Skip to main content

Topological states in quasicrystals


With the rapid development of topological states in crystals, the study of topological states has been extended to quasicrystals in recent years. In this review, we summarize the recent progress of topological states in quasicrystals, particularly focusing on one-dimensional (1D) and 2D systems. We first give a brief introduction to quasicrystalline structures. Then, we discuss topological phases in 1D quasicrystals where the topological nature is attributed to the synthetic dimensions associated with the quasiperiodic order of quasicrystals. We further present the generalization of various types of crystalline topological states to 2D quasicrystals, where real-space expressions of corresponding topological invariants are introduced due to the lack of translational symmetry in quasicrystals. Finally, since quasicrystals possess forbidden symmetries in crystals such as five-fold and eight-fold rotation, we provide an overview of unique quasicrystalline symmetry-protected topological states without crystalline counterpart.

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


  1. 1.

    X. L. Qi and S. C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83(4), 1057 (2011)

    ADS  Article  Google Scholar 

  2. 2.

    M. Z. Hasan and C. L. Kane, Topological insulators, Rev. Mod. Phys. 82(4), 3045 (2010)

    ADS  Article  Google Scholar 

  3. 3.

    B. A. Bernevig, Topological Insulators and Topological Superconductors, Princeton University Press, 2013

  4. 4.

    K. Klitzing, G. Dorda, and M. Pepper, New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance, Phys. Rev. Lett. 45(6), 494 (1980)

    ADS  Article  Google Scholar 

  5. 5.

    D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett. 49(6), 405 (1982)

    ADS  Article  Google Scholar 

  6. 6.

    A. Altland and M. R. Zirnbauer, Nonstandard symmetry classes in mesoscopic normal superconducting hybrid structures, Phys. Rev. B 55(2), 1142 (1997)

    ADS  Article  Google Scholar 

  7. 7.

    A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Classification of topological insulators and superconductors in three spatial dimensions, Phys. Rev. B 78(19), 195125 (2008)

    ADS  Article  Google Scholar 

  8. 8.

    A. Kitaev, Periodic table for topological insulators and superconductors, in: AIP Conference Proceedings, Vol. 1134, pp 22–30, American Institute of Physics, 2009

  9. 9.

    S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, Topological insulators and superconductors: Tenfold way and dimensional hierarchy, New J. Phys. 12(6), 065010 (2010)

    ADS  Article  Google Scholar 

  10. 10.

    H. Zhang and S. C. Zhang, Topological insulators from the perspective of first-principles calculations, Phys. Status Solidi Rapid Res. Lett. 7(1–2), 72 (2013)

    ADS  Article  Google Scholar 

  11. 11.

    Y. Ando, Topological insulator materials, J. Phys. Soc. Jpn. 82(10), 102001 (2013)

    ADS  Article  Google Scholar 

  12. 12.

    M. Sato and Y. Ando, Topological superconductors: A review, Rep. Prog. Phys. 80(7), 076501 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  13. 13.

    H. Huang, Y. Xu, J. Wang, and W. Duan, Emerging topological states in quasi-two-dimensional materials, WIRES: Comp. Mol. Sci, 7(4), el296 (2017)

    Google Scholar 

  14. 14.

    H. Huang, J. Liu, and W. Duan, Nontrivial Z2 topology in bismuth-based iii-v compounds, Phys. Rev. B 90(19), 195105 (2014)

    ADS  Article  Google Scholar 

  15. 15.

    H. Huang, Z. Liu, H. Zhang, W. Duan, and D. Vanderbilt, Emergence of a Chern insulating state from a semi-Dirac dispersion, Phys. Rev. B 92(16), 161115 (2015)

    ADS  Article  Google Scholar 

  16. 16.

    H. Huang and F. Liu, A unified view of topological phase transition in band theory, Research 2020, 7832610 (2020)

    Google Scholar 

  17. 17.

