Topological phases in acoustic and mechanical systems

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.

Access options

from\$8.99

All prices are NET prices.

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. 1.

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

2. 2.

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

3. 3.

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

4. 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).

5. 5.

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

6. 6.

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

7. 7.

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

8. 8.

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

9. 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).

10. 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).

11. 11.

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

12. 12.

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

13. 13.

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

14. 14.

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

15. 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).

16. 16.

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

17. 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).

18. 18.

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

19. 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.

20. 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).

21. 21.

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

22. 22.

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

23. 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).

24. 24.

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

25. 25.

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

26. 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).

27. 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).

28. 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).

29. 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).

30. 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. 31.

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

32. 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).

33. 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).

34. 34.

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

35. 35.

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

36. 36.

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

37. 37.

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

38. 38.

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

39. 39.

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

40. 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).

41. 41.

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

42. 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).

43. 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).

44. 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).

45. 45.

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

46. 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).

47. 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).

48. 48.

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

49. 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).

50. 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).

51. 51.

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

52. 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).

53. 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.

54. 54.

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

55. 55.

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

56. 56.

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

57. 57.

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

58. 58.

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

59. 59.

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

60. 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.

61. 61.

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

62. 62.

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

63. 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).

64. 64.

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

65. 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.

66. 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).

67. 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.

68. 68.

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

69. 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).

70. 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).

71. 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).

72. 72.

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

73. 73.

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

74. 74.

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

75. 75.

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

76. 76.

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

77. 77.

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

78. 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).

79. 79.

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

80. 80.

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

81. 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.

82. 82.

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

83. 83.

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

84. 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).

85. 85.

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

86. 86.

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

87. 87.

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

88. 88.

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

89. 89.

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

90. 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.

91. 91.

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

92. 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).

93. 93.

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

94. 94.

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

95. 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.

96. 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).

97. 97.

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

98. 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. 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.

100. 100.

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

101. 101.

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

102. 102.

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

103. 103.

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

104. 104.

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

105. 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).

106. 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).

107. 107.

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

108. 108.

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

109. 109.

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

110. 110.

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

111. 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).

112. 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).

113. 113.

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

114. 114.

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

115. 115.

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

116. 116.

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

117. 117.

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

118. 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).

119. 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).

120. 120.

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

121. 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).

122. 122.

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

123. 123.

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

124. 124.

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

125. 125.

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

126. 126.

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

127. 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).

128. 128.

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

129. 129.

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

130. 130.

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

131. 131.

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

132. 132.

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

133. 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).

134. 134.

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

135. 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).

136. 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).

137. 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).

138. 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).

139. 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).

140. 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).

141. 141.

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

142. 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. 143.

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

144. 144.

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

145. 145.

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

146. 146.

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

147. 147.

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

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).

Author information

All authors contributed to the writing of the manuscript.

Correspondence to Guancong Ma or Meng Xiao or C. T. Chan.

Ethics declarations

Competing interests

The authors declare no competing interests.

Publisher’s note

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

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.

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.

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 εμ, με, EB 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.

Rights and permissions

Reprints and Permissions

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

• Optimal quantum valley Hall insulators by rationally engineering Berry curvature and band structure

• Zongliang Du
• , Hui Chen
•  & Guoliang Huang

Journal of the Mechanics and Physics of Solids (2020)

• Observation of Edge Waves in a Two-Dimensional Su-Schrieffer-Heeger Acoustic Network

• Li-Yang Zheng
• , Vassos Achilleos
• , Olivier Richoux
• , Georgios Theocharis
•  & Vincent Pagneux

Physical Review Applied (2019)

• Observation of a topological nodal surface and its surface-state arcs in an artificial acoustic crystal

• Yihao Yang
• , Jian-ping Xia
• , Hong-xiang Sun
• , Yong Ge
• , Ding Jia
• , Shou-qi Yuan
• , Shengyuan A. Yang
• , Yidong Chong
•  & Baile Zhang

Nature Communications (2019)

• Robust and High-Capacity Phononic Communications through Topological Edge States by Discrete Degree-of-Freedom Multiplexing

• Jun Mei
• , Jiqian Wang
• , Xiujuan Zhang
• , Siyuan Yu
• , Zhen Wang
•  & Ming-Hui Lu

Physical Review Applied (2019)

• Higher-Order Topological Corner States Induced by Gain and Loss

• Xi-Wang Luo
•  & Chuanwei Zhang

Physical Review Letters (2019)