Skip to main content

Thank you for visiting 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.

Higher-order band topology


A conventional topological insulator (TI) has gapped bulk states but gapless edge states. The emergence of the gapless edge states is dictated by the bulk topological invariant of the insulator and the preservation of relevant symmetries. Over the past four years, a new type of TI has been found, which hosts gapless hinge or corner states, rather than edge states. These unconventional TIs, termed higher-order TIs (HOTIs), are common among crystalline and quasi-crystalline materials. Higher-order band topology expands our previous understanding of topological phases and provides unprecedented lower-dimensional boundary states for devices. Here, we review the principles, theories and experimental realizations of HOTIs for both electrons and classical waves. There is an emphasis on the development of HOTIs in photonic, phononic and circuit systems owing to their special contributions to these fields. From these discussions, we remark on trends and challenges in the field and the impact of higher-order band topology on other scientific disciplines.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Classification of higher-order topological insulators with multidimensional topological boundary states.
Fig. 2: Higher-order topological insulators with quantized multipole moments.
Fig. 3: Higher-order topological insulators without quantized multipole moments.
Fig. 4: Experimental realizations of higher-order topological insulators.


  1. 1.

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

    ADS  Google Scholar 

  2. 2.

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

    ADS  Google Scholar 

  3. 3.

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

    MathSciNet  ADS  Google Scholar 

  4. 4.

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

    MathSciNet  MATH  ADS  Google Scholar 

  5. 5.

    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).

    ADS  Google Scholar 

  6. 6.

    Langbehn, J., Peng, Y., Trifunovic, L., von Oppen, F. & Brouwer, P. W. Reflection-symmetric second-order topological insulators and superconductors. Phys. Rev. Lett. 119, 246401 (2017).

    ADS  Google Scholar 

  7. 7.

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

    ADS  Google Scholar 

  8. 8.

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

    ADS  Google Scholar 

  9. 9.

    Geier, M., Trifunovic, L., Hoskam, M. & Brouwer, P. W. Second-order topological insulators and superconductors with an order-two crystalline symmetry. Phys. Rev. B 97, 205135 (2018).

    ADS  Google Scholar 

  10. 10.

    Hsu, C.-H., Stano, P., Klinovaja, J. & Loss, D. Majorana Kramers pairs in higher-order topological insulators. Phys. Rev. Lett. 121, 196801 (2018).

    Google Scholar 

  11. 11.

    Zhang, W. et al. Low-threshold topological nanolasers based on the second-order corner state. Light Sci. Appl. 9, 109 (2020).

    ADS  Google Scholar 

  12. 12.

    Kim, H.-R. et al. Multipolar lasing modes from topological corner states. Nat. Commun. 11, 5758 (2020).

    ADS  Google Scholar 

  13. 13.

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

    ADS  Google Scholar 

  14. 14.

    T. Neupert, F. A. M. Higher-order topological insulators and superconductors. APS (2017).

  15. 15.

    Sitte, M., Rosch, A., Altman, E. & Fritz, L. Topological insulators in magnetic fields: Quantum Hall effect and edge channels with a nonquantized θ term. Phys. Rev. Lett. 108, 126807 (2012).

    ADS  Google Scholar 

  16. 16.

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

    ADS  Google Scholar 

  17. 17.

    Slager, R.-J., Rademaker, L., Zaanen, J. & Balents, L. Impurity-bound states and Green’s function zeros as local signatures of topology. Phys. Rev. B 92, 085126 (2015).

    ADS  Google Scholar 

  18. 18.

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

    ADS  Google Scholar 

  19. 19.

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

    ADS  Google Scholar 

  20. 20.

    Ezawa, M. Topological switch between second-order topological insulators and topological crystalline insulators. Phys. Rev. Lett. 121, 116801 (2018).

    ADS  Google Scholar 

  21. 21.

    Park, M. J., Kim, Y., Cho, G. Y. & Lee, S. Higher-order topological insulator in twisted bilayer graphene. Phys. Rev. Lett. 123, 216803 (2019).

    ADS  Google Scholar 

  22. 22.

    Khalaf, E. Higher-order topological insulators and superconductors protected by inversion symmetry. Phys. Rev. B 97, 205136 (2018).

    ADS  Google Scholar 

  23. 23.

    Hsu, C.-H., Stano, P., Klinovaja, J. & Loss, D. Majorana Kramers pairs in higher-order topological insulators. Phys. Rev. Lett. 121, 196801 (2018).

    Google Scholar 

  24. 24.

    Matsugatani, A. & Watanabe, H. Connecting higher-order topological insulators to lower-dimensional topological insulators. Phys. Rev. B 98, 205129 (2018).

    ADS  Google Scholar 

  25. 25.

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

    Google Scholar 

  26. 26.

    Van Miert, G. & Ortix, C. Higher-order topological insulators protected by inversion and rotoinversion symmetries. Phys. Rev. B 98, 081110 (2018).

    ADS  Google Scholar 

  27. 27.

    Yan, Z. Higher-order topological odd-parity superconductors. Phys. Rev. Lett. 123, 177001 (2019).

    MathSciNet  ADS  Google Scholar 

  28. 28.

    Kudo, K., Yoshida, T. & Hatsugai, Y. Higher-order topological Mott insulators. Phys. Rev. Lett. 123, 196402 (2019).

    MathSciNet  ADS  Google Scholar 

  29. 29.

