Topological states in quasicrystals

Abstract

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.

Graphical abstract

Keywords

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

References

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

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

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

[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)

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

[6] A. Altland and M. R. Zirnbauer, Nonstandard symmetry classes in mesoscopic normal superconducting hybrid structures, Phys. Rev. B55(2), 1142 (1997)

[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)

[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] 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)

[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)

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

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

[13] H. Huang, Y. Xu, J. Wang, and W. Duan, Emerging topological states in quasi-two-dimensional materials, WIRES: Comp. Mol. Sci, 7(4), e1296 (2017)

[14] H. Huang, J. Liu, and W. Duan, Nontrivial Z2 topology in bismuth-based iii–v compounds, Phys. Rev. B90(19), 195105 (2014)

[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. B92(16), 161115 (2015)

[16] H. Huang and F. Liu, A unified view of topological phase transition in band theory, Research2020, 7832610 (2020)

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

[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)

[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)

[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)

[21] H. C. Po, A. Vishwanath, and H. Watanabe, Symmetrybased indicators of band topology in the 230 space groups, Nat. Commun. 8(1), 50 (2017)

[22] B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, Topological quantum chemistry, Nature547(7663), 298 (2017)

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

[24] T. Zhang, Y. Jiang, Z. Song, H. Huang, Y. He, Z. Fang, H. Weng, and C. Fang, Catalogue of topological electronic materials, Nature566(7745), 475 (2019)

[25] F. Tang, H. C. Po, A. Vishwanath, and X. Wan, Comprehensive search for topological materials using symmetry indicators, Nature566(7745), 486 (2019)

[26] M. G. Vergniory, L. Elcoro, C. Felser, N. Regnault, B. A. Bernevig, and Z. Wang, A complete catalogue of high-quality topological materials, Nature566(7745), 480 (2019)

[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)

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

[29] C. Janot, Quasicrystals, in: Neutron and Synchrotron Radiation for Condensed MatterStudies, pp 197–211, Springer, 1994

[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)

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

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

[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)

[34] D. T. Tran, A. Dauphin, N. Goldman, and P. Gaspard, Topological Hofstadter insulators in a two-dimensional quasicrystal, Phys. Rev. B91(8), 085125 (2015)

[35] D. R. Hofstadter, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B14(6), 2239 (1976)

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

[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. B100(16), 165101 (2019)

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

[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)

[40] H. Huang and F. Liu, Theory of spin Bott index for quantum spin hall states in nonperiodic systems, Phys. Rev. B98(12), 125130 (2018)

[41] H. Huang and F. Liu, Comparison of quantum spin Hall states in quasicrystals and crystals, Phys. Rev. B100(8), 085119 (2019)

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

[43] R. Chen, D. H. Xu, and B. Zhou, Topological Anderson insulator phase in a quasicrystal lattice, Phys. Rev. B100(11), 115311 (2019)

[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)

[45] A. L. He, L. R. Ding, Y. Zhou, Y. F. Wang, and C. D. Gong, Quasicrystalline Chern insulators, Phys. Rev. B100(21), 214109 (2019)

[46] H. Huang, Y. S. Wu, and F. Liu, Aperiodic topological crystalline insulators, Phys. Rev. B101(4), 041103 (2020)

[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)

[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)

[49] S. Spurrier and N. R. Cooper, Kane-Mele with a twist: Quasicrystalline higher-order topological insulators with fractional mass kinks, Phys. Rev. Research2(3), 033071 (2020)

[50] C. B. Hua, R. Chen, B. Zhou, and D. H. Xu, Higher-order topological insulator in a dodecagonal quasicrystal, Phys. Rev. B102(24), 241102 (2020)

[51] T. Kitagawa, E. Berg, M. Rudner, and E. Demler, Topological characterization of periodically driven quantum systems, Phys. Rev. B82(23), 235114 (2010)

[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)

[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)

[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)

[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)

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

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

[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)

[59] Z. Li and Z. F. Wang, Quantum anomalous Hall effect in twisted bilayer graphene quasicrystal, Chin. Phys. B29(10), 107101 (2020)

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

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

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

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

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

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

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

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

[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)

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

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

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

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

[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)

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

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

[76] L. Guidoni, B. Dépret, A. Di Stefano, and P. Verkerk, Atomic diffusion in an optical quasicrystal with five-fold symmetry, Phys. Rev. A60(6), R4233 (1999)

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

[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)

[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)

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

[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, Nanotechnology11(4), 274 (2000)

[82] B. Freedman, G. Bartal, M. Segev, R. Lifshitz, and N. Demetrios, Wave and defect dynamics in nonlinear photonic quasicrystals, Nature440(7088), 1166 (2006)

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

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

[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)

[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)

[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)

[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)

[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)

[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)

[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, Nature404(6779), 740 (2000)

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

[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)

[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, Nature496(7444), 196 (2013)

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

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

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

[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)

[99] D. N. Christodoulides, F. Lederer, and Y. Silberberg, Discretizing light behaviour in linear and nonlinear waveguide lattices, Nature424(6950), 817 (2003)

[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. Express13(26), 10552 (2005)

[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)

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

[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)

[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)

[105] R. Bianco and R. Resta, Mapping topological order in coordinate space, Phys. Rev. B84(24), 241106 (2011)

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

[107] M. A. Bandres, M. C. Rechtsman, and M. Segev, Topological photonic quasicrystals: Fractal topological spectrum and protected transport, Phys. Rev. X6(1), 011016 (2016)

[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)

[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] 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)

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

[112] M. Brzezińska, A. M. Cook, and T. Neupert, Topology in the Sierpiński–Hofstadter problem, Phys. Rev. B98(20), 205116 (2018)

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

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

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

[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)

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

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

[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)

[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. B75(12), 121403 (2007)

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

[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)

[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)

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

[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)

[126] J. C. Y. Teo, L. Fu, and C. L. Kane, Surface states and topological invariants in three-dimensional topological insulators: Application to Bi_1−xSb_x, Phys. Rev. B78(4), 045426 (2008)

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

[128] Z. Ringel, Y. E. Kraus, and A. Stern, Strong side of weak topological insulators, Phys. Rev. B86(4), 045102 (2012)

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

[130] A. Y. Kitaev, Unpaired Majorana fermions in quantum wires, Phys. Uspekhi44(10S), 131 (2001)

[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. B61(15), 10267 (2000)

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

[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)

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

RIGHTS & PERMISSIONS

2022 Higher Education Press