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Introduction

  • Hiroki IsobeEmail author
Chapter
  • 668 Downloads
Part of the Springer Theses book series (Springer Theses)

Abstract

To begin with the thesis, which focuses on how electron interactions play in topological matters, we first look at the stage; we review the basic idea of topological electronic states and their material realizations, after giving the scope and the outline of the thesis. Starting from the celebrated quantum Hall effect, we introduce the basic notion of the topology of electron wave functions. This evolves to the important discovery in the recent decade of topological insulators. A characteristic feature of topological insulators is the gapless linear dispersion on the surface. It is described by the Weyl or Dirac equation, originated in the high-energy physics. Not only on a surface, such linear energy dispersion can emerge in a bulk; those materials are dubbed as Weyl or Dirac semimetals, whose band crossings are protected by the topological characters of the electron wave function. The protection of the topological properties is dependent on the symmetry of systems. We give the classification of topological electronic states for noninteracting cases, and also state on the realization of topological states driven by electron correlations.

Keywords

Topological insulators Dirac electrons Weyl/Dirac semimetals Topological crystalline insulators 

References

  1. 1.
    K.v. Klitzing, G. Dorda, M. Pepper, Phys. Rev. Lett. 45, 494 (1980)Google Scholar
  2. 2.
    D.J. Thouless, M. Kohmoto, M.P. Nightingale, M. den Nijs, Phys. Rev. Lett. 49, 405 (1982)ADSCrossRefGoogle Scholar
  3. 3.
    M.Z. Hasan, C.L. Kane, Rev. Mod. Phys. 82, 3045 (2010)ADSCrossRefGoogle Scholar
  4. 4.
    X.-L. Qi, S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011)ADSCrossRefGoogle Scholar
  5. 5.
    M.V. Berry, Proc. R. Soc. A. 392, 45 (1984)ADSCrossRefGoogle Scholar
  6. 6.
    M. Nakahara, Geometry, Topology and Physics, 2nd edn. (Taylor & Francis Group, Boca Raton, 2003)zbMATHGoogle Scholar
  7. 7.
    R. Jackiw, C. Rebbi, Phys. Rev. D 13, 3398 (1976)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    C.L. Kane, E.J. Mele, Phys. Rev. Lett. 95, 226801 (2005)ADSCrossRefGoogle Scholar
  9. 9.
    C.L. Kane, E.J. Mele, Phys. Rev. Lett. 95, 146802 (2005)ADSCrossRefGoogle Scholar
  10. 10.
    B.A. Bernevig, S.-C. Zhang, Phys. Rev. Lett. 96, 106802 (2006)ADSCrossRefGoogle Scholar
  11. 11.
    L. Fu, C.L. Kane, E.J. Mele, Phys. Rev. Lett. 98, 106803 (2007)ADSCrossRefGoogle Scholar
  12. 12.
    J.E. Moore, L. Balents, Phys. Rev. B 75, 121306 (2007)ADSCrossRefGoogle Scholar
  13. 13.
    R. Roy, Phys. Rev. B 79, 195322 (2009)ADSCrossRefGoogle Scholar
  14. 14.
    S. Murakami, N. Nagaosa, S.-C. Zhang, Science 301, 1348 (2003)ADSCrossRefGoogle Scholar
  15. 15.
    S. Murakami, N. Nagaosa, in Comprehensive Semiconductor Science and Technology, ed. by P. Bhattacharya, R. Fornari, H. Kamimura (Elsevier, Amsterdam, 2011), p. 222CrossRefGoogle Scholar
  16. 16.
    F.D.M. Haldane, Phys. Rev. Lett. 61, 2015 (1988)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    X.-L. Qi, T.L. Hughes, S.-C. Zhang, Phys. Rev. B 78, 195424 (2008)ADSCrossRefGoogle Scholar
  18. 18.
    L. Fu, C.L. Kane, Phys. Rev. B 74, 195312 (2006)ADSCrossRefGoogle Scholar
  19. 19.
    L. Fu, C.L. Kane, Phys. Rev. B 76, 045302 (2007)ADSCrossRefGoogle Scholar
  20. 20.
    H. Nielsen, M. Ninomiya, Phys. Lett. B 105, 219 (1981)ADSCrossRefGoogle Scholar
  21. 21.
    H. Nielsen, M. Ninomiya, Phys. Lett. B 130, 389 (1983)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    H. Min, J.E. Hill, N.A. Sinitsyn, B.R. Sahu, L. Kleinman, A.H. MacDonald, Phys. Rev. B 74, 165310 (2006)ADSCrossRefGoogle Scholar
  23. 23.