    Queiroz, R. & Stern, A. Splitting the hinge mode of higher-order topological insulators. Phys. Rev. Lett. 123, 036802 (2019).

    ADS  Google Scholar 

  30. 30.

    Sheng, X.-L. et al. Two-dimensional second-order topological insulator in graphdiyne. Phys. Rev. Lett. 123, 256402 (2019).

    ADS  Google Scholar 

  31. 31.

    Araki, H., Mizoguchi, T. & Hatsugai, Y. ZQ Berry phase for higher-order symmetry-protected topological phases. Phys. Rev. Res. 2, 012009 (2020).

    Google Scholar 

  32. 32.

    Yang, Y.-B., Li, K., Duan, L.-M. & Xu, Y. Type-II quadrupole topological insulators. Phys. Rev. Res. 2, 033029 (2020).

    Google Scholar 

  33. 33.

    You, Y., Burnell, F. & Hughes, T. L. Multipolar topological field theories: bridging higher order topological insulators and fractons. Preprint at (2019).

  34. 34.

    Trifunovic, L. & Brouwer, P. W. Higher-order bulk-boundary correspondence for topological crystalline phases. Phys. Rev. X 9, 011012 (2019).

    Google Scholar 

  35. 35.

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

    ADS  Google Scholar 

  36. 36.

    Fu, L. Topological crystalline insulators. Phys. Rev. Lett. 106, 106802 (2011).

    ADS  Google Scholar 

  37. 37.

    Po, H. C., Vishwanath, A. & Watanabe, H. Symmetry-based indicators of band topology in the 230 space groups. Nat. Commun. 8, 50 (2017).

    ADS  Google Scholar 

  38. 38.

    Tang, F., Po, H. C., Vishwanath, A. & Wan, X. Efficient topological materials discovery using symmetry indicators. Nat. Phys. 15, 470–476 (2019).

    Google Scholar 

  39. 39.

    Po, H. C. Symmetry indicators of band topology. J. Phys. Condens. Matter 32, 263001 (2020).

    ADS  Google Scholar 

  40. 40.

    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).

    ADS  Google Scholar 

  41. 41.

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

    ADS  Google Scholar 

  42. 42.

    Peterson, C. W., Benalcazar, W. A., Hughes, T. L. & Bahl, G. A quantized microwave quadrupole insulator with topologically protected corner states. Nature 555, 346–350 (2018).

    ADS  Google Scholar 

  43. 43.

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

    Google Scholar 

  44. 44.

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

    ADS  Google Scholar 

  45. 45.

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

    ADS  Google Scholar 

  46. 46.

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

    ADS  Google Scholar 

  47. 47.

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

    ADS  Google Scholar 

  48. 48.

    Liu, S. et al. Octupole corner state in a three-dimensional topological circuit. Light Sci. Appl. 9, 1–9 (2020).

    Google Scholar 

  49. 49.

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

    ADS  Google Scholar 

  50. 50.

    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).

    ADS  Google Scholar 

  51. 51.

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

    ADS  Google Scholar 

  52. 52.

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

    ADS  Google Scholar 

  53. 53.

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

    ADS  Google Scholar 

  54. 54.

    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).

    Google Scholar 

  55. 55.

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

    Google Scholar 

  56. 56.

    Xie, B.-Y. et al. Visualization of higher-order topological insulating phases in two-dimensional dielectric photonic crystals. Phys. Rev. Lett. 122, 233903 (2019).

    ADS  Google Scholar 

  57. 57.

    Chen, X.-D. et al. Direct observation of corner states in second-order topological photonic crystal slabs. Phys. Rev. Lett. 122, 233902 (2019).

    ADS  Google Scholar 

  58. 58.

    Ota, Y. et al. Photonic crystal nanocavity based on a topological corner state. Optica 6, 786–789 (2019).

    ADS  Google Scholar 

  59. 59.

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

    ADS  Google Scholar 

  60. 60.

    Zhou, X. et al. Twisted quadrupole topological photonic crystals. Laser Photonics Rev. 14, 2000010 (2020).

    ADS  Google Scholar 

  61. 61.

    Lin, Z.-K., Wang, H.-X., Xiong, Z., Lu, M.-H. & Jiang, J.-H. Anomalous quadrupole topological insulators in two-dimensional nonsymmorphic sonic crystals. Phys. Rev. B 102, 035105 (2020).

    ADS  Google Scholar 

  62. 62.

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

    ADS  Google Scholar 

  63. 63.

    Chen, Y., Lin, Z.-K., Chen, H. & Jiang, J.-H. Plasmon-polaritonic quadrupole topological insulators. Phys. Rev. B 101, 041109 (2020).

    ADS  Google Scholar 

  64. 64.

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

    ADS  Google Scholar 

  65. 65.

    Kim, M. & Rho, J. Topological edge and corner states in a two-dimensional photonic Su-Schrieffer-Heeger lattice. Nanophotonics 9, 3227–3234 (2020).

    Google Scholar 

  66. 66.

    He, L., Addison, Z., Mele, E. J. & Zhen, B. Quadrupole topological photonic crystals. Nat. Commun. 11, 3119 (2020).

    ADS  Google Scholar 

  67. 67.

    Zhang, L. et al. Higher-order topological states in surface-wave photonic crystals. Adv. Sci. 7, 1902724 (2020).

    Google Scholar 

  68. 68.

