, 993–998 (2018).

    ADS  Google Scholar 

  88. Liu, T.-W. & Semperlotti, F. Tunable acoustic valley hall edge states in reconfigurable phononic elastic waveguides. Phys. Rev. Appl. 9, 014001 (2018).

    ADS  Google Scholar 

  89. Kitagawa, T., Berg, E., Rudner, M. & Demler, E. Topological characterization of periodically driven quantum systems. Phys. Rev. B 82, 235114 (2010).

    ADS  Google Scholar 

  90. Fleury, R., Khanikaev, A. B. & Alu, A. Floquet topological insulators for sound. Nat. Commun. 7, 11744 (2016). This is a theoretical and numerical study of a Floquet topological insulator for sound.

    ADS  Google Scholar 

  91. Rechtsman, M. C. et al. Photonic Floquet topological insulators. Nature 496, 196–200 (2013).

    ADS  Google Scholar 

  92. 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).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  94. Hermann, W. Elektron und Gravitation. I [German]. Z. für Phys. 56, 330–352 (1929).

    Google Scholar 

  95. 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). This paper discusses how to realize Weyl points in phononic crystals.

    Google Scholar 

  96. Chang, M.-L., Xiao, M., Chen, W.-J. & Chan, C. T. Multiple Weyl points and the sign change of their topological charges in woodpile photonic crystals. Phys. Rev. B 95, 125136 (2017).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  98. Liu, T., Zheng, S., Dai, H., Yu, D. & Xia, B. Acoustic semimetal with Weyl points and surface states. Preprint atarXiv https://arxiv.org/abs/1803.04284 (2018).

  99. 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). This is the first experimental realization of acoustic Weyl points.

    Google Scholar 

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

    ADS  Google Scholar 

  101. Soluyanov, A. A. et al. Type-II Weyl semimetals. Nature 527, 495–498 (2015).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  103. Fruchart, M. et al. Soft self-assembly of Weyl materials for light and sound. Proc. Natl. Acad. Sci. USA 115, E3655 (2018).

    Google Scholar 

  104. Yao-Ting, W. & Ya-Wen, T. Multiple Weyl and double-Weyl points in an elastic chiral lattice. New J. Phys. 20, 083031 (2018).

    Google Scholar 

  105. 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).

    ADS  Google Scholar 

  106. Chen, W.-J., Xiao, M. & Chan, C. T. Photonic crystals possessing multiple Weyl points and the experimental observation of robust surface states. Nat. Commun. 7, 13038 (2016).

    ADS  Google Scholar 

  107. Zhang, T. et al. Double-Weyl phonons in transition-metal monosilicides. Phys. Rev. Lett. 120, 016401 (2018).

    ADS  Google Scholar 

  108. Xiao, M. & Fan, S. Topologically charged nodal surface. Preprint at arXiv https://arxiv.org/abs/1709.02363 (2017).

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

    ADS  Google Scholar 

  110. Kane, C. L. & Lubensky, T. C. Topological boundary modes in isostatic lattices. Nat. Phys. 10, 39–45 (2014).

    Google Scholar 

  111. Rocklin, D. Z., Chen, B. Gg, Falk, M., Vitelli, V. & Lubensky, T. C. Mechanical Weyl modes in topological Maxwell lattices. Phys. Rev. Lett. 116, 135503 (2016).

    ADS  Google Scholar 

  112. Paulose, J., Meeussen, A. S. & Vitelli, V. Selective buckling via states of self-stress in topological metamaterials. Proc. Natl. Acad. Sci. USA 112, 7639–7644 (2015).

    ADS  Google Scholar 

  113. Meeussen, A. S., Paulose, J. & Vitelli, V. Geared topological metamaterials with tunable mechanical stability. Phys. Rev. X 6, 041029 (2016).

    Google Scholar 

  114. Huber, S. D. Topological mechanics. Nat. Phys. 12, 621 (2016).

    Google Scholar 

  115. Bertoldi, K., Vitelli, V., Christensen, J. & van Hecke, M. Flexible mechanical metamaterials. Nat. Rev. Mater. 2, 17066 (2017).

    ADS  Google Scholar 

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

    ADS  MathSciNet  MATH  Google Scholar 

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

    ADS  Google Scholar 

  118. Xue, H., Yang, Y., Gao, F., Chong, Y. & Zhang, B. Acoustic higher-order topological insulator on a kagome lattice. Nat. Mater. 18, 108–112 (2018).

    ADS  Google Scholar 

  119. 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 (2018).

    ADS  Google Scholar 

  120. Zhang, X., et al Acoustic hierarchical topological insulators. Preprint at arXiv https://arxiv.org/abs/1811.05514 (2018).

