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Majorana States

  • Jeffrey C. Y. TeoEmail author
Chapter
Part of the Springer Series in Solid-State Sciences book series (SSSOL, volume 192)

Abstract

The main goal of this chapter is to introduce the basic concepts of Majorana fermions and zero energy Majorana bound states, and their origin from topology, magnetism and superconductivity. This chapter gears towards young researchers at their early developing stage in their career, and for the most parts, the central ideas are presented in a self-contained manner without assuming professional background knowledge other than fundamental quantum mechanics and solid state physics.

Notes

Acknowledgements

I am in debt to Prof. Charlie Kane, who was my Ph.D advisor and introduced the concepts of Majorana and topological insulators and superconductors to me during my graduate years. In addition, I also thank all my research collaborators, in particular Ching-Kai Chiu, Eduardo Fradkin, Liang Fu, Taylor Hughes, and Shinsei Ryu, who not only made significant contributions to the advancement in Majorana physics but also had valuable impact on my academic development. Last but not least, I am grateful to all my students, especially Syed Raza, who went through the manuscript and provided helpful feedback. This chapter is supported by the National Science Foundation under Grant No. DMR 1653535.

References

  1. 1.
    J. Alicea, Rep. Prog. Phys. 75, 076501 (2012).  https://doi.org/10.1088/0034-4885/75/7/076501ADSCrossRefGoogle Scholar
  2. 2.
    C.W.J. Beenakker, Annu. Rev. Conden. Matter Phys. 4, 113 (2013).  https://doi.org/10.1146/annurev-conmatphys-030212-184337ADSCrossRefGoogle Scholar
  3. 3.
    C.-K. Chiu, J.C.Y. Teo, A.P. Schnyder, S. Ryu, Rev. Mod. Phys. 88, 035005 (2016).  https://doi.org/10.1103/RevModPhys.88.035005ADSCrossRefGoogle Scholar
  4. 4.
    S. Das Sarma, M. Freedman, C. Nayak, Npj Quantum Inf. 1, 15001 (2015).  https://doi.org/10.1038/npjqi.2015.1
  5. 5.
    S.R. Elliott, M. Franz, Rev. Mod. Phys. 87, 137 (2015).  https://doi.org/10.1103/RevModPhys.87.137ADSCrossRefGoogle Scholar
  6. 6.
    M.Z. Hasan, C.L. Kane, Rev. Mod. Phys. 82, 3045 (2010).  https://doi.org/10.1103/RevModPhys.82.3045ADSCrossRefGoogle Scholar
  7. 7.
    M. Leijnse, K. Flensberg, Semicond. Sci. Technol. 27(12), 124003 (2012).  https://doi.org/10.1088/0268-1242/27/12/124003ADSCrossRefGoogle Scholar
  8. 8.
    X.-L. Qi, S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).  https://doi.org/10.1103/RevModPhys.83.1057ADSCrossRefGoogle Scholar
  9. 9.
    T.D. Stanescu, S. Tewari, J. Phys. Condens. Matter 25, 233201 (2013), http://stacks.iop.org/0953-8984/25/i=23/a=233201ADSGoogle Scholar
  10. 10.
    E. Schrödinger, Phys. Rev. 28, 1049 (1926).  https://doi.org/10.1103/PhysRev.28.1049ADSCrossRefGoogle Scholar
  11. 11.
    A. Einstein, Ann. der Phys. 322(10), 891 (1905).  https://doi.org/10.1002/andp.19053221004ADSCrossRefGoogle Scholar
  12. 12.
    P.A.M. Dirac, Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 117(778), 610 (1928).  https://doi.org/10.1098/rspa.1928.0023ADSCrossRefGoogle Scholar
  13. 13.
    E. Majorana, Il Nuovo Cimento (1924–1942) 14(4), 171 (1937).  https://doi.org/10.1007/BF02961314ADSCrossRefGoogle Scholar
  14. 14.
    A. Kitaev, AIP Conf. Proc. 1134, 22 (2008).  https://doi.org/10.1063/1.3149495ADSCrossRefGoogle Scholar
  15. 15.
    X.-L. Qi, T.L. Hughes, S. Raghu, S.-C. Zhang, Phys. Rev. Lett. 102, 187001 (2009).  https://doi.org/10.1103/PhysRevLett.102.187001ADSCrossRefGoogle Scholar
  16. 16.
    A.P. Schnyder, S. Ryu, A. Furusaki, A.W.W. Ludwig, Phys. Rev. B 78, 195125 (2008).  https://doi.org/10.1103/PhysRevB.78.195125ADSCrossRefGoogle Scholar
  17. 17.
  18. 18.
    A. Cook, M. Franz, Phys. Rev. B 84, 201105 (2011).  https://doi.org/10.1103/PhysRevB.84.201105ADSCrossRefGoogle Scholar
  19. 19.
    R.M. Lutchyn, J.D. Sau, S. Das Sarma, Phys. Rev. Lett. 105, 077001 (2010).  https://doi.org/10.1103/PhysRevLett.105.077001ADSCrossRefGoogle Scholar
  20. 20.
    S. Nadj-Perge, I.K. Drozdov, J. Li, H. Chen, S. Jeon, J. Seo, A.H. MacDonald, B.A. Bernevig, A. Yazdani, Science 346, 602 (2014).  https://doi.org/10.1126/science.1259327ADSCrossRefGoogle Scholar
  21. 21.