    Li, M. et al. Higher-order topological states in photonic kagome crystals with long-range interactions. Nat. Photonics 14, 89–94 (2020).

    ADS  Google Scholar 

  69. 69.

    Chen, Y., Lu, X. & Chen, H. Effect of truncation on photonic corner states in a Kagome lattice. Optics Lett. 44, 4251–4254 (2019).

    ADS  Google Scholar 

  70. 70.

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

    Google Scholar 

  71. 71.

    Yang, Y. et al. Gapped topological kink states and topological corner states in honeycomb lattice. Sci. Bull. 65, 531−537 (2020).

  72. 72.

    Fan, H., Xia, B., Tong, L., Zheng, S. & Yu, D. Elastic higher-order topological insulator with topologically protected corner states. Phys. Rev. Lett. 122, 204301 (2019).

    ADS  Google Scholar 

  73. 73.

    Wang, Z., Wei, Q., Xu, H.-Y. & Wu, D.-J. A higher-order topological insulator with wide bandgaps in Lamb-wave systems. J. Appl. Phys. 127, 075105 (2020).

    ADS  Google Scholar 

  74. 74.

    Yang, H., Li, Z.-X., Liu, Y., Cao, Y. & Yan, P. Observation of symmetry-protected zero modes in topolectrical circuits. Phys. Rev. Res. 2, 022028 (2020).

    Google Scholar 

  75. 75.

    Wakao, H., Yoshida, T., Araki, H., Mizoguchi, T. & Hatsugai, Y. Higher-order topological phases in a spring-mass model on a breathing Kagome lattice. Phys. Rev. B 101, 094107 (2020).

    ADS  Google Scholar 

  76. 76.

    Pelegrí, G., Marques, A., Ahufinger, V., Mompart, J. & Dias, R. Second-order topological corner states with ultracold atoms carrying orbital angular momentum in optical lattices. Phys. Rev. B 100, 205109 (2019).

    ADS  Google Scholar 

  77. 77.

    Wang, Y. et al. Protecting quantum superposition and entanglement with photonic higher-order topological crystalline insulator. Preprint at (2020).

  78. 78.

    Chen, C.-W., Chaunsali, R., Christensen, J., Theocharis, G. & Yang, J. Corner states in second-order mechanical topological insulator. Preprint at (2020).

  79. 79.

    Wu, Y., Yan, M., Wang, H.-X., Li, F. & Jiang, J.-H. On-chip higher-order topological micromechanical metamaterials. Preprint at (2020).

  80. 80.

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

    Google Scholar 

  81. 81.

    Banerjee, R., Mandal, S. & Liew, T. Coupling between exciton-polariton corner modes through edge states. Phys. Rev. Lett. 124, 063901 (2020).

    ADS  Google Scholar 

  82. 82.

    Xiong, Z. et al. Corner states and topological transitions in two-dimensional higher-order topological sonic crystals with inversion symmetry. Phys. Rev. B 102, 125144 (2020).

    ADS  Google Scholar 

  83. 83.

    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).

    ADS  Google Scholar 

  84. 84.

    Joannopoulos, J. D., Johnson, S. G., Winn, J. N. & Meade, R. D. Molding the Flow of Light (Princeton Univ. Press, 2008).

  85. 85.

    Lu, M.-H., Feng, L. & Chen, Y.-F. Phononic crystals and acoustic metamaterials. Mater. Today 12, 34–42 (2009).

    Google Scholar 

  86. 86.

    Kim, M., Jacob, Z. & Rho, J. Recent advances in 2D, 3D and higher-order topological photonics. Light Sci. Appl. 9, 1–30 (2020).

    Google Scholar 

  87. 87.

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

    ADS  Google Scholar 

  88. 88.

    Khanikaev, A. B. & Shvets, G. Two-dimensional topological photonics. Nat. Photonics 11, 763–773 (2017).

    ADS  Google Scholar 

  89. 89.

    Ozawa, T. et al. Topological photonics. Rev. Mod. Phys. 91, 015006 (2019).

    MathSciNet  ADS  Google Scholar 

  90. 90.

    Xie, B.-Y. et al. Photonics meets topology. Opt. Express 26, 24531–24550 (2018).

    ADS  Google Scholar 

  91. 91.

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

    ADS  Google Scholar 

  92. 92.

    Zhang, X., Xiao, M., Cheng, Y., Lu, M.-H. & Christensen, J. Topological sound. Commun. Phys. 1, 1–13 (2018).

    Google Scholar 

  93. 93.

    Liu, Y., Chen, X. & Xu, Y. Topological phononics: from fundamental models to real materials. Adv. Funct. Mater. 30, 1904784 (2020).

    Google Scholar 

  94. 94.

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

    Google Scholar 

  95. 95.

    Kunst, F. K., van Miert, G. & Bergholtz, E. J. Lattice models with exactly solvable topological hinge and corner states. Phys. Rev. B 97, 241405 (2018).

    ADS  Google Scholar 

  96. 96.

    Sheng, X.-L. et al. Two-dimensional second-order topological insulator in graphdiyne. Phys. Rev. Lett. 123, 256402 (2019).

    ADS  Google Scholar 

  97. 97.

    Ren, Y., Qiao, Z. & Niu, Q. Engineering corner states from two-dimensional topological insulators. Phys. Rev. Lett. 124, 166804 (2020).

    ADS  Google Scholar 

  98. 98.