    Y. Yao, F. Ye, X.-L. Qi, S.-C. Zhang, Z. Fang, Phys. Rev. B 75, 041401 (2007)ADSCrossRefGoogle Scholar
  24. 24.
    B.A. Bernevig, T.L. Hughes, S.-C. Zhang, Science 314, 1757 (2006)ADSCrossRefGoogle Scholar
  25. 25.
    M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L.W. Molenkamp, X.-L. Qi, S.-C. Zhang, Science 318, 766 (2007)ADSCrossRefGoogle Scholar
  26. 26.
    D. Hsieh, D. Qian, L. Wray, Y. Xia, Y.S. Hor, R.J. Cava, M.Z. Hasan, Nature 452, 970 (2008)ADSCrossRefGoogle Scholar
  27. 27.
    H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, S.-C. Zhang, Nat. Phys. 5, 438 (2009)CrossRefGoogle Scholar
  28. 28.
    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)CrossRefGoogle Scholar
  29. 29.
    Z. Ren, A.A. Taskin, S. Sasaki, K. Segawa, Y. Ando, Phys. Rev. B 82, 241306 (2010)ADSCrossRefGoogle Scholar
  30. 30.
    J. Xiong, A. Petersen, D. Qu, Y. Hor, R. Cava, N. Ong, Phys. E 44, 917 (2012)CrossRefGoogle Scholar
  31. 31.
    Z. Ren, A.A. Taskin, S. Sasaki, K. Segawa, Y. Ando, Phys. Rev. B 84, 165311 (2011)ADSCrossRefGoogle Scholar
  32. 32.
    Y. Ando, J. Phys. Soc. Jpn. 82, 102001 (2013)ADSCrossRefGoogle Scholar
  33. 33.
    S.-Y. Xu, Y. Xia, L.A. Wray, S. Jia, F. Meier, J.H. Dil, J. Osterwalder, B. Slomski, A. Bansil, H. Lin, R.J. Cava, M.Z. Hasan, Science 332, 560 (2011)ADSCrossRefGoogle Scholar
  34. 34.
    T. Sato, K. Segawa, K. Kosaka, S. Souma, K. Nakayama, K. Eto, T. Minami, Y. Ando, T. Takahashi, Nat. Phys. 7, 840 (2011)CrossRefGoogle Scholar
  35. 35.
    J. Von Neumann, E. Wigner, Phys. Z. 30, 467 (1929)Google Scholar
  36. 36.
    C. Herring, Phys. Rev. 52, 365 (1937)ADSCrossRefGoogle Scholar
  37. 37.
    O. Vafek, A. Vishwanath, Annu. Rev. Condens. Matter Phys. 5, 83 (2014)CrossRefGoogle Scholar
  38. 38.
    X. Wan, A.M. Turner, A. Vishwanath, S.Y. Savrasov, Phys. Rev. B 83, 205101 (2011)ADSCrossRefGoogle Scholar
  39. 39.
    W. Witczak-Krempa, Y.B. Kim, Phys. Rev. B 85, 045124 (2012)ADSCrossRefGoogle Scholar
  40. 40.
    X. Wan, A. Vishwanath, S.Y. Savrasov, Phys. Rev. Lett. 108, 146601 (2012)ADSCrossRefGoogle Scholar
  41. 41.
    G. Xu, H. Weng, Z. Wang, X. Dai, Z. Fang, Phys. Rev. Lett. 107, 186806 (2011)ADSCrossRefGoogle Scholar
  42. 42.
    S.-M. Huang, S.-Y. Xu, I. Belopolski, C.-C. Lee, G. Chang, B. Wang, N. Alidoust, G. Bian, M. Neupane, C. Zhang, S. Jia, A. Bansil, H. Lin, M.Z. Hasan, Nat. Commun. 6, 7373 (2015)ADSCrossRefGoogle Scholar
  43. 43.
    A.A. Burkov, L. Balents, Phys. Rev. Lett. 107, 127205 (2011)ADSCrossRefGoogle Scholar
  44. 44.
    G.B. Halász, L. Balents, Phys. Rev. B 85, 035103 (2012)ADSCrossRefGoogle Scholar
  45. 45.
    W. Witczak-Krempa, G. Chen, Y.B. Kim, L. Balents, Annu. Rev. Condens. Matter Phys. 5, 57 (2014)CrossRefGoogle Scholar
  46. 46.
    A.M. Turner, Y. Zhang, R.S.K. Mong, A. Vishwanath, Phys. Rev. B 85, 165120 (2012)ADSCrossRefGoogle Scholar
  47. 47.