    Y. Oreg, G. Refael, F. von Oppen, Phys. Rev. Lett. 105, 177002 (2010).  https://doi.org/10.1103/PhysRevLett.105.177002ADSCrossRefGoogle Scholar
  22. 22.
    V. Gurarie, L. Radzihovsky, Phys. Rev. B 75, 212509 (2007b).  https://doi.org/10.1103/PhysRevB.75.212509ADSCrossRefGoogle Scholar
  23. 23.
    D.A. Ivanov, Phys. Rev. Lett. 86, 268 (2001).  https://doi.org/10.1103/PhysRevLett.86.268ADSCrossRefGoogle Scholar
  24. 24.
  25. 25.
    N. Read, D. Green, Phys. Rev. B 61, 10267 (2000).  https://doi.org/10.1103/PhysRevB.61.10267ADSCrossRefGoogle Scholar
  26. 26.
    G.E. Volovik, Pisma Zh. Eksp. Teor. Fiz. 70, 601 (1999).  https://doi.org/10.1134/1.568223ADSCrossRefGoogle Scholar
  27. 27.
    G.E. Volovik, The Universe in a Helium Droplet (Oxford University Press, 2003)Google Scholar
  28. 28.
    C.-K. Chiu, M.J. Gilbert, T.L. Hughes, Phys. Rev. B 84(14), 144507 (2011).  https://doi.org/10.1103/PhysRevB.84.144507ADSCrossRefGoogle Scholar
  29. 29.
    L. Fu, C.L. Kane, Phys. Rev. Lett. 100, 096407 (2008).  https://doi.org/10.1103/PhysRevLett.100.096407ADSCrossRefGoogle Scholar
  30. 30.
    L. Fu, C.L. Kane, Phys. Rev. B 79, 161408(R) (2009a).  https://doi.org/10.1103/PhysRevB.79.161408ADSCrossRefGoogle Scholar
  31. 31.
    P. Hosur, P. Ghaemi, R.S.K. Mong, A. Vishwanath, Phys. Rev. Lett. 107(9), 097001 (2011).  https://doi.org/10.1103/PhysRevLett.107.097001ADSCrossRefGoogle Scholar
  32. 32.
    J.C.Y. Teo, C.L. Kane, Phys. Rev. Lett. 104, 046401 (2010a).  https://doi.org/10.1103/PhysRevLett.104.046401ADSCrossRefGoogle Scholar
  33. 33.
    J.C.Y. Teo, C.L. Kane, Phys. Rev. B 82, 115120 (2010b).  https://doi.org/10.1103/PhysRevB.82.115120ADSCrossRefGoogle Scholar
  34. 34.
    J.-P. Xu, C. Liu, M.-X. Wang, J. Ge, Z.-L. Liu, X. Yang, Y. Chen, Y. Liu, Z.-A. Xu, C.-L. Gao, D. Qian, F.-C. Zhang, J.-F. Jia, Phys. Rev. Lett. 112, 217001 (2014).  https://doi.org/10.1103/PhysRevLett.112.217001ADSCrossRefGoogle Scholar
  35. 35.
    P.G. De Gennes, Superconductivity Of Metals And Alloys (Westview Press, 1999)Google Scholar
  36. 36.
    A.J. Leggett, Quantum Liquids: Bose Condensation and Cooper Pairing in Condensed-Matter Systems (Oxford University Press, 2006)Google Scholar
  37. 37.
    A. Altland, M.R. Zirnbauer, Phys. Rev. B 55, 1142 (1997).  https://doi.org/10.1103/PhysRevB.55.1142ADSCrossRefGoogle Scholar
  38. 38.
    F. Wilczek, Nat. Phys. 5, 614 (2009).  https://doi.org/10.1038/nphys1380CrossRefGoogle Scholar
  39. 39.
    M. Freedman, A. Kitaev, M. Larsen, Z. Wang, Bull. Am. Math. Soc. 40, 31 (2002).  https://doi.org/10.1090/S0273-0979-02-00964-3CrossRefGoogle Scholar
  40. 40.
  41. 41.
    C. Nayak, S.H. Simon, A. Stern, M. Freedman, S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008).  https://doi.org/10.1103/RevModPhys.80.1083ADSCrossRefGoogle Scholar
  42. 42.
    J. Preskill, Topological Quantum Computation, http://www.theory.caltech.edu/~preskill/ph219/topological.pdf (2004)
  43. 43.
    R. Walter Ogburn, J. Preskill, Topological quantum computation, in Quantum Computing and Quantum Communications: First NASA International Conference, QCQC’98 Palm Springs, California, USA February 17–20, 1998 Selected Papers, ed. by C.P. Williams (Springer, Berlin, Heidelberg, 1999), pp. 341–356.  https://doi.org/10.1007/3-540-49208-9_31CrossRefGoogle Scholar
  44. 44.
    Z. Wang, Topological Quantum Computation (American Mathematics Society, 2010)Google Scholar
  45. 45.
    W.P. Su, J.R. Schrieffer, A.J. Heeger, Phys. Rev. B 22, 2099 (1980).  https://doi.org/10.1103/PhysRevB.22.2099ADSCrossRefGoogle Scholar
  46. 46.