    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).

    ADS  Google Scholar 

  99. 99.

    Wang, Q., Liu, C.-C., Lu, Y.-M. & Zhang, F. High-temperature Majorana corner states. Phys. Rev. Lett. 121, 186801 (2018).

    ADS  Google Scholar 

  100. 100.

    Liu, T., He, J. J. & Nori, F. et al. Majorana corner states in a two-dimensional magnetic topological insulator on a high-temperature superconductor. Phys. Rev. B 98, 245413 (2018).

    ADS  Google Scholar 

  101. 101.

    Zhu, X. Tunable Majorana corner states in a two-dimensional second-order topological superconductor induced by magnetic fields. Phys. Rev. B 97, 205134 (2018).

    ADS  Google Scholar 

  102. 102.

    Pahomi, T. E., Sigrist, M. & Soluyanov, A. A. Braiding Majorana corner modes in a second-order topological superconductor. Phys. Rev. Res. 2, 032068 (2020).

    Google Scholar 

  103. 103.

    You, Y., Litinski, D. & Von Oppen, F. Higher-order topological superconductors as generators of quantum codes. Phys. Rev. B 100, 054513 (2019).

    ADS  Google Scholar 

  104. 104.

    Hsu, Y.-T., Cole, W. S., Zhang, R.-X. & Sau, J. D. Inversion-protected higher-order topological superconductivity in monolayer WTe2. Phys. Rev. Lett. 125, 097001 (2020).

    ADS  Google Scholar 

  105. 105.

    Xie, Y.-M., Zhou, B. T. & Law, K. T. Spin-orbit-parity-coupled superconductivity in topological monolayer WTe2. Phys. Rev. Lett. 125, 107001 (2020).

    ADS  Google Scholar 

  106. 106.

    Ghorashi, S. A. A., Hughes, T. L. & Rossi, E. Vortex and surface phase transitions in superconducting higher-order topological insulators. Phys. Rev. Lett. 125, 037001 (2020).

    ADS  Google Scholar 

  107. 107.

    Zhang, R.-X., Cole, W. S., Wu, X. & Das Sarma, S. Higher-order topology and nodal topological superconductivity in Fe(Se, Te) heterostructures. Phys. Rev. Lett. 123, 167001 (2019).

    ADS  Google Scholar 

  108. 108.

    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).

    Google Scholar 

  109. 109.

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

    ADS  Google Scholar 

  110. 110.

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

    ADS  Google Scholar 

  111. 111.

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

    ADS  Google Scholar 

  112. 112.

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

    ADS  Google Scholar 

  113. 113.

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

  114. 114.

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

    ADS  Google Scholar 

  115. 115.

    Liu, F. & Wakabayashi, K. Novel topological phase with a zero Berry curvature. Phys. Rev. Lett. 118, 076803 (2017).

    ADS  Google Scholar 

  116. 116.

    Liu, S. et al. Topologically protected edge state in two-dimensional Su–Schrieffer–Heeger circuit. Research 2019, 8609875 (2019).

    Google Scholar 

  117. 117.

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

  118. 118.

    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 

  119. 119.

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

    Google Scholar 

  120. 120.

    Yang, Y. et al. Visualization of a unidirectional electromagnetic waveguide using topological photonic crystals made of dielectric materials. Phys. Rev. Lett. 120, 217401 (2018).

    ADS  Google Scholar 

  121. 121.

    Xie, B. et al. Higher-order quantum spin Hall effect in a photonic crystal. Nat. Commun. 11, 3768 (2020).

    ADS  Google Scholar 

  122. 122.

    Lin, Z.-K., Wu, S.-Q., Wang, H.-X. & Jiang, J.-H. Higher-order topological spin Hall effect of sound. Chin. Phys. Lett. 37, 074302 (2020).

    ADS  Google Scholar 

  123. 123.

    Zhang, Z. et al. Pseudospin induced topological corner state at intersecting sonic lattices. Phys. Rev. B 101, 220102 (2020).

    ADS  Google Scholar 

  124. 124.

    Wu, S., Jiang, B., Liu, Y. & Jiang, J.-H. All-dielectric photonic crystal with unconventional higher-order topology. Photonics Res. 9, 668–677 (2021).

    Google Scholar 

  125. 125.

    Wang, H.-X. et al. Higher-order topological phases in tunable C3-symmetric photonic crystals. Preprint at (2021).

  126. 126.

    Rachel, S. Interacting topological insulators: a review. Rep. Prog. Phys. 81, 116501 (2018).

    ADS  Google Scholar 

  127. 127.

    Ezawa, M. Higher-order topological electric circuits and topological corner resonance on the breathing kagome and pyrochlore lattices. Phys. Rev. B 98, 201402 (2018).

    ADS  Google Scholar 

  128. 128.

    Fang, C. & Fu, L. New classes of topological crystalline insulators having surface rotation anomaly. Sci. Adv. 5, eaat2374 (2019).

    ADS  Google Scholar 

  129. 129.

    Kooi, S. H., Van Miert, G. & Ortix, C. Inversion-symmetry protected chiral hinge states in stacks of doped quantum Hall layers. Phys. Rev. B 98, 245102 (2018).

    ADS  Google Scholar 

  130. 130.

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

    ADS  Google Scholar 

  131. 131.

    Varnava, N. & Vanderbilt, D. Surfaces of axion insulators. Phys. Rev. B 98, 245117 (2018).