    D. Pesin, L. Balents, Nat. Phys. 6, 376 (2010)CrossRefGoogle Scholar
  48. 48.
    B.-J. Yang, Y.B. Kim, Phys. Rev. B 82, 085111 (2010)ADSCrossRefGoogle Scholar
  49. 49.
    M. Kargarian, J. Wen, G.A. Fiete, Phys. Rev. B 83, 165112 (2011)ADSCrossRefGoogle Scholar
  50. 50.
    K.-Y. Yang, Y.-M. Lu, Y. Ran, Phys. Rev. B 84, 075129 (2011)ADSCrossRefGoogle Scholar
  51. 51.
    D. Yanagishima, Y. Maeno, J. Phys. Soc. Jpn. 70, 2880 (2001)ADSCrossRefGoogle Scholar
  52. 52.
    N. Taira, M. Wakeshima, Y. Hinatsu, J. Phys. Condens. Matter 13, 5527 (2001)ADSCrossRefGoogle Scholar
  53. 53.
    K. Matsuhira, M. Wakeshima, R. Nakanishi, T. Yamada, A. Nakamura, W. Kawano, S. Takagi, Y. Hinatsu, J. Phys. Soc. Jpn. 76, 043706 (2007)ADSCrossRefGoogle Scholar
  54. 54.
    S. Zhao, J.M. Mackie, D.E. MacLaughlin, O.O. Bernal, J.J. Ishikawa, Y. Ohta, S. Nakatsuji, Phys. Rev. B 83, 180402 (2011)ADSCrossRefGoogle Scholar
  55. 55.
    K. Ueda, J. Fujioka, Y. Takahashi, T. Suzuki, S. Ishiwata, Y. Taguchi, Y. Tokura, Phys. Rev. Lett. 109, 136402 (2012)ADSCrossRefGoogle Scholar
  56. 56.
    C. Zhang, Z. Yuan, S. Xu, Z. Lin, B. Tong, M. Zahid Hasan, J. Wang, C. Zhang, S. Jia, arXiv:1502.00251
  57. 57.
    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, 613 (2015)ADSCrossRefGoogle Scholar
  58. 58.
    S.-Y. Xu, N. Alidoust, I. Belopolski, Z. Yuan, G. Bian, T.-R. Chang, H. Zheng, V.N. Strocov, D.S. Sanchez, G. Chang, C. Zhang, D. Mou, Y. Wu, L. Huang, C.-C. Lee, S.-M. Huang, B. Wang, A. Bansil, H.-T. Jeng, T. Neupert, A. Kaminski, H. Lin, S. Jia, M. Zahid Hasan, Nat. Phys. 11, 748 (2015)CrossRefGoogle Scholar
  59. 59.
    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, 864 (2014)Google Scholar
  60. 60.
    S.M. Young, S. Zaheer, J.C.Y. Teo, C.L. Kane, E.J. Mele, A.M. Rappe, Phys. Rev. Lett. 108, 140405 (2012)ADSCrossRefGoogle Scholar
  61. 61.
    Z. Wang, Y. Sun, X.-Q. Chen, C. Franchini, G. Xu, H. Weng, X. Dai, Z. Fang, Phys. Rev. B 85, 195320 (2012)ADSCrossRefGoogle Scholar
  62. 62.
    Z. Wang, H. Weng, Q. Wu, X. Dai, Z. Fang, Phys. Rev. B 88, 125427 (2013)ADSCrossRefGoogle Scholar
  63. 63.
    S. Murakami, New J. Phys. 9, 356 (2007)ADSCrossRefGoogle Scholar
  64. 64.
    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)ADSGoogle Scholar
  65. 65.
    S. Borisenko, Q. Gibson, D. Evtushinsky, V. Zabolotnyy, B. Büchner, R.J. Cava, Phys. Rev. Lett. 113, 027603 (2014)ADSCrossRefGoogle Scholar
  66. 66.
    B.-J. Yang, N. Nagaosa, Nat. Commun. 5, 4898 (2014)ADSCrossRefGoogle Scholar
  67. 67.
    H. Seo, C. Hotta, H. Fukuyama, Chem. Rev. 104, 5005 (2004)CrossRefGoogle Scholar
  68. 68.
    K. Bender, I. Hennig, D. Schweitzer, K. Dietz, H. Endres, H.J. Keller, Mol. Cryst. Liq. Cryst. 108, 359 (1984)CrossRefGoogle Scholar
  69. 69.
    N. Tajima, K. Kajita, Sci. Tech. Adv. Mater. 10, 024308 (2009)CrossRefGoogle Scholar
  70. 70.