    C.L. Kane, E.J. Mele, Phys. Rev. Lett. 95, 226801 (2005a).  https://doi.org/10.1103/PhysRevLett.95.226801ADSCrossRefGoogle Scholar
  47. 47.
    C.L. Kane, E.J. Mele, Phys. Rev. Lett. 95, 146802 (2005b).  https://doi.org/10.1103/PhysRevLett.95.146802ADSCrossRefGoogle Scholar
  48. 48.
    I. Knez, C.T. Rettner, S.-H. Yang, S.S.P. Parkin, L. Du, R.-R. Du, G. Sullivan, Phys. Rev. Lett. 112, 026602 (2014).  https://doi.org/10.1103/PhysRevLett.112.026602ADSCrossRefGoogle Scholar
  49. 49.
    M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. Molenkamp, X.-L. Qi, S. Zhang, Science 318, 766 (2007).  https://doi.org/10.1126/science.1148047ADSCrossRefGoogle Scholar
  50. 50.
    C. Liu, T.L. Hughes, X.-L. Qi, K. Wang, S.-C. Zhang, Phys. Rev. Lett. 100, 236601 (2008).  https://doi.org/10.1103/PhysRevLett.100.236601ADSCrossRefGoogle Scholar
  51. 51.
    V. Mourik, K. Zuo, S. Frolov, S. Plissard, E. Bakkers, L. Kouwenhoven, Science 336, 1003 (2012).  https://doi.org/10.1126/science.1222360ADSCrossRefGoogle Scholar
  52. 52.
    A.R. Akhmerov, J. Nilsson, C.W.J. Beenakker, Phys. Rev. Lett. 102, (2009).  https://doi.org/10.1103/PhysRevLett.102.216404
  53. 53.
    A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, H. Shtrikman, Nat. Phys. 8, 887 (2012).  https://doi.org/10.1038/nphys2479CrossRefGoogle Scholar
  54. 54.
    M.T. Deng, C.L. Yu, G.Y. Huang, M. Larsson, P. Caroff, H.Q. Xu, Nano Lett. 12, 6414 (2012).  https://doi.org/10.1021/nl303758wADSCrossRefGoogle Scholar
  55. 55.
    L. Fu, C.L. Kane, Phys. Rev. Lett. 102, 216403 (2009b).  https://doi.org/10.1103/PhysRevLett.102.216403ADSCrossRefGoogle Scholar
  56. 56.
    K.T. Law, P.A. Lee, T.K. Ng, Phys. Rev. Lett. 103, 237001 (2009).  https://doi.org/10.1103/PhysRevLett.103.237001ADSCrossRefGoogle Scholar
  57. 57.
    L.P. Rokhinson, X. Liu, J. K. Furdyna, Nat. Phys. 8, 795 (2012).  https://doi.org/10.1038/nphys2429ADSCrossRefGoogle Scholar
  58. 58.
    H. Zhang, C.-X. Liu, S. Gazibegovic, D. Xu, J.A. Logan, G. Wang, N. van Loo, J.D.S. Bommer, M.W.A. de Moor, D. Car, R.L.M.O. het Veld, P.J. van Veldhoven, S. Koelling, M.A. Verheijen, M. Pendharkar, D.J. Pennachio, B. Shojaei, J.S. Lee, C.J. Palmstrom, E.P.A.M. Bakkers, S. Das Sarma, L.P. Kouwenhoven (2017), arXiv:1006.5454 [cond-mat.mes-hall]
  59. 59.
  60. 60.
    B. van Heck, R.M. Lutchyn, L.I. Glazman, Phys. Rev. B 93, 235431 (2016).  https://doi.org/10.1103/PhysRevB.93.235431ADSCrossRefGoogle Scholar
  61. 61.
    R. Hützen, A. Zazunov, B. Braunecker, A.L. Yeyati, R. Egger, Phys. Rev. Lett. 109, 166403 (2012).  https://doi.org/10.1103/PhysRevLett.109.166403ADSCrossRefGoogle Scholar
  62. 62.
    J.D. Sau, B. Swingle, S. Tewari, Phys. Rev. B 92, 020511 (2015).  https://doi.org/10.1103/PhysRevB.92.020511ADSCrossRefGoogle Scholar
  63. 63.
    S.M. Albrecht, A.P. Higginbotham, M. Madsen, F. Kuemmeth, T.S. Jespersen, J. Nygård, P. Krogstrup, C.M. Marcus, Nature 531, 206 (2016).  https://doi.org/10.1038/nature17162ADSCrossRefGoogle Scholar
  64. 64.
    G.E. Volovik, JETP Lett. 55, 368 (1992), http://www.jetpletters.ac.ru/ps/1273/article_19263.shtml
  65. 65.
    A. Cappelli, M. Huerta, G.R. Zemba, Nucl. Phys. B 636(3), 568 (2002).  https://doi.org/10.1016/S0550-3213(02)00340-1ADSCrossRefGoogle Scholar
  66. 66.
    C.L. Kane, M.P.A. Fisher, Phys. Rev. B 55, 15832 (1997).  https://doi.org/10.1103/PhysRevB.55.15832ADSCrossRefGoogle Scholar
  67. 67.