    ADS  Google Scholar 

  132. 132.

    Hackenbroich, A., Hudomal, A., Schuch, N., Bernevig, B. A. & Regnault, N. Fractional chiral hinge insulator. Phys. Rev. B 103, 161110 (2021).

    ADS  Google Scholar 

  133. 133.

    Pozo, O., Repellin, C. & Grushin, A. G. Quantization in Chiral higher order topological insulators: circular dichroism and local Chern marker. Phys. Rev. Lett. 123, 247401 (2019).

    ADS  Google Scholar 

  134. 134.

    Park, M. J., Lee, S. & Kim, Y. B. Hinge magnons from non-collinear magnetic order in honeycomb antiferromagnet. Preprint at (2021).

  135. 135.

    Xu, Y., Song, Z., Wang, Z., Weng, H. & Dai, X. Higher-order topology of the axion insulator EuIn2As2. Phys. Rev. Lett. 122, 256402 (2019).

    ADS  Google Scholar 

  136. 136.

    Yue, C. et al. Symmetry-enforced chiral hinge states and surface quantum anomalous Hall effect in the magnetic axion insulator Bi2−xSmxSe3. Nat. Phys. 15, 577–581 (2019).

    Google Scholar 

  137. 137.

    Wang, Z., Wieder, B. J., Li, J., Yan, B. & Bernevig, B. A. Higher-order topology, monopole nodal lines, and the origin of large Fermi arcs in transition metal dichalcogenides XTe2(X = Mo, W). Phys. Rev. Lett. 123, 186401 (2019).

    ADS  Google Scholar 

  138. 138.

    Kim, K. C. F., Ali, M. N., Law, K. T. & Lee, G.-H. Evidence of higher order topology in multilayer WTe2 from Josephson coupling through anisotropic hinge states. Nat. Mater. 19, 974–979 (2020).

    ADS  Google Scholar 

  139. 139.

    Gray, M. J. et al. Evidence for helical hinge zero modes in an Fe-based superconductor. Nano Lett. 19, 4890–4896 (2019).

    ADS  Google Scholar 

  140. 140.

    Tiwari, A., Li, M.-H., Bernevig, B. A., Neupert, T. & Parameswaran, S. A. Unhinging the surfaces of higher-order topological insulators and superconductors. Phys. Rev. Lett. 124, 046801 (2020).

    ADS  Google Scholar 

  141. 141.

    Zhang, R.-X., Cole, W. S. & Sarma, S. D. Helical hinge Majorana modes in iron-based superconductors. Phys. Rev. Lett. 122, 187001 (2019).

    ADS  Google Scholar 

  142. 142.

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

    Google Scholar 

  143. 143.

    Dutt, A., Minkov, M., Williamson, I. A. & Fan, S. Higher-order topological insulators in synthetic dimensions. Light Sci. Appl. 9, 1–9 (2020).

    Google Scholar 

  144. 144.

    Petrides, I. & Zilberberg, O. Higher-order topological insulators, topological pumps and the quantum Hall effect in high dimensions. Phys. Rev. Res. 2, 022049 (2020).

    Google Scholar 

  145. 145.

    Zhang, W. & Zhang, X. Photonic quadrupole topological phases in zero-dimensional cavity with synthetic dimensions. Preprint at (2019).

  146. 146.

    Araki, H., Mizoguchi, T. & Hatsugai, Y. Phase diagram of a disordered higher-order topological insulator: a machine learning study. Phys. Rev. B 99, 085406 (2019).

    ADS  Google Scholar 

  147. 147.

    Agarwala, A., Juričić, V. & Roy, B. Higher-order topological insulators in amorphous solids. Phys. Rev. Res. 2, 012067 (2020).

    Google Scholar 

  148. 148.

    Chen, R., Chen, C.-Z., Gao, J.-H., Zhou, B. & Xu, D.-H. Higher-order topological insulators in quasicrystals. Phys. Rev. Lett. 124, 036803 (2020).

    MathSciNet  ADS  Google Scholar 

  149. 149.

    Spurrier, S. & Cooper, N. R. Kane-Mele with a twist: quasicrystalline higher-order topological insulators with fractional mass kinks. Phys. Rev. Res. 2, 033071 (2020).

    Google Scholar 

  150. 150.

    Varjas, D. et al. Topological phases without crystalline counterparts. Phys. Rev. Lett. 123, 196401 (2019).

    ADS  Google Scholar 

  151. 151.

    Hua, C.-B., Chen, R., Zhou, B. & Xu, D.-H. Higher-order topological insulator in a dodecagonal quasicrystal. Phys. Rev. B 102, 241102 (2020).

    ADS  Google Scholar 

  152. 152.

    Peng, Y. Floquet higher-order topological insulators and superconductors with space-time symmetries. Phys. Rev. Res. 2, 013124 (2020).

    Google Scholar 

  153. 153.

    Rasmussen, A. & Lu, Y.-M. Classification and construction of higher-order symmetry-protected topological phases of interacting bosons. Phys. Rev. B 101, 085137 (2020).

    ADS  Google Scholar 

  154. 154.

    Okuma, N., Sato, M. & Shiozaki, K. Topological classification under nonmagnetic and magnetic point group symmetry: application of real-space Atiyah-Hirzebruch spectral sequence to higher-order topology. Phys. Rev. B 99, 085127 (2019).