  121. Mitchell, N. P., Nash, L. M., Hexner, D., Turner, A. M. & Irvine, W. T. M. Amorphous topological insulators constructed from random point sets. Nat. Phys. 14, 380–385 (2018).

    Google Scholar 

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

    ADS  Google Scholar 

  123. Yang, Y. et al. Realization of a three-dimensional photonic topological insulator. Nature 565, 622–626, doi: (2019).

    ADS  Google Scholar 

  124. Li, J., Chu, R.-L., Jain, J. K. & Shen, S.-Q. Topological anderson insulator. Phys. Rev. Lett. 102, 136806 (2009).

    ADS  Google Scholar 

  125. Stützer, S. et al. Photonic topological Anderson insulators. Nature 560, 461 (2018).

    ADS  Google Scholar 

  126. Aubry, S. & André, G. Analyticity breaking and anderson localization in incommensurate lattices. Ann. Isr. Phys. Soc. 3, 18 (1980).

    MathSciNet  MATH  Google Scholar 

  127. 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).

    ADS  Google Scholar 

  128. Kitaev, A. Y. Unpaired Majorana fermions in quantum wires. Sov. Phys. Usp. 44, 131 (2001).

    Google Scholar 

  129. El-Ganainy, R. et al. Non-Hermitian physics and PTsymmetry. Nat. Phys. 14, 11 (2018).

    Google Scholar 

  130. Feng, L., El-Ganainy, R. & Ge, L. Non-Hermitian photonics based on parity–time symmetry. Nat. Photon. 11, 752–762 (2017).

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  132. Li, G., Zhang, S. & Zentgraf, T. Nonlinear photonic metasurfaces. Nat. Rev. Mater. 2, 17010 (2017).

    ADS  Google Scholar 

  133. Fang, A., Zhang, Z. Q., Louie, S. G. & Chan, C. T. Anomalous Anderson localization behaviors in disordered pseudospin systems. Proc. Natl. Acad. Sci. USA 114, 4087–4092 (2017).

    MathSciNet  MATH  Google Scholar 

  134. Chan, C. T., Hang, Z. H. & Huang, X. Dirac dispersion in two-dimensional photoniccrystals. Adv. Optoelectron. 2012, 11 (2012).

    Google Scholar 

  135. Liu, F., Huang, X. & Chan, C. T. Dirac cones at k in acoustic crystals and zero refractive index acoustic materials. Appl. Phys. Lett. 100, 071911–071914 (2012).

    ADS  Google Scholar 

  136. Huang, X., Lai, Y., Hang, Z. H., Zheng, H. & Chan, C. T. Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials. Nat. Mater. 10, 582–586 (2011).

    ADS  Google Scholar 

  137. Mei, J., Wu, Y., Chan, C. T. & Zhang, Z.-Q. First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals. Phys. Rev. B 86, 035141 (2012).

    ADS  Google Scholar 

  138. Dubois, M., Shi, C., Zhu, X., Wang, Y. & Zhang, X. Observation of acoustic Dirac-like cone and double zero refractive index. Nat. Commun. 8, 14871 (2017).

    ADS  Google Scholar 

  139. Liu, F. & Liu, Z. Elastic waves scattering without conversion in metamaterials with simultaneous zero indices for longitudinal and transverse waves. Phys. Rev. Lett. 115, 175502 (2015).

    ADS  Google Scholar 

  140. Chan, C. T., Huang, X., Liu, F. & Hang, Z. H. Dirac dispersion and zero-index in two dimensional and three dimensional photonic and phononic systems. Prog. Electromagn. Res. B 44, 163–190 (2012).

    Google Scholar 

  141. Liu, F., Lai, Y., Huang, X. & Chan, C. T. Dirac cones at k = 0 in phononic crystals. Phys. Rev. B 84, 224113 (2011).

    ADS  Google Scholar 

  142. Huang, X., Xiao, M., Chan, C. T. & Liu, F. in World Scientific Handbook of Metamaterials and Plasmonics World Scientific Series in Nanoscience and Nanotechnology 553-597 (World Scientific, 2017).

  143. Wu, Y. A semi-Dirac point and an electromagnetic topological transition in a dielectric photonic crystal. Opt. Express 22, 1906–1917 (2014).

    ADS  Google Scholar 

  144. Sakoda, K. Proof of the universality of mode symmetries in creating photonic Dirac cones. Opt. Express 20, 25181–25194 (2012).