    L. Fu, Topological crystalline insulators, Phys. Rev. Lett. 106(10), 106802 (2011)

    ADS  Article  Google Scholar 

  18. 18.

    Y. Ando and L. Fu, Topological crystalline insulators and topological superconductors: From concepts to materials, Annu. Rev. Condens. Matter Phys. 6(1), 361 (2015)

    ADS  Article  Google Scholar 

  19. 19.

    F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. P. Parkin, B. A. Bernevig, and T. Neupert, Higher-order topological insulators, Sci. Adv. 4(6), eaat0346 (2018)

    ADS  Article  Google Scholar 

  20. 20.

    J. Langbehn, Y. Peng, L. Trifunovic, F. von Oppen, and P. W. Brouwer, Reflection-symmetric second-order topological insulators and superconductors, Phys. Rev. Lett. 119(24), 246401 (2017)

    ADS  Article  Google Scholar 

  21. 21.

    H. C. Po, A. Vishwanath, and H. Watanabe, Symmetry-based indicators of band topology in the 230 space groups, Nat. Commun. 8(1), 50 (2017)

    ADS  Article  Google Scholar 

  22. 22.

    B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, Topological quantum chemistry, Nature 547(7663), 298 (2017)

    ADS  Article  Google Scholar 

  23. 23.

    Z. Song, T. Zhang, Z. Fang, and C. Fang, Quantitative mappings between symmetry and topology in solids, Nat. Commun. 9, 3530 (2018)

    ADS  Article  Google Scholar 

  24. 24.

    T. Zhang, Y. Jiang, Z. Song, H. Huang, Y. He, Z. Fang, H. Weng, and C. Fang, Catalogue of topological electronic materials, Nature 566(7745), 475 (2019)

    ADS  Article  Google Scholar 

  25. 25.

    F. Tang, H. C. Po, A. Vishwanath, and X. Wan, Comprehensive search for topological materials using symmetry indicators, Nature 566(7745), 486 (2019)

    ADS  Article  Google Scholar 

  26. 26.

    M. G. Vergniory, L. Elcoro, C. Felser, N. Regnault, B. A. Bernevig, and Z. Wang, A complete catalogue of high-quality topological materials, Nature 566(7745), 480 (2019)

    ADS  Article  Google Scholar 

  27. 27.

    D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett. 53(20), 1951 (1984)

    ADS  Article  Google Scholar 

  28. 28.

    P. J. Steinhardt and S. Ostlund, The Physics of Quasicrystals, World Scientific, 1987

  29. 29.

    C. Janot, Quasicrystals, in: Neutron and Synchrotron Radiation for Condensed Matter Studies, pp 197–211, Springer, 1994

  30. 30.

    Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, Topological states and adiabatic pumping in quasicrystals, Phys. Rev. Lett. 109(10), 106402 (2012)

    ADS  Article  Google Scholar 

  31. 31.

    Y. E. Kraus and O. Zilberberg, Topological equivalence between the Fibonacci quasicrystal and the Harper model, Phys. Rev. Lett. 109(11), 116404 (2012)

    ADS  Article  Google Scholar 

  32. 32.

    Y. E. Kraus, Z. Ringel, and O. Zilberberg, Four-dimensional quantum hall effect in a two-dimensional quasicrystal, Phys. Rev. Lett. 111(22), 226401 (2013)

    ADS  Article  Google Scholar 

  33. 33.

    M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, Observation of topological phase transitions in photonic quasicrystals, Phys. Rev. Lett. 110(7), 076403 (2013)

    ADS  Article  Google Scholar 

  34. 34.

    D. T. Tran, A. Dauphin, N. Goldman, and P. Gaspard, Topological Hofstadter insulators in a two-dimensional quasicrystal, Phys. Rev. B 91(8), 085125 (2015)

    ADS  Article  Google Scholar 

  35. 35.

    D. R. Hofstadter, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B 14(6), 2239 (1976)

    ADS  Article  Google Scholar 

  36. 36.