    ADS  Google Scholar 

  155. 155.

    Chen, Z.-G., Xu, C., Al Jahdali, R., Mei, J. & Wu, Y. Corner states in a second-order acoustic topological insulator as bound states in the continuum. Phys. Rev. B 100, 075120 (2019).

    ADS  Google Scholar 

  156. 156.

    Benalcazar, W. A. & Cerjan, A. Bound states in the continuum of higher-order topological insulators. Phys. Rev. B 101, 161116 (2020).

    ADS  Google Scholar 

  157. 157.

    Liu, Y. et al. Bulk-disclination correspondence in topological crystalline insulators. Nature 589, 381–385 (2021).

    ADS  Google Scholar 

  158. 158.

    Peterson, C. W., Li, T., Jiang, W., Hughes, T. L. & Bahl, G. Trapped fractional charges at bulk defects in topological insulators. Nature 589, 376–380 (2021).

    ADS  Google Scholar 

  159. 159.

    Liu, F., Deng, H.-Y. & Wakabayashi, K. Helical topological edge states in a quadrupole phase. Phys. Rev. Lett. 122, 086804 (2019).

    ADS  Google Scholar 

  160. 160.

    Wei, Q. et al. Higher-order topological semimetal in acoustic crystals. Nat. Mater. (2021).

  161. 161.

    Hoeller, J. & Alexandradinata, A. Topological Bloch oscillations. Phys. Rev. B 98, 024310 (2018).

    ADS  Google Scholar 

  162. 162.

    Barut, A. & Bracken, A. Zitterbewegung and the internal geometry of the electron. Phys. Rev. D 23, 2454 (1981).

    MathSciNet  ADS  Google Scholar 

  163. 163.

    Jiang, X. et al. Direct observation of Klein tunneling in phononic crystals. Science 370, 1447–1450 (2020).

    ADS  Google Scholar 

  164. 164.

    Lohse, M., Schweizer, C., Zilberberg, O., Aidelsburger, M. & Bloch, I. A Thouless quantum pump with ultracold Bosonic atoms in an optical superlattice. Nat. Phys. 12, 350–354 (2016).

    Google Scholar 

  165. 165.

    Nakajima, S. et al. Topological Thouless pumping of ultracold fermions. Nat. Phys. 12, 296–300 (2016).

    Google Scholar 

  166. 166.

    Di Liberto, M., Goldman, N. & Palumbo, G. Non-Abelian Bloch oscillations in higher-order topological insulators. Nat. Commun. 11, 5942 (2020).

    Google Scholar 

  167. 167.

    Benalcazar, W. A. et al. Higher-order topological pumping. Preprint at (2020).

  168. 168.

    Bi-Ye, X., Oubo, Y. & Zhang, S. Topological disclination pump. Preprint at (2021).

  169. 169.

    El-Ganainy, R. et al. Non-Hermitian physics and PT symmetry. Nat. Phys. 14, 11–19 (2018).

    Google Scholar 

  170. 170.

    Yao, S., Song, F. & Wang, Z. Non-Hermitian Chern bands. Phys. Rev. Lett. 121, 136802 (2018).

    ADS  Google Scholar 

  171. 171.

    Ashida, Y., Gong, Z. & Ueda, M. Non-Hermitian physics. Adv. Phys. 69, 3 (2020).

    Google Scholar 

  172. 172.

    Lee, C. H., Li, L. & Gong, J. Hybrid higher-order skin-topological modes in nonreciprocal systems. Phys. Rev. Lett. 123, 016805 (2019).

    ADS  Google Scholar 

  173. 173.

    Liu, T. et al. Second-order topological phases in non-Hermitian systems. Phys. Rev. Lett. 122, 076801 (2019).

    ADS  Google Scholar 

  174. 174.

    Edvardsson, E., Kunst, F. K. & Bergholtz, E. J. Non-Hermitian extensions of higher-order topological phases and their biorthogonal bulk-boundary correspondence. Phys. Rev. B 99, 081302 (2019).

    ADS  Google Scholar 

  175. 175.

    Luo, X.-W. & Zhang, C. Higher-order topological corner states induced by gain and loss. Phys. Rev. Lett. 123, 073601 (2019).

    MathSciNet  ADS  Google Scholar 

  176. 176.

    Liu, T. et al. Second-order topological phases in non-Hermitian systems. Phys. Rev. Lett. 122, 076801 (2019).

    ADS  Google Scholar 

  177. 177.

    Yu, Y., Jung, M. & Shvets, G. Zero-energy corner states in a non-Hermitian quadrupole insulator. Phys. Rev. B 103, L041102 (2021).

    ADS  Google Scholar 

  178. 178.

    Edvardsson, E., Kunst, F. K. & Bergholtz, E. J. Non-Hermitian extensions of higher-order topological phases and their biorthogonal bulk-boundary correspondence. Phys. Rev. B 99, 081302 (2019).

    ADS  Google Scholar 

  179. 179.

    Wu, Y.-J., Liu, C.-C. & Hou, J. Wannier-type photonic higher-order topological corner states induced solely by gain and loss. Phys. Rev. A 101, 043833 (2020).

    ADS  Google Scholar 

  180. 180.

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

    ADS  Google Scholar 

  181. 181.

    Ezawa, M. Non-Hermitian higher-order topological states in nonreciprocal and reciprocal systems with their electric-circuit realization. Phys. Rev. B 99, 201411 (2019).