    V. Gurarie, L. Radzihovsky, Ann. Phys. 322(1), 2 (2007).  https://doi.org/10.1016/j.aop.2006.10.009ADSMathSciNetCrossRefGoogle Scholar
  68. 68.
    M. Sigrist, K. Ueda, Rev. Mod. Phys. 63, 239 (1991).  https://doi.org/10.1103/RevModPhys.63.239ADSCrossRefGoogle Scholar
  69. 69.
    D.I. Uzunov, Theory of ferromagnetic unconventional superconductors with spin-triplet electron pairing, in Superconductors—Materials, Properties and Applications, ed. by A. Gabovich, Chapter 17 (InTech, 2003), pp. 415–440.  https://doi.org/10.5772/48579Google Scholar
  70. 70.
    M. Atiyah, K-Theory (Westview Press, 1994)Google Scholar
  71. 71.
    H.B. Lawson, M.-L. Michelsohn, Spin Geometry (Princeton University Press, 1990)Google Scholar
  72. 72.
    M. Nakahara, Geometry, Topology and Physics, 2nd edn. Graduate Student Series in Physics (Taylor & Francis, 2003), http://www.amazon.ca/exec/obidos/redirect?tag=citeulike09-20&;path=ASIN/0750306068
  73. 73.
    J.D. Jackson, Classical Electrodynamics 3rd edn. (Wiley, New York, 1999)Google Scholar
  74. 74.
    P. Di Francesco, P. Mathieu, D. Senechal, Conformal Field Theory (Springer, New York, 1999)Google Scholar
  75. 75.
    M.F. Atiyah, V.K. Patodi, I.M. Singer, Math. Proc. Camb. Philos. Soc. 77(1), 43 (1975a).  https://doi.org/10.1017/S0305004100049410CrossRefGoogle Scholar
  76. 76.
    M.F. Atiyah, V.K. Patodi, I.M. Singer, Math. Proc. Camb. Philos. Soc. 78(3), 405 (1975b).  https://doi.org/10.1017/S0305004100051872CrossRefGoogle Scholar
  77. 77.
    M.F. Atiyah, V.K. Patodi, I.M. Singer, Math. Proc. Camb. Philos. Soc. 79(1), 71 (1976).  https://doi.org/10.1017/S0305004100052105CrossRefGoogle Scholar
  78. 78.
    B.I. Halperin, Phys. Rev. B 25, 2185 (1982).  https://doi.org/10.1103/PhysRevB.25.2185ADSCrossRefGoogle Scholar
  79. 79.
  80. 80.
    R.B. Laughlin, Phys. Rev. B 23, 5632 (1981).  https://doi.org/10.1103/PhysRevB.23.5632ADSCrossRefGoogle Scholar
  81. 81.
    H. Schulz-Baldes, J. Kellendonk, T. Richter, J. Phys. A: Math. Gen. 33(2), L27 (2000), http://stacks.iop.org/0305-4470/33/i=2/a=102
  82. 82.
    D.J. Thouless, M. Kohmoto, M.P. Nightingale, M. den Nijs, Phys. Rev. Lett. 49, 405 (1982).  https://doi.org/10.1103/PhysRevLett.49.405ADSCrossRefGoogle Scholar
  83. 83.
    J. Bellissard, A. van Elst, H. Schulz-Baldes, J. Math. Phys. 35(10), 5373 (1994).  https://doi.org/10.1063/1.530758ADSMathSciNetCrossRefGoogle Scholar
  84. 84.
    T.A. Loring, M.B. Hastings, EPL (Europhysics Lett.) 92(6), 67004 (2010), http://stacks.iop.org/0295-5075/92/i=6/a=67004ADSCrossRefGoogle Scholar
  85. 85.
    E. Prodan, H. Schulz-Baldes, J. Funct. Anal. 271(5), 1150 (2016).  https://doi.org/10.1016/j.jfa.2016.06.001MathSciNetCrossRefGoogle Scholar
  86. 86.
    P.W. Anderson, P. Morel, Phys. Rev. 123, 1911 (1961).  https://doi.org/10.1103/PhysRev.123.1911ADSMathSciNetCrossRefGoogle Scholar
  87. 87.
    A.J. Leggett, Rev. Mod. Phys. 47, 331 (1975).  https://doi.org/10.1103/RevModPhys.47.331ADSCrossRefGoogle Scholar
  88. 88.
    G.E. Volovik, JETP Lett. 93(2), 66 (2011).  https://doi.org/10.1134/S0021364011020147ADSCrossRefGoogle Scholar
  89. 89.
    G.M. Luke, Y. Fudamoto, K.M. Kojima, M.I. Larkin, J. Merrin, B. Nachumi, Y.J. Uemura, Y. Maeno, Z.Q. Mao, Y. Mori, H. Nakamura, M. Sigrist, Nature 394, 558 (1998).  https://doi.org/10.1038/29038ADSCrossRefGoogle Scholar
  90. 90.
    K.D. Nelson, Z.Q. Mao, Y. Maeno, Y. Liu, Science 306(5699), 1151 (2004).  https://doi.org/10.1126/science.1103881ADSCrossRefGoogle Scholar
  91. 91.
    T.M. Rice, M. Sigrist, J. Phys.: Condens. Matter 7, L643 (1995).  https://doi.org/10.1088/0953-8984/7/47/002ADSCrossRefGoogle Scholar
  92. 92.