    J. N. Fuchs and J. Vidal, Hofstadter butterfly of a quasicrystal, Phys. Rev. B 94(20), 205437 (2016)

    ADS  Article  Google Scholar 

  37. 37.

    G. Naumis, Higher-dimensional quasicrystalline approach to the Hofstadter butterfly topological-phase band conductances: Symbolic sequences and self-similar rules at all magnetic fluxes, Phys. Rev. B 100(16), 165101 (2019)

    ADS  Article  Google Scholar 

  38. 38.

    C. W. Duncan, S. Manna, and A. E. B. Nielsen, Topological models in rotationally symmetric quasicrystals, Phys. Rev. B 101(11), 115413 (2020)

    ADS  Article  Google Scholar 

  39. 39.

    H. Huang and F. Liu, Quantum spin Hall effect and spin Bott index in a quasicrystal lattice, Phys. Rev. Lett. 121(12), 126401 (2018)

    ADS  Article  Google Scholar 

  40. 40.

    H. Huang and F. Liu, Theory of spin Bott index for quantum spin hall states in nonperiodic systems, Phys. Rev. B 98(12), 125130 (2018)

    ADS  Article  Google Scholar 

  41. 41.

    H. Huang and F. Liu, Comparison of quantum spin Hall states in quasicrystals and crystals, Phys. Rev. B 100(8), 085119 (2019)

    ADS  Article  Google Scholar 

  42. 42.

    J. Li, R. L. Chu, J. K. Jain, and S. Q. Shen, Topological Anderson insulator, Phys. Rev. Lett. 102(13), 136806 (2009)

    ADS  Article  Google Scholar 

  43. 43.

    R. Chen, D. H. Xu, and B. Zhou, Topological Anderson insulator phase in a quasicrystal lattice, Phys. Rev. B 100(11), 115311 (2019)

    ADS  Article  Google Scholar 

  44. 44.

    T. Peng, C. B. Hua, R. Chen, D. H. Xu, and B. Zhou, Topological Anderson insulators in an Ammann-Beenker quasicrystal and a snub-square crystal, Phys. Rev. B 103(8), 085307 (2021)

    ADS  Article  Google Scholar 

  45. 45.

    A. L. He, L. R. Ding, Y. Zhou, Y. F. Wang, and C. D. Gong, Quasicrystalline Chern insulators, Phys. Rev. B 100(21), 214109 (2019)

    ADS  Article  Google Scholar 

  46. 46.

    H. Huang, Y. S. Wu, and F. Liu, Aperiodic topological crystalline insulators, Phys. Rev. B 101(4), 041103 (2020)

    ADS  Article  Google Scholar 

  47. 47.

    D. Varjas, A. Lau, K. Pöyhönen, A. R. Akhmerov, D. I. Pikulin, and I. C. Fulga, Topological phases without crystalline counterparts, Phys. Rev. Lett. 123(19), 196401 (2019)

    ADS  Article  Google Scholar 

  48. 48.

    R. Chen, C. Z. Chen, J. H. Gao, B. Zhou, and D. H. Xu, Higher-order topological insulators in quasicrystals, Phys. Rev. Lett. 124(3), 036803 (2020)

    ADS  MathSciNet  Article  Google Scholar 

  49. 49.

    S. Spurrier and N. R. Cooper, Kane-Mele with a twist: Quasicrystalline higher-order topological insulators with fractional mass kinks, Phys. Rev. Research 2(3), 033071 (2020)

    ADS  Article  Google Scholar 

  50. 50.

    C. B. Hua, R. Chen, B. Zhou, and D. H. Xu, Higher-order topological insulator in a dodecagonal quasicrystal, Phys. Rev. B 102(24), 241102 (2020)

    ADS  Article  Google Scholar 

  51. 51.

    T. Kitagawa, E. Berg, M. Rudner, and E. Demler, Topological characterization of periodically driven quantum systems, Phys. Rev. B 82(23), 235114 (2010)

    ADS  Article  Google Scholar 

  52. 52.