    ADS  Google Scholar 

  182. 182.

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

    ADS  Google Scholar 

  183. 183.

    Cayssol, J., Dóra, B., Simon, F. & Moessner, R. Floquet topological insulators. Phys. Status Solidi RRL 7, 101–108 (2013).

    Google Scholar 

  184. 184.

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

    ADS  Google Scholar 

  185. 185.

    Hu, H., Huang, B., Zhao, E. & Liu, W. V. Dynamical singularities of Floquet higher-order topological insulators. Phys. Rev. Lett. 124, 057001 (2020).

    ADS  Google Scholar 

  186. 186.

    Peng, Y. & Refael, G. Floquet second-order topological insulators from nonsymmorphic space-time symmetries. Phys. Rev. Lett. 123, 016806 (2019).

    ADS  Google Scholar 

  187. 187.

    Huang, B. & Liu, W. V. Floquet higher-order topological insulators with anomalous dynamical polarization. Phys. Rev. Lett. 124, 216601 (2020).

    ADS  Google Scholar 

  188. 188.

    Bomantara, R. W., Zhou, L., Pan, J. & Gong, J. Coupled-wire construction of static and Floquet second-order topological insulators. Phys. Rev. B 99, 045441 (2019).

    ADS  Google Scholar 

  189. 189.

    Rodriguez-Vega, M., Kumar, A. & Seradjeh, B. Higher-order Floquet topological phases with corner and bulk bound states. Phys. Rev. B 100, 085138 (2019).

    ADS  Google Scholar 

  190. 190.

    Seshadri, R., Dutta, A. & Sen, D. Generating a second-order topological insulator with multiple corner states by periodic driving. Phys. Rev. B 100, 115403 (2019).

    ADS  Google Scholar 

  191. 191.

    Ghosh, A. K., Paul, G. C. & Saha, A. Higher order topological insulator via periodic driving. Phys. Rev. B 101, 235403 (2020).

    ADS  Google Scholar 

  192. 192.

    Mukherjee, S. et al. Experimental observation of anomalous topological edge modes in a slowly driven photonic lattice. Nat. Commun. 8, 13918 (2017).

    ADS  Google Scholar 

  193. 193.

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

    ADS  Google Scholar 

  194. 194.

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

    Google Scholar 

  195. 195.

    Zhu, W., Xue, H., Gong, J., Chong, Y. & Zhang, B. Time-periodic corner states from Floquet higher-order topology. Preprint at (2020).

  196. 196.

    Meng, Y., Chen, G. & Jia, S. Second-order topological insulator in a coinless discrete-time quantum walk. Phys. Rev. A 102, 012203 (2020).

    ADS  Google Scholar 

  197. 197.

    Dubinkin, O. & Hughes, T. L. Entanglement signatures of multipolar higher order topological phases. Preprint at (2020).

  198. 198.

    Kitagawa, T. et al. Observation of topologically protected bound states in photonic quantum walks. Nat. Commun. 3, 882 (2012).

    ADS  Google Scholar 

  199. 199.

    Blanco-Redondo, A., Bell, B., Oren, D., Eggleton, B. J. & Segev, M. Topological protection of biphoton states. Science 362, 568–571 (2018).

    MathSciNet  MATH  ADS  Google Scholar 

  200. 200.

    Barik, S. et al. A topological quantum optics interface. Science 359, 666–668 (2018).

    MathSciNet  MATH  ADS  Google Scholar 

  201. 201.

    Kadic, M., Milton, G. W., van Hecke, M. & Wegener, M. 3D metamaterials. Nat. Rev. Phys. 1, 198–210 (2019).

    Google Scholar 

  202. 202.

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

    ADS  Google Scholar 

  203. 203.

    Yang, B. et al. Ideal Weyl points and helicoid surface states in artificial photonic crystal structures. Science 359, 1013–1016 (2018).

    MathSciNet  MATH  ADS  Google Scholar 

  204. 204.

    Jia, H. et al. Observation of chiral zero mode in inhomogeneous three-dimensional Weyl metamaterials. Science 363, 148–151 (2019).

    MathSciNet  MATH  ADS  Google Scholar 

  205. 205.

    Li, Z.-X., Cao, Y., Wang, X. R. & Yan, P. Symmetry-protected zero modes in metamaterials based on topological spin texture. Phys. Rev. Appl. 13, 064058 (2020).

    ADS  Google Scholar 

  206. 206.

    Vakulenko, A. et al. Near-field characterization of higher-order topological photonic states at optical frequencies. Adv. Mater. 33, 2004376 (2021).

  207. 207.

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

    ADS  Google Scholar 

  208. 208.

    Salerno, G., Palumbo, G., Goldman, N. & Di Liberto, M. Interaction-induced lattices for bound states: designing flat bands, quantized pumps, and higher-order topological insulators for doublons. Phys. Rev. Res. 2, 013348 (2020).

    Google Scholar 

  209. 209.

    You, Y., Devakul, T., Burnell, F. J. & Neupert, T. Higher-order symmetry-protected topological states for interacting bosons and fermions. Phys. Rev. B 98, 235102 (2018).

    ADS  Google Scholar 

  210. 210.

    Zhang, Y., Kartashov, Y., Torner, L., Li, Y. & Ferrando, A. Nonlinear higher-order polariton topological insulator. Opt. Lett. 45, 4710–4713 (2020).