    J. Xia, Y. Maeno, P.T. Beyersdorf, M.M. Fejer, A. Kapitulnik, Phys. Rev. Lett. 97, 167002 (2006).  https://doi.org/10.1103/PhysRevLett.97.167002ADSCrossRefGoogle Scholar
  93. 93.
    J.R. Kirtley, C. Kallin, C.W. Hicks, E.-A. Kim, Y. Liu, K.A. Moler, Y. Maeno, K.D. Nelson, Phys. Rev. B 76, 014526 (2007).  https://doi.org/10.1103/PhysRevB.76.014526ADSCrossRefGoogle Scholar
  94. 94.
    Y. Maeno, T.M. Rice, M. Sigrist, Phys. Today 54, 42 (2001).  https://doi.org/10.1063/1.1349611ADSCrossRefGoogle Scholar
  95. 95.
    S. Raghu, A. Kapitulnik, S.A. Kivelson, Phys. Rev. Lett. 105, 136401 (2010).  https://doi.org/10.1103/PhysRevLett.105.136401ADSCrossRefGoogle Scholar
  96. 96.
    F.D.M. Haldane, Phys. Rev. Lett. 61, 2015 (1988).  https://doi.org/10.1103/PhysRevLett.61.2015ADSCrossRefGoogle Scholar
  97. 97.
    C.-X. Liu, S.-C. Zhang, X.-L. Qi, Annu. Rev. Condens. Matter Phys. 7, 301 (2016).  https://doi.org/10.1146/annurev-conmatphys-031115-011417ADSCrossRefGoogle Scholar
  98. 98.
    Q.L. He, L. Pan, A.L. Stern, E.C. Burks, X. Che, G. Yin, J. Wang, B. Lian, Q. Zhou, E.S. Choi, K. Murata, X. Kou, Z. Chen, T. Nie, Q. Shao, Y. Fan, S.-C. Zhang, K. Liu, J. Xia, K.L. Wang, Science 357(6348), 294 (2017).  https://doi.org/10.1126/science.aag2792ADSMathSciNetCrossRefGoogle Scholar
  99. 99.
    A. Ii, K. Yada, M. Sato, Y. Tanaka, Phys. Rev. B 83, 224524 (2011).  https://doi.org/10.1103/PhysRevB.83.224524ADSCrossRefGoogle Scholar
  100. 100.
    X.-L. Qi, T.L. Hughes, S.-C. Zhang, Phys. Rev. B 82, 184516 (2010).  https://doi.org/10.1103/PhysRevB.82.184516ADSCrossRefGoogle Scholar
  101. 101.
    J. Wang, Q. Zhou, B. Lian, S.-C. Zhang, Phys. Rev. B 92, 064520 (2015).  https://doi.org/10.1103/PhysRevB.92.064520ADSCrossRefGoogle Scholar
  102. 102.
    M. Stone, R. Roy, Phys. Rev. B 69, 184511 (2004).  https://doi.org/10.1103/PhysRevB.69.184511ADSCrossRefGoogle Scholar
  103. 103.
    C. Caroli, P.G. de Gennes, J. Matricon, Phys. Lett. 9, 307 (1964).  https://doi.org/10.1016/0031-9163(64)90375-0ADSCrossRefGoogle Scholar
  104. 104.
    J.J. Sakurai, Modern Quantum Mechanics, 2nd edn. (Addison Wesley, 1994), http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0201539292
  105. 105.
    M.E. Cage, K. Klitzing, A. Chang, F. Duncan, M. Haldane, R. Laughlin, A. Pruisken, D. Thouless, R.E. Prange, S.M. Girvin, The Quantum Hall Effect (Springer Science & Business Media, Berlin, 2012)Google Scholar
  106. 106.
    E. Fradkin, Field Theories of Condensed Matter Physics, 2nd edn. (Cambridge University Press, 2013)Google Scholar
  107. 107.
    N. Read, G. Moore, Nucl. Phys. B 360, 362 (1991).  https://doi.org/10.1016/0550-3213(91)90407-OADSCrossRefGoogle Scholar
  108. 108.
    P.M. Chaikin, T.C. Lubensky, Principles of Condensed Matter Physics (Cambridge University Press, 2000)Google Scholar
  109. 109.
    D.R. Nelson, Defects and Geometry in Condensed Matter Physics (Cambridge University Press, 2002)Google Scholar
  110. 110.
    D. Asahi, N. Nagaosa, Phys. Rev. B 86, 100504(R) (2012).  https://doi.org/10.1103/PhysRevB.86.100504ADSCrossRefGoogle Scholar
  111. 111.
    T.L. Hughes, H. Yao, X.-L. Qi, Phys. Rev. B 90, 235123 (2014).  https://doi.org/10.1103/PhysRevB.90.235123ADSCrossRefGoogle Scholar
  112. 112.
    Ran, Y. (2010), arXiv:1006.5454 [cond-mat.str-el]
  113. 113.
    J.C. Teo, T.L. Hughes, Annu. Rev. Condens. Matter Phys. 8(1), 211 (2017).  https://doi.org/10.1146/annurev-conmatphys-031016-025154ADSCrossRefGoogle Scholar
  114. 114.
    W.A. Benalcazar, J.C.Y. Teo, T.L. Hughes, Phys. Rev. B 89, 224503 (2014).  https://doi.org/10.1103/PhysRevB.89.224503ADSCrossRefGoogle Scholar
  115. 115.