    Z. Gu, H. A. Fertig, D. P. Arovas, and A. Auerbach, Floquet spectrum and transport through an irradiated graphene ribbon, Phys. Rev. Lett. 107(21), 216601 (2011)

    ADS  Article  Google Scholar 

  53. 53.

    M. Tezuka and N. Kawakami, Reentrant topological transitions in a quantum wire/superconductor system with quasiperiodic lattice modulation, Phys. Rev. B 85(14), 140508 (2012)

    ADS  Article  Google Scholar 

  54. 54.

    W. DeGottardi, D. Sen, and S. Vishveshwara, Majorana fermions in superconducting 1D systems having periodic, quasiperiodic, and disordered potentials, Phys. Rev. Lett. 110(14), 146404 (2013)

    ADS  Article  Google Scholar 

  55. 55.

    R. Ghadimi, T. Sugimoto, and T. Tohyama, Majorana zero-energy mode and fractal structure in Fibonacci-Kitaev chain, J. Phys. Soc. Jpn. 86(11), 114707 (2017)

    ADS  Article  Google Scholar 

  56. 56.

    I. C. Fulga, D. I. Pikulin, and T. A. Loring, Aperiodic weak topological superconductors, Phys. Rev. Lett. 116(25), 257002 (2016)

    ADS  Article  Google Scholar 

  57. 57.

    R. Ghadimi, T. Sugimoto, K. Tanaka, and T. Tohyama, Topological superconductivity in quasicrystals, arXiv: 2006.06952 (2020)

  58. 58.

    Y. Cao, Y. Zhang, Y. B. Liu, C. C. Liu, W. Q. Chen, and F. Yang, Kohn-Luttinger mechanism driven exotic topological superconductivity on the Penrose lattice, Phys. Rev. Lett. 125(1), 017002 (2020)

    ADS  MathSciNet  Article  Google Scholar 

  59. 59.

    Z. Li and Z. F. Wang, Quantum anomalous Hall effect in twisted bilayer graphene quasicrystal, Chin. Phys. B 29(10), 107101 (2020)

    ADS  Article  Google Scholar 

  60. 60.

    W. Yao, E. Wang, C. Bao, Y. Zhang, K. Zhang, et al., Quasicrystalline 30 twisted bilayer graphene as an incommensurate superlattice with strong interlayer coupling, Proceedings of the National Academy of Sciences, 115(27), 6928 (2018)

    ADS  Article  Google Scholar 

  61. 61.

    A. Bansil, H. Lin, and T. Das, Topological band theory, Rev. Mod. Phys. 88(2), 021004 (2016)

    ADS  Article  Google Scholar 

  62. 62.

    C. L. Kane and E. J. Mele, Z2 topological order and the quantum spin hall effect, Phys. Rev. Lett. 95(14), 146802 (2005)

    ADS  Article  Google Scholar 

  63. 63.

    L. Fu and C. L. Kane, Topological insulators with inversion symmetry, Phys. Rev. B 76(4), 045302 (2007)

    ADS  Article  Google Scholar 

  64. 64.

    A. Jagannathan, The Fibonacci quasicrystal: Case study of hidden dimensions and multifractality, arXiv: 2012.14744 (2020)

  65. 65.

    E. Prodan, Virtual topological insulators with real quantized physics, Phys. Rev. B 91(24), 245104 (2015)

    ADS  Article  Google Scholar 

  66. 66.

    D. Levine and P. J. Steinhardt, Quasicrystals: A new class of ordered structures, Phys. Rev. Lett. 53(26), 2477 (1984)

    ADS  Article  Google Scholar 

  67. 67.

    N. Wang, H. Chen, and K. H. Kuo, Two-dimensional quasicrystal with eightfold rotational symmetry, Phys. Rev. Lett. 59(9), 1010 (1987)

    ADS  Article  Google Scholar 

  68. 68.