    A. Kobayashi, S. Katayama, Y. Suzumura, Sci. Tech. Adv. Mater. 10, 024309 (2009)CrossRefGoogle Scholar
  71. 71.
    N. Tajima, S. Sugawara, M. Tamura, Y. Nishio, K. Kajita, J. Phys. Soc. Jpn. 75, 051010 (2006)ADSCrossRefGoogle Scholar
  72. 72.
    H. Kino, H. Fukuyama, J. Phys. Soc. Jpn. 64, 1877 (1995)ADSCrossRefGoogle Scholar
  73. 73.
    B. Rothaemel, L. Forró, J.R. Cooper, J.S. Schilling, M. Weger, P. Bele, H. Brunner, D. Schweitzer, H.J. Keller, Phys. Rev. B 34, 704 (1986)ADSCrossRefGoogle Scholar
  74. 74.
    S. Katayama, A. Kobayashi, Y. Suzumura, J. Phys. Soc. Jpn. 75, 054705 (2006)ADSCrossRefGoogle Scholar
  75. 75.
    R. Kondo, S. Kagoshima, J. Harada, Rev. Sci. Instrum. 76, 093902 (2005)ADSCrossRefGoogle Scholar
  76. 76.
    H. Kino, T. Miyazaki, J. Phys. Soc. Jpn. 75, 034704 (2006)ADSCrossRefGoogle Scholar
  77. 77.
    A. Kobayashi, S. Katayama, Y. Suzumura, H. Fukuyama, J. Phys. Soc. Jpn. 76, 034711 (2007)ADSCrossRefGoogle Scholar
  78. 78.
    A. Kobayashi, Y. Suzumura, H. Fukuyama, J. Phys. Soc. Jpn. 77, 064718 (2008)ADSCrossRefGoogle Scholar
  79. 79.
    S. Katayama, A. Kobayashi, Y. Suzumura, Eur. Phys. J. B 67, 139 (2009)ADSCrossRefGoogle Scholar
  80. 80.
    M.O. Goerbig, J.-N. Fuchs, G. Montambaux, F. Piéchon, Phys. Rev. B 78, 045415 (2008)ADSCrossRefGoogle Scholar
  81. 81.
    R. Kondo, S. Kagoshima, N. Tajima, R. Kato, J. Phys. Soc. Jpn. 78, 114714 (2009)ADSCrossRefGoogle Scholar
  82. 82.
    Y. Suzumura, A. Kobayashi, J. Phys. Soc. Jpn. 82, 044709 (2013)ADSCrossRefGoogle Scholar
  83. 83.
    K. Kajita, T. Ojiro, H. Fujii, Y. Nishio, H. Kobayashi, A. Kobayashi, R. Kato, J. Phys. Soc. Jpn. 61, 23 (1992)ADSCrossRefGoogle Scholar
  84. 84.
    K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos, A.A. Firsov, Nature 438, 197 (2005)ADSCrossRefGoogle Scholar
  85. 85.
    Y. Zhang, Y.-W. Tan, H.L. Stormer, P. Kim, Nature 438, 201 (2005)ADSCrossRefGoogle Scholar
  86. 86.
    M. Hirata, K. Ishikawa, K. Miyagawa, M. Tamura, C. Berthier, D. Basko, A. Kobayashi, G. Matsuno, K. Kanoda, Nat. Commun. 7, 12666 (2016)ADSCrossRefGoogle Scholar
  87. 87.
    S. Raghu, X.-L. Qi, C. Honerkamp, S.-C. Zhang, Phys. Rev. Lett. 100, 156401 (2008)ADSCrossRefGoogle Scholar
  88. 88.
    Y. Zhang, Y. Ran, A. Vishwanath, Phys. Rev. B 79, 245331 (2009)ADSCrossRefGoogle Scholar
  89. 89.
    J. Wen, A. Rüegg, C.-C.J. Wang, G.A. Fiete, Phys. Rev. B 82, 075125 (2010)ADSCrossRefGoogle Scholar
  90. 90.
    M. Kurita, Y. Yamaji, M. Imada, J. Phys. Soc. Jpn. 80, 044708 (2011)ADSCrossRefGoogle Scholar
  91. 91.
    K. Sun, H. Yao, E. Fradkin, S.A. Kivelson, Phys. Rev. Lett. 103, 046811 (2009)ADSCrossRefGoogle Scholar
  92. 92.
    L. Fu, Phys. Rev. Lett. 106, 106802 (2011)ADSCrossRefGoogle Scholar
  93. 93.