    J.C.Y. Teo, T.L. Hughes, Phys. Rev. Lett. 111, 047006 (2013).  https://doi.org/10.1103/PhysRevLett.111.047006ADSCrossRefGoogle Scholar
  116. 116.
    S.B. Chung, H. Bluhm, E.-A. Kim, Phys. Rev. Lett. 99, 197002 (2007).  https://doi.org/10.1103/PhysRevLett.99.197002ADSCrossRefGoogle Scholar
  117. 117.
    S. Das Sarma, C. Nayak, S. Tewari, Phys. Rev. B 73, 220502 (2006).  https://doi.org/10.1103/PhysRevB.73.220502CrossRefGoogle Scholar
  118. 118.
    J. Jang, D.G. Ferguson, V. Vakaryuk, R. Budakian, S.B. Chung, P.M. Goldbart, Y. Maeno, Science 331(6014), 186 (2011).  https://doi.org/10.1126/science.1193839ADSCrossRefGoogle Scholar
  119. 119.
    M.M. Salomaa, G.E. Volovik, Phys. Rev. Lett. 55, 1184 (1985).  https://doi.org/10.1103/PhysRevLett.55.1184ADSCrossRefGoogle Scholar
  120. 120.
    S.B. Chung, S.-C. Zhang, Phys. Rev. Lett. 103, 235301 (2009).  https://doi.org/10.1103/PhysRevLett.103.235301ADSCrossRefGoogle Scholar
  121. 121.
    S. Murakawa, Y. Tamura, Y. Wada, M. Wasai, M. Saitoh, Y. Aoki, R. Nomura, Y. Okuda, Y. Nagato, M. Yamamoto, S. Higashitani, K. Nagai, Phys. Rev. Lett. 103, 155301 (2009).  https://doi.org/10.1103/PhysRevLett.103.155301ADSCrossRefGoogle Scholar
  122. 122.
    S. Murakawa, Y. Wada, Y. Tamura, M. Wasai, M. Saitoh, Y. Aoki, R. Nomura, Y. Okuda, Y. Nagato, M. Yamamoto, S. Higashitani, K. Nagai, J. Phys. Soc. Jpn. 80, 013602 (2011).  https://doi.org/10.1143/JPSJ.80.013602ADSCrossRefGoogle Scholar
  123. 123.
    S. Ryu, A.P. Schnyder, A. Furusaki, A.W.W. Ludwig, New J. Phys 12, 065010 (2010), http://stacks.iop.org/1367-2630/12/i=6/a=065010
  124. 124.
    Y. Wada, S. Murakawa, Y. Tamura, M. Saitoh, Y. Aoki, R. Nomura, Y. Okuda, Phys. Rev. B 78, 214516 (2008).  https://doi.org/10.1103/PhysRevB.78.214516ADSCrossRefGoogle Scholar
  125. 125.
    L. Fu, E. Berg, Phys. Rev. Lett. 105, 097001 (2010).  https://doi.org/10.1103/PhysRevLett.105.097001ADSCrossRefGoogle Scholar
  126. 126.
    Y.S. Hor, A.J. Williams, J.G. Checkelsky, P. Roushan, J. Seo, Q. Xu, H.W. Zandbergen, A. Yazdani, N.P. Ong, R.J. Cava, Phys. Rev. Lett. 104, 057001 (2010).  https://doi.org/10.1103/PhysRevLett.104.057001ADSCrossRefGoogle Scholar
  127. 127.
    S. Sasaki, M. Kriener, K. Segawa, K. Yada, Y. Tanaka, M. Sato, Y. Ando, Phys. Rev. Lett. 107, 217001 (2011).  https://doi.org/10.1103/PhysRevLett.107.217001ADSCrossRefGoogle Scholar
  128. 128.
    L.A. Wray, S.-Y. Xu, Y. Xia, Y.S. Hor, D. Qian, A.V. Fedorov, H. Lin, A. Bansil, R.J. Cava, M.Z. Hasan, Nat. Phys. 6, 855 (2010).  https://doi.org/10.1038/nphys1762CrossRefGoogle Scholar
  129. 129.
    B.A. Bernevig, T.L. Hughes, S.-C. Zhang, Science 314(5806), 1757 (2006).  https://doi.org/10.1126/science.1133734ADSCrossRefGoogle Scholar
  130. 130.
    J. Nilsson, A.R. Akhmerov, C.W.J. Beenakker, Phys. Rev. Lett. 101, 120403 (2008).  https://doi.org/10.1103/PhysRevLett.101.120403ADSCrossRefGoogle Scholar
  131. 131.
    R. Jackiw, C. Rebbi, Phys. Rev. D 13, 3398 (1976).  https://doi.org/10.1103/PhysRevD.13.3398ADSMathSciNetCrossRefGoogle Scholar
  132. 132.
    J.C. Gallop, SQUIDs, the Josephson Effects and Superconducting Electronics (CRC Press, 1991)Google Scholar
  133. 133.
  134. 134.
    L. Fu, C.L. Kane, E.J. Mele, Phys. Rev. Lett. 98, 106803 (2007).  https://doi.org/10.1103/PhysRevLett.98.106803ADSCrossRefGoogle Scholar
  135. 135.