    N. I. N. G. Wang, K. K. Fung, and K. H. Kuo, Symmetry study of the Mn-Si-Al octagonal quasicrystal by convergent beam electron diffraction, Appl. Phys. Lett. 52(25), 2120 (1988)

    ADS  Article  Google Scholar 

  69. 69.

    R. Lifshitz, Quasicrystals: A matter of definition, Found. Phys. 33(12), 1703 (2003)

    MathSciNet  Article  Google Scholar 

  70. 70.

    P. Bak, Phenomenological theory of icosahedral incommensurate (“quasiperiodic”) order in Mn-Al alloys, Phys. Rev. Lett. 54(14), 1517 (1985)

    ADS  Article  Google Scholar 

  71. 71.

    M. Duneau and A. Katz, Quasiperiodic patterns, Phys. Rev. Lett. 54(25), 2688 (1985)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  72. 72.

    V. Elser and C. L. Henley, Crystal and quasicrystal structures in Al-Mn-Si alloys, Phys. Rev. Lett. 55(26), 2883 (1985)

    ADS  Article  Google Scholar 

  73. 73.

    J. E. S. Socolar, T. C. Lubensky, and P. J. Steinhardt, Phonons, phasons, and dislocations in quasicrystals, Phys. Rev. B 34(5), 3345 (1986)

    ADS  Article  Google Scholar 

  74. 74.

    S. J. Poon, Electronic properties of quasicrystals an experimental review, Adv. Phys. 41(4), 303 (1992)

    ADS  Article  Google Scholar 

  75. 75.

    L. Guidoni, C. Triché, P. Verkerk, and G. Grynberg, Quasiperiodic optical lattices, Phys. Rev. Lett. 79(18), 3363 (1997)

    ADS  Article  Google Scholar 

  76. 76.

    L. Guidoni, B. Dépret, A. Di Stefano, and P. Verkerk, Atomic diffusion in an optical quasicrystal with five-fold symmetry, Phys. Rev. A 60(6), R4233 (1999)

    ADS  Article  Google Scholar 

  77. 77.

    T. A. Corcovilos and J. Mittal, Two-dimensional optical quasicrystal potentials for ultracold atom experiments, Appl. Opt. 58(9), 2256 (2019)

    ADS  Article  Google Scholar 

  78. 78.

    K. Viebahn, M. Sbroscia, E. Carter, J. C. Yu, and U. Schneider, Matter-wave diffraction from a quasicrystalline optical lattice, Phys. Rev. Lett. 122(11), 110404 (2019)

    ADS  Article  Google Scholar 

  79. 79.

    M. Sbroscia, K. Viebahn, E. Carter, J.-C. Yu, A. Gaunt, and U. Schneider, Observing localization in a 2D quasicrystalline optical lattice, Phys. Rev. Lett. 125, 200604 (2020)

    ADS  Article  Google Scholar 

  80. 80.

    W. Steurer and D. Sutter-Widmer, Photonic and phononic quasicrystals, J. Phys. D Appl. Phys. 40(13), R229 (2007)

    ADS  Article  Google Scholar 

  81. 81.

    M. A. Kaliteevski, S. Brand, R. A. Abram, T. F. Krauss, R. DeLa Rue, and P. Millar, Two-dimensional penrosetiled photonic quasicrystals: From diffraction pattern to band structure, Nanotechnology 11(4), 274 (2000)

    ADS  Article  Google Scholar 

  82. 82.

    B. Freedman, G. Bartal, M. Segev, R. Lifshitz, and N. Demetrios, Wave and defect dynamics in nonlinear photonic quasicrystals, Nature 440(7088), 1166 (2006)

    ADS  Article  Google Scholar 

  83. 83.

    A. Jagannathan and M. Duneau, An eightfold optical quasicrystal with cold atoms, EPL 104(6), 66003 (2014)

    ADS  Article  Google Scholar 

  84. 84.