    ADS  Google Scholar 

  145. Li, Y., Wu, Y. & Mei, J. Double Dirac cones in phononic crystals. Appl. Phys. Lett. 105, 014107 (2014).

    ADS  Google Scholar 

  146. Chen, Z.-G. et al. Accidental degeneracy of double Dirac cones in a phononic crystal. Sci. Rep. 4, 4613 (2014).

    Google Scholar 

  147. Zhou, X. et al. Photonic spin Hall effect in topological insulators. Phys. Rev. A 88, 053840 (2013).

    ADS  Google Scholar 

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Acknowledgements

The authors are grateful to R.-Y. Zhang for proofreading the manuscript. G.M. thanks W.-J. Chen, Shubo Wang and Changqing Xu for helpful discussions. G.M. is supported by the Hong Kong Research Grants Council (grant no. RGC-ECS 22302718, ANR-RGC A-HKUST601/18 and CRF C6013-18GF), the National Natural Science Foundation of China (grant no. 11802256) and the Hong Kong Baptist University through FRG2/17-18/056. M.X. is supported by the startup funding of Wuhan University and the US National Science Foundation (grant no. CBET-1641069). C.T.C. is supported by the Hong Kong Research Grants Council (grant no. AoE/P-02/12).

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Author notes

  1. These authors contributed equally: Guancong Ma, Meng Xiao.

Authors and Affiliations

  1. Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China

    Guancong Ma

  2. Key Laboratory of Artificial Micro- and Nano-structures of Ministry of Education and School of Physics and Technology, Wuhan University, Wuhan, China

    Meng Xiao

  3. Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China

    C. T. Chan

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  1. Guancong Ma
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Glossary

Weyl points

Point degeneracies on the band structure, with local properties describable by the Weyl Hamiltonian. They can be regarded mathematically as a magnetic monopole in momentum space.

Fermi arcs

In classical wave crystals, a type of isofrequency cut present as arcs in the momentum space. In condensed matter physics, the cut of eigen-spectra is at the Fermi energy.

Adiabatic cyclic evolution

A cyclic change in a parameter that is slow enough that the physical system remains in its instantaneous eigenstate, the eigenvalue of which is not degenerate with any other eigenvalue of the Hamiltonian.

Berry phase

A phase difference acquired over an adiabatic cyclic evolution, resulting solely from the geometric properties of the parameter space of the system Hamiltonian.

Poincaré sphere

A graphical representation of all the polarization states of a pure state of light on the surface of a 3D sphere.

Berry connection

The local gauge potential for Berry curvature \({\boldsymbol{A}}=i\left\langle {u}_{{\boldsymbol{R}}}\left|{\nabla }_{{\boldsymbol{R}}}\right|{u}_{{\boldsymbol{R}}}\right\rangle \); \(\left|{u}_{{\boldsymbol{R}}}\right\rangle \) is the instantaneous eigenstate of the system with parameter R. Viewable as a momentum-space analogy to the vector potential for a magnetic field.

Surface impedance

The surface property of a truncated bulk for waves. For acoustic waves, surface impedance is defined as the ratio of the sound pressure variation to the local fluid velocity.

Good quantum number

The eigenvalue of an operator that remains unchanged as the system evolves. In the example in the main text, the system changes from a boundary-less periodic system to a truncated bulk.

Berry curvature

A local gauge field defined as \({\boldsymbol{\Omega }}=\nabla \times {\boldsymbol{A}}\), where A is the Berry connection.

Kramers pairs

In time-reversal-symmetric systems with half-integer total spins, every energy eigenstate is at least two-fold degenerate. This degeneracy is protected by time-reversal symmetry, and the two eigenstates form a Kramers pair.

Hofstadter model

A mathematical model describing the behaviour of electrons in a magnetic field in a 2D lattice. The energy levels form a fractal set.

Duality symmetry

Symmetry such that Maxwell equations of a source-free system are invariant under ε→μ, μ→ε, E→B and B→−E, where ε is the dielectric constant, μ is the permeability, E is the electric field and B is the magnetic field.

Lamb modes

Two types of mode of waves in a solid plate, one symmetric about the plate mid-plane (characterized by breathing vibrations) and one antisymmetric (characterized by bending vibrations).

Poynting vectors

Vector fields representing the direction and amplitude of local energy flow. In acoustics, the vector field is defined as S = Pv, where P is the pressure variation and v is the local fluid velocity.

Time-ordering operator

A mathematical representation of a procedure that orders the product of a series of operators according to the time sequence of these operations.

Screw symmetry

A combination of rotation about an axis and a translation parallel to that axis that leaves a crystal unchanged.

Nodal surface

Two or more states form a nodal surface when they are degenerate over a continuous range of momenta that form a surface in the momentum space.

Isostatic lattices

Lattices consisting of point masses connected by rigid bonds or central-force springs, in which the number of bonds equals the total number of degrees of freedom of the masses.

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Ma, G., Xiao, M. & Chan, C.T. Topological phases in acoustic and mechanical systems. Nat Rev Phys 1, 281–294 (2019). https://doi.org/10.1038/s42254-019-0030-x

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