    D. Hsieh, D. Qian, L. Wray, Y. Xia, Y.S. Hor, R.J. Cava, M.Z. Hasan, Nature 452, 970 (2008).  https://doi.org/10.1038/nature06843ADSCrossRefGoogle Scholar
  136. 136.
    J.E. Moore, L. Balents, Phys. Rev. B 75, 121306(R) (2007).  https://doi.org/10.1103/PhysRevB.75.121306ADSCrossRefGoogle Scholar
  137. 137.
    X.-L. Qi, T.L. Hughes, S.-C. Zhang, Phys. Rev. B 78, 195424 (2008).  https://doi.org/10.1103/PhysRevB.78.195424ADSCrossRefGoogle Scholar
  138. 138.
  139. 139.
    Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y.S. Hor, R.J. Cava, M.Z. Hasan, Nat. Phys. 5, 398 (2009).  https://doi.org/10.1038/nphys1274CrossRefGoogle Scholar
  140. 140.
    R. Jackiw, P. Rossi, Nucl. Phys. B 190, 681 (1981).  https://doi.org/10.1016/0550-3213(81)90044-4ADSCrossRefGoogle Scholar
  141. 141.
    M.F. Atiyah, G.B. Segal, Ann. Math. 87, 531 (1968).  https://doi.org/10.2307/1970716MathSciNetCrossRefGoogle Scholar
  142. 142.
    M.F. Atiyah, I.M. Singer, Bull. Am. Math. Soc. 69, 422 (1963), http://www.ams.org/journals/bull/1963-69-03/S0002-9904-1963-10957-X/home.html
  143. 143.
    M.F. Atiyah, I.M. Singer, Ann. Math. 87, 484 (1968a).  https://doi.org/10.2307/1970715MathSciNetCrossRefGoogle Scholar
  144. 144.
    M.F. Atiyah, I.M. Singer, Ann. Math. 87, 546 (1968b).  https://doi.org/10.2307/1970717MathSciNetCrossRefGoogle Scholar
  145. 145.
    M.F. Atiyah, I.M. Singer, Ann. Math. 93, 119 (1971a).  https://doi.org/10.2307/1970756MathSciNetCrossRefGoogle Scholar
  146. 146.
    M.F. Atiyah, I.M. Singer, Ann. Math. 93, 139 (1971b).  https://doi.org/10.2307/1970757MathSciNetCrossRefGoogle Scholar
  147. 147.
    M. Freedman, M.B. Hastings, C. Nayak, X.-L. Qi, K. Walker, Z. Wang, Phys. Rev. B 83, 115132 (2011).  https://doi.org/10.1103/PhysRevB.83.115132ADSCrossRefGoogle Scholar
  148. 148.
    X.-L. Qi, E. Witten, S.-C. Zhang, Phys. Rev. B 87, 134519 (2013).  https://doi.org/10.1103/PhysRevB.87.134519ADSCrossRefGoogle Scholar
  149. 149.
    A.A. Burkov, L. Balents, Phys. Rev. Lett. 107, 127205 (2011).  https://doi.org/10.1103/PhysRevLett.107.127205ADSCrossRefGoogle Scholar
  150. 150.
    A.A. Burkov, M.D. Hook, L. Balents, Phys. Rev. B 84, 235126 (2011).  https://doi.org/10.1103/PhysRevB.84.235126ADSCrossRefGoogle Scholar
  151. 151.
  152. 152.
    O. Vafek, A. Vishwanath, Annu. Rev. Condens. Matter Phys. 5(1), 83 (2014).  https://doi.org/10.1146/annurev-conmatphys-031113-133841ADSCrossRefGoogle Scholar
  153. 153.
    X. Wan, A.M. Turner, A. Vishwanath, S.Y. Savrasov, Phys. Rev. B 83, 205101 (2011).  https://doi.org/10.1103/PhysRevB.83.205101ADSCrossRefGoogle Scholar
  154. 154.
    Z. Wang, Y. Sun, X.-Q. Chen, C. Franchini, G. Xu, H. Weng, X. Dai, Z. Fang, Phys. Rev. B 85, 195320 (2012).  https://doi.org/10.1103/PhysRevB.85.195320ADSCrossRefGoogle Scholar
  155. 155.
    S.M. Young, S. Zaheer, J.C.Y. Teo, C.L. Kane, E.J. Mele, A.M. Rappe, Phys. Rev. Lett. 108, 140405 (2012).  https://doi.org/10.1103/PhysRevLett.108.140405ADSCrossRefGoogle Scholar
  156. 156.
    S. Borisenko, Q. Gibson, D. Evtushinsky, V. Zabolotnyy, B. Büchner, R.J. Cava, Phys. Rev. Lett. 113, 027603 (2014).  https://doi.org/10.1103/PhysRevLett.113.027603ADSCrossRefGoogle Scholar
  157. 157.
    X. Huang, L. Zhao, Y. Long, P. Wang, D. Chen, Z. Yang, H. Liang, M. Xue, H. Weng, Z. Fang, X. Dai, G. Chen, Phys. Rev. X 5, 031023 (2015).  https://doi.org/10.1103/PhysRevX.5.031023CrossRefGoogle Scholar
  158. 158.
    S. Jeon, B.B. Zhou, A. Gyenis, B.E. Feldman, I. Kimchi, A.C. Potter, Q.D. Gibson, R.J. Cava, A. Vishwanath, A. Yazdani, Nat. Mater. 13, 851 (2014).  https://doi.org/10.1038/nmat4023ADSCrossRefGoogle Scholar
  159. 159.
    T. Liang, Q. Gibson, M.N. Ali, M. Liu, R.J. Cava, N.P. Ong, Nat. Mater. 14(3), 280 (2015).  https://doi.org/10.1038/nmat4143ADSCrossRefGoogle Scholar
  160. 160.
    Z.K. Liu, J. Jiang, B. Zhou, Z.J. Wang, Y. Zhang, H.M. Weng, D. Prabhakaran, S.-K. Mo, H. Peng, P. Dudin, T. Kim, M. Hoesch, Z. Fang, X. Dai, Z.X. Shen, D.L. Feng, Z. Hussain, Y.L. Chen, Nat. Mater. 13, 677 (2014a).  https://doi.org/10.1038/nmat3990ADSCrossRefGoogle Scholar
  161. 161.
    Z.K. Liu, B. Zhou, Y. Zhang, Z.J. Wang, H.M. Weng, D. Prabhakaran, S.-K. Mo, Z.X. Shen, Z. Fang, X. Dai, Z. Hussain, Y.L. Chen, Science 343(6173), 864 (2014b).  https://doi.org/10.1126/science.1245085ADSCrossRefGoogle Scholar
  162. 162.
    B.Q. Lv, H.M. Weng, B.B. Fu, X.P. Wang, H. Miao, J. Ma, P. Richard, X.C. Huang, L.X. Zhao, G.F. Chen, Z. Fang, X. Dai, T. Qian, H. Ding, Phys. Rev. X 5, 031013 (2015a).  https://doi.org/10.1103/PhysRevX.5.031013CrossRefGoogle Scholar
  163. 163.
    B.Q. Lv, N. Xu, H.M. Weng, J.Z. Ma, P. Richard, X.C. Huang, L.X. Zhao, G.F. Chen, C.E. Matt, F. Bisti, V.N. Strocov, J. Mesot, Z. Fang, X. Dai, T. Qian, M. Shi, H. Ding, Nat. Phys. 11(9), 724 (2015b).  https://doi.org/10.1038/nphys3426CrossRefGoogle Scholar
  164. 164.
    M. Neupane, S.-Y. Xu, R. Sankar, N. Alidoust, G. Bian, C. Liu, I. Belopolski, T.-R. Chang, H.-T. Jeng, H. Lin, A. Bansil, F. Chou, M.Z. Hasan, Nat. Commun. 5, 3786 (2014).  https://doi.org/10.1038/ncomms4786CrossRefGoogle Scholar
  165. 165.
    Z. Wang, H. Weng, Q. Wu, X. Dai, Z. Fang, Phys. Rev. B 88, 125427 (2013).  https://doi.org/10.1103/PhysRevB.88.125427ADSCrossRefGoogle Scholar
  166. 166.
    S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, S.-M. Huang, H. Zheng, J. Ma, D.S. Sanchez, B. Wang, A. Bansil, F. Chou, P.P. Shibayev, H. Lin, S. Jia, M.Z. Hasan, Science 349(6248), 613 (2015a).  https://doi.org/10.1126/science.aaa9297ADSCrossRefGoogle Scholar
  167. 167.
    S.-Y. Xu, C. Liu, S.K. Kushwaha, R. Sankar, J.W. Krizan, I. Belopolski, M. Neupane, G. Bian, N. Alidoust, T.-R. Chang, H.-T. Jeng, C.-Y. Huang, W.-F. Tsai, H. Lin, P.P. Shibayev, F.-C. Chou, R.J. Cava, M.Z. Hasan, Science 347(6219), 294 (2015b).  https://doi.org/10.1126/science.1256742ADSCrossRefGoogle Scholar
  168. 168.
    C. Zhang, S.-Y. Xu, I. Belopolski, Z. Yuan, Z. Lin, B. Tong, N. Alidoust, C.-C. Lee, S.-M. Huang, H. Lin, M. Neupane, D.S. Sanchez, H. Zheng, G. Bian, J. Wang, C. Zhang, T. Neupert, M. Zahid Hasan, S. Jia, Nat. Commun. 7, 10735 (2016).  https://doi.org/10.1038/ncomms10735ADSCrossRefGoogle Scholar
  169. 169.
    L. Aggarwal, A. Gaurav, G.S. Thakur, Z. Haque, A.K. Ganguli, G. Sheet, Nat. Mater. 15, 32 (2016).  https://doi.org/10.1038/nmat4455ADSCrossRefGoogle Scholar
  170. 170.
    S. Kobayashi, M. Sato, Phys. Rev. Lett. 115, 187001 (2015).  https://doi.org/10.1103/PhysRevLett.115.187001ADSCrossRefGoogle Scholar
  171. 171.
    P.L.e.S. Lopes, J.C.Y. Teo, S. Ryu, Phys. Rev. B 95, 235134 (2017).  https://doi.org/10.1103/PhysRevB.95.235134
  172. 172.
    M.-J. Park, J.C.Y. Teo, M.J. Gilbert, To appear soon (2017)Google Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of VirginiaCharlottesvilleUSA

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