    M. Verbin, O. Zilberberg, Y. Lahini, and E. Yaacov, Topological pumping over a photonic Fibonacci quasicrystal, Phys. Rev. B 91(6), 064201 (2015)

    ADS  Article  Google Scholar 

  85. 85.

    M. Bayindir, E. Cubukcu, I. Bulu, and E. Ozbay, Photonic band-gap effect, localization, and waveguiding in the two-dimensional Penrose lattice, Phys. Rev. B 63(16), 161104 (2001)

    ADS  Article  Google Scholar 

  86. 86.

    A. Della Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, Band gap formation and multiple scattering in photonic quasicrystals with a Penrose-type lattice, Phys. Rev. Lett. 94(18), 183903 (2005)

    ADS  Article  Google Scholar 

  87. 87.

    P. Bordia, H. Lüschen, S. Scherg, S. Gopalakrishnan, M. Knap, U. Schneider, and I. Bloch, Probing slow relaxation and many-body localization in two-dimensional quasiperiodic systems, Phys. Rev. X 7(4), 041047 (2017)

    Google Scholar 

  88. 88.

    H. P. Lüschen, P. Bordia, S. Scherg, F. Alet, E. Altman, U. Schneider, and I. Bloch, Observation of slow dynamics near the many-body localization transition in one-dimensional quasiperiodic systems, Phys. Rev. Lett. 119(26), 260401 (2017)

    ADS  Article  Google Scholar 

  89. 89.

    Y. S. Chan, C. T. Chan, and Z. Y. Liu, Photonic band gaps in two dimensional photonic quasicrystals, Phys. Rev. Lett. 80(5), 956 (1998)

    ADS  Article  Google Scholar 

  90. 90.

    L. Dal Negro, C. J. Oton, Z. Gaburro, L. Pavesi, P. Johnson, A. Lagendijk, R. Righini, M. Colocci, and D. S. Wiersma, Light transport through the band-edge states of Fibonacci quasicrystals, Phys. Rev. Lett. 90(5), 055501 (2003)

    ADS  Article  Google Scholar 

  91. 91.

    M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, Complete photonic bandgaps in 12-fold symmetric quasicrystals, Nature 404(6779), 740 (2000)

    ADS  Article  Google Scholar 

  92. 92.

    I. Bloch, Ultracold quantum gases in optical lattices, Nat. Phys. 1(1), 23 (2005)

    Article  Google Scholar 

  93. 93.

    T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, Topological photonics, Rev. Mod. Phys. 91(1), 015006 (2019)

    ADS  MathSciNet  Article  Google Scholar 

  94. 94.

    M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Photonic floquet topological insulators, Nature 496(7444), 196 (2013)

    ADS  Article  Google Scholar 

  95. 95.

    O. Zilberberg, Topology in quasicrystals, arXiv: 2012. 03644 (2020)

  96. 96.

    S. Aubry and G. André, Analyticity breaking and Anderson localization in incommensurate lattices, Ann. Isr. Phys. Soc. 3(133), 18 (1980)

    MathSciNet  MATH  Google Scholar 

  97. 97.

    J. Zak, Magnetic translation group, Phys. Rev. 134(6A), A1602 (1964)

    ADS  MATH  Article  Google Scholar 

  98. 98.

    I. Dana, Y. Avron, and J. Zak, Quantised Hall conductance in a perfect crystal, J. Phys. C Solid State Phys. 18(22), L679 (1985)

    ADS  Article  Google Scholar 

  99. 99.

    D. N. Christodoulides, F. Lederer, and Y. Silberberg, Discretizing light behaviour in linear and nonlinear waveguide lattices, Nature 424(6950), 817 (2003)

    ADS  Article  Google Scholar 

  100. 100.

    A. Szameit, D. Blömer, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, Discrete nonlinear localization in femtosecond laser written waveguides in fused silica, Opt. Express 13(26), 10552 (2005)

    ADS  Article  Google Scholar 

  101. 101.

    Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson, and Y. Silberberg, Observation of a localization transition in quasiperiodic photonic lattices, Phys. Rev. Lett. 103, 013901 (2009)

    ADS  Article  Google Scholar 

  102. 102.

    I. Petrides, H. M. Price, and O. Zilberberg, Six-dimensional quantum hall effect and three-dimensional topological pumps, Phys. Rev. B 98, 125431 (2018)

    ADS  Article  Google Scholar 

  103. 103.

    T. Fukui, Y. Hatsugai, and H. Suzuki, Chern numbers in discretized Brillouin zone: Efficient method of computing (spin) Hall conductances, J. Phys. Soc. Jpn. 74(6), 1674 (2005)

    ADS  Article  Google Scholar 

  104. 104.

    Y. Hatsugai, T. Fukui, and H. Aoki, Topological analysis of the quantum hall effect in graphene: Dirac-Fermi transition across van hove singularities and edge versus bulk quantum numbers, Phys. Rev. B 74(20), 205414 (2006)

    ADS  Article  Google Scholar 

  105. 105.

    R. Bianco and R. Resta, Mapping topological order in coordinate space, Phys. Rev. B 84(24), 241106 (2011)

    ADS  Article  Google Scholar 

  106. 106.

    N. H. Lindner, G. Refael, and V. Galitski, Floquet topological insulator in semiconductor quantum wells, Nat. Phys. 7(6), 490 (2011)

    Article  Google Scholar 

  107. 107.

    M. A. Bandres, M. C. Rechtsman, and M. Segev, Topological photonic quasicrystals: Fractal topological spectrum and protected transport, Phys. Rev. X 6(1), 011016 (2016)

    Google Scholar 

  108. 108.

    Z. Gu, H. A. Fertig, and P. Daniel, Floquet spectrum and transport through an irradiated graphene ribbon, Phys. Rev. Lett. 107(21), 216601 (2011)

    ADS  Article  Google Scholar 

  109. 109.

    D. Toniolo, On the equivalence of the Bott index and the Chern number on a torus, and the quantization of the Hall conductivity with a real space Kubo formula, arXiv: 1708.05912 (2017)

  110. 110.

    F. D. M. Haldane, Model for a quantum hall effect without landau levels: Condensed-matter realization of the parity anomaly, Phys. Rev. Lett. 61(18), 2015 (1988)

    ADS  Article  Google Scholar 

  111. 111.

    A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Phys. 321(1), 2 (2006)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  112. 112.

    M. Brzezińska, A. M. Cook, and T. Neupert, Topology in the Sierpiński-Hofstadter problem, Phys. Rev. B 98(20), 205116 (2018)

    ADS  Article  Google Scholar 

  113. 113.

    C. L. Kane and E. J. Mele, Quantum spin Hall effect in graphene, Phys. Rev. Lett. 95(22), 226801 (2005)

    ADS  Article  Google Scholar 

  114. 114.

    B. A. Bernevig and S. C. Zhang, Quantum spin Hall effect, Phys. Rev. Lett. 96(10), 106802 (2006)

    ADS  Article  Google Scholar 

  115. 115.

    J. Maciejko, T. L. Hughes, and S.-C. Zhang, The quantum spin Hall effect, Annu. Rev. Condens. Matter Phys. 2(1), 31 (2011)

    ADS  Article  Google Scholar 

  116. 116.

    M. König, H. Buhmann, L. W. Molenkamp, T. Hughes, C. X. Liu, X. L. Qi, and S. C. Zhang, The quantum spin Hall effect: Theory and experiment, J. Phys. Soc. Jpn. 77(3), 031007 (2008)

    ADS  Article  Google Scholar 

  117. 117.

    J. C. Slater and G. F. Koster, Simplified LCAO method for the periodic potential problem, Phys. Rev. 94(6), 1498 (1954)

    ADS  MATH  Article  Google Scholar 

  118. 118.