    ADS  Google Scholar 

  211. 211.

    Zangeneh-Nejad, F. & Fleury, R. Nonlinear second-order topological insulators. Phys. Rev. Lett. 123, 053902 (2019).

    ADS  Google Scholar 

  212. 212.

    Manzeli, S., Ovchinnikov, D., Pasquier, D., Yazyev, O. V. & Kis, A. 2D transition metal dichalcogenides. Nat. Rev. Mater. 2, 17033 (2017).

    ADS  Google Scholar 

  213. 213.

    Wang, H.-X., Wang, Q., Zhou, K.-G. & Zhang, H.-L. Graphene in light: design, synthesis and applications of photo-active graphene and graphene-like materials. Small 9, 1266–1283 (2013).

    ADS  Google Scholar 

  214. 214.

    Quan, L. N., García de Arquer, F. P., Sabatini, R. P. & Sargent, E. H. Perovskites for light emission. Adv. Mater. 30, 1801996 (2018).

    Google Scholar 

  215. 215.

    Xia, F., Wang, H., Xiao, D., Dubey, M. & Ramasubramaniam, A. Two-dimensional material nanophotonics. Nat. Photonics 8, 899–907 (2014).

    ADS  Google Scholar 

  216. 216.

    Mak, K. F. & Shan, J. Photonics and optoelectronics of 2D semiconductor transition metal dichalcogenides. Nat. Photonics 10, 216–226 (2016).

    ADS  Google Scholar 

  217. 217.

    Bhimanapati, G. R. et al. Recent advances in two-dimensional materials beyond graphene. ACS Nano 9, 11509–11539 (2015).

    Google Scholar 

  218. 218.

    Gan, X. et al. Controlling the spontaneous emission rate of monolayer MoS2 in a photonic crystal nanocavity. Appl. Phys. Lett. 103, 181119 (2013).

    ADS  Google Scholar 

  219. 219.

    Wu, S. et al. Monolayer semiconductor nanocavity lasers with ultralow thresholds. Nature 520, 69–72 (2015).

    ADS  Google Scholar 

  220. 220.

    Xie, X. et al. Cavity quantum electrodynamics with second-order topological corner state. Laser Photonics Rev. 14, 1900425 (2020).

    ADS  Google Scholar 

  221. 221.

    Bomantara, R. W. & Gong, J. Measurement-only quantum computation with Floquet Majorana corner modes. Phys. Rev. B 101, 085401 (2020).

    ADS  Google Scholar 

  222. 222.

    Ezawa, M. Braiding of Majorana-like corner states in electric circuits and its non-Hermitian generalization. Phys. Rev. B 100, 045407 (2019).

    ADS  Google Scholar 

  223. 223.

    Iadecola, T., Schuster, T. & Chamon, C. Non-Abelian braiding of light. Phys. Rev. Lett. 117, 073901 (2016).

    ADS  Google Scholar 

  224. 224.

    Noh, J. et al. Braiding photonic topological zero modes. Nat. Phys. 16, 989–993 (2020).

    Google Scholar 

  225. 225.

    Karzig, T. et al. Scalable designs for quasiparticle-poisoning-protected topological quantum computation with Majorana zero modes. Phys. Rev. B 95, 235305 (2017).

    ADS  Google Scholar 

  226. 226.

    Gao, P. et al. Majorana-like zero modes in Kekulé distorted sonic lattices. Phys. Rev. Lett. 123, 196601 (2019).

    ADS  Google Scholar 

Download references


This work was supported by the National Key R & D Program of China (grant numbers 2017YFA0303702, 2018YFA0306200 and 2016YFB0700301), the National Natural Science Foundation of China (grant numbers 11625418, 11474158, 11890700, 51732006, 11904060 and 12074281) and the Jiangsu specially appointed professor funding.

Author information




All authors worked together on preparing and writing this Perspective.

Corresponding authors

Correspondence to Jian-Hua Jiang, Minghui Lu or Yanfeng Chen.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information

Nature Reviews Physics thanks Baile Zhang and other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note

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


Ammann−Beenker tiling

A tiling generated by a set of 45−135-degree rhombuses and 45−45−90 degree triangles, which has a non-periodic and hierarchical structure.

Atiyah−Hirzebruch spectral sequence

A type of spectral sequence for calculating generalized cohomology.

Bulk-boundary correspondence

The correspondence between the topological phases of bulk materials and the emergence of gapless boundary states.


The dimension of bulk states minus the dimension of some specific states. Therefore, higher-codimensional states have lower dimensionality in one system.


A homology theory, dual to K-theory and used to classify the elliptic pseudo-differential operators acting on the vector bundles over a space.


A branch of mathematics that studies a ring generated by vector bundles over a topological space or scheme; it can be applied to classify topological phases with respect to different symmetries.

Multistep driving

Spatial control of the tunnelling amplitudes between lattice sites in multiple steps within each driving period in a Floquet system.

Non-Hermitian skin effect

A unique feature of non-Hermitian systems, in which an extensive number of eigenstates are exponentially localized at the boundary of the system under the open boundary conditions.

Transmission line networks

Transmission lines are coaxial cables that can be regarded as 1D waveguides; they can be connected at nodes to form a network that can be used to mimic tight-binding models.

Zak phase

Berry phase gained by a particle during the adiabatic motion across the Brillouin zone, which is quantized and can be regarded as the topological invariant of the band.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation


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