    S.A. Parameswaran, A.M. Turner, D.P. Arovas, A. Vishwanath, Nat. Phys. 9, 299 (2013)CrossRefGoogle Scholar
  94. 94.
    C.-X. Liu, R.-X. Zhang, B.K. VanLeeuwen, Phys. Rev. B 90, 085304 (2014)ADSCrossRefGoogle Scholar
  95. 95.
    K. Shiozaki, M. Sato, K. Gomi, Phys. Rev. B 91, 155120 (2015)ADSCrossRefGoogle Scholar
  96. 96.
    C. Fang, L. Fu, Phys. Rev. B 91, 161105 (2015)ADSCrossRefGoogle Scholar
  97. 97.
    T.H. Hsieh, H. Lin, J. Liu, W. Duan, A. Bansil, L. Fu, Nat. Commun. 3, 982 (2012)ADSCrossRefGoogle Scholar
  98. 98.
    Y. Tanaka, Z. Ren, T. Sato, K. Nakayama, S. Souma, T. Takahashi, K. Segawa, Y. Ando, Nat. Phys. 8, 800 (2012)CrossRefGoogle Scholar
  99. 99.
    P. Dziawa, B.J. Kowalski, K. Dybko, R. Buczko, A. Szczerbakow, M. Szot, E. Lusakowska, T. Balasubramanian, B.M. Wojek, M.H. Berntsen, O. Tjernberg, T. Story, Nat. Mater. 11, 1023 (2012)ADSGoogle Scholar
  100. 100.
    S.-Y. Xu, C. Liu, N. Alidoust, M. Neupane, D. Qian, I. Belopolski, J. Denlinger, Y. Wang, H. Lin, L. Wray, G. Landolt, B. Slomski, J. Dil, A. Marcinkova, E. Morosan, Q. Gibson, R. Sankar, F. Chou, R. Cava, A. Bansil, M. Hasan, Nat. Commun. 3, 1192 (2012)ADSCrossRefGoogle Scholar
  101. 101.
    Y. Ando, L. Fu, Annu. Rev. Condens. Matter Phys. 6, 361 (2015)ADSCrossRefGoogle Scholar
  102. 102.
    J.C.Y. Teo, L. Fu, C.L. Kane, Phys. Rev. B 78, 045426 (2008)ADSCrossRefGoogle Scholar
  103. 103.
    J.O. Dimmock, I. Melngailis, A.J. Strauss, Phys. Rev. Lett. 16, 1193 (1966)ADSCrossRefGoogle Scholar
  104. 104.
    J. Liu, W. Duan, L. Fu, Phys. Rev. B 88, 241303 (2013)ADSCrossRefGoogle Scholar
  105. 105.
    Y. Tanaka, T. Shoman, K. Nakayama, S. Souma, T. Sato, T. Takahashi, M. Novak, K. Segawa, Y. Ando, Phys. Rev. B 88, 235126 (2013)ADSCrossRefGoogle Scholar
  106. 106.
    A.A. Taskin, F. Yang, S. Sasaki, K. Segawa, Y. Ando, Phys. Rev. B 89, 121302 (2014)ADSCrossRefGoogle Scholar
  107. 107.
    J. Liu, T.H. Hsieh, P. Wei, W. Duan, J. Moodera, L. Fu, Nat. Mater. 13, 178 (2014)ADSCrossRefGoogle Scholar
  108. 108.
    E.O. Wrasse, T.M. Schmidt, Nano Lett. 14, 5717 (2014)ADSCrossRefGoogle Scholar
  109. 109.
    J. Liu, X. Qian, L. Fu, Nano Lett. 15, 2657 (2015)ADSCrossRefGoogle Scholar
  110. 110.
    C. Niu, P.M. Buhl, G. Bihlmayer, D. Wortmann, S. Blügel, Y. Mokrousov, Phys. Rev. B 91, 201401 (2015)ADSCrossRefGoogle Scholar
  111. 111.
    Y. Okada, M. Serbyn, H. Lin, D. Walkup, W. Zhou, C. Dhital, M. Neupane, S. Xu, Y.J. Wang, R. Sankar, F. Chou, A. Bansil, M.Z. Hasan, S.D. Wilson, L. Fu, V. Madhavan, Science 341, 1496 (2013)ADSCrossRefGoogle Scholar
  112. 112.
    I. Zeljkovic, Y. Okada, M. Serbyn, R. Sankar, D. Walkup, W. Zhou, J. Liu, G. Chang, Y.J. Wang, M.Z. Hasan, F. Chou, H. Lin, A. Bansil, L. Fu, V. Madhavan