    W. A. Harrison, Electronic structure and the properties of solids: the physics of the chemical bond, Courier Corporation, 2012

  119. 119.

    D. N. Sheng, Z. Y. Weng, L. Sheng, and F. D. M. Haldane, Quantum spin-Hall effect and topologically invariant Chern numbers, Phys. Rev. Lett. 97(3), 036808 (2006)

    ADS  Article  Google Scholar 

  120. 120.

    T. Fukui and Y. Hatsugai, Topological aspects of the quantum spin-Hall effect in graphene: Z2 topological order and spin Chern number, Phys. Rev. B 75(12), 121403 (2007)

    ADS  Article  Google Scholar 

  121. 121.

    E. Prodan, Robustness of the spin-Chern number, Phys. Rev. B 80(12), 125327 (2009)

    ADS  Article  Google Scholar 

  122. 122.

    J. Bellissard, A. van Elst, and H. Schulz-Baldes, The noncommutative geometry of the quantum Hall effect, J. Math. Phys. 35(10), 5373 (1994)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  123. 123.

    M. B. Hastings and T. A. Loring, Almost commuting matrices, localized Wannier functions, and the quantum hall effect, J. Math. Phys. 51(1), 015214 (2010)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  124. 124.

    R. Exel and A. Terry, Invariants of almost commuting unitaries, J. Funct. Anal. 95(2), 364 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  125. 125.

    H. Katsura and T. Koma, The noncommutative index theorem and the periodic table for disordered topological insulators and superconductors, J. Math. Phys. 59(3), 031903 (2018)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  126. 126.

    J. C. Y. Teo, L. Fu, and C. L. Kane, Surface states and topological invariants in three-dimensional topological insulators: Application to Bi1−xSbx, Phys. Rev. B 78(4), 045426 (2008)

    ADS  Article  Google Scholar 

  127. 127.

    T. A. Loring, K-theory and pseudospectra for topological insulators, Ann. Phys. 356, 383 (2015)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  128. 128.

    Z. Ringel, Y. E. Kraus, and A. Stern, Strong side of weak topological insulators, Phys. Rev. B 86(4), 045102 (2012)

    ADS  Article  Google Scholar 

  129. 129.

    I. C. Fulga, B. van Heck, J. M. Edge, and A. R. Akhmerov, Statistical topological insulators, Phys. Rev. B 89(15), 155424 (2014)

    ADS  Article  Google Scholar 

  130. 130.

    A. Y. Kitaev, Unpaired Majorana fermions in quantum wires, Phys. Uspekhi 44(10S), 131 (2001)

    ADS  Article  Google Scholar 

  131. 131.

    N. Read and D. Green, Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect, Phys. Rev. B 61(15), 10267 (2000)

    ADS  Article  Google Scholar 

  132. 132.

    R. Jackiw and C. Rebbi, Solitons with fermion number 1/2, Phys. Rev. D 13(12), 3398 (1976)

    ADS  MathSciNet  Article  Google Scholar 

  133. 133.

    J. C. Y. Teo and T. L. Hughes, Existence of majoranafermion bound states on disclinations and the classification of topological crystalline superconductors in two dimensions, Phys. Rev. Lett. 111(4), 047006 (2013)

    ADS  Article  Google Scholar 

  134. 134.

    M. Baake and U. Grimm, Aperiodic Order, Vol. 1, Cambridge University Press, 2013

Download references


This work was supported by the National Natural Science Foundation of China (Grant No. 12074006) and the Start-up Funds from Peking University.

Author information



Corresponding author

Correspondence to Huaqing Huang.

Additional information

This article can also be found at

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fan, J., Huang, H. Topological states in quasicrystals. Front. Phys. 17, 13203 (2022).

Download citation


  • topological states
  • quasicrystals
  • quantum Hall effect
  • topological insulator
  • topological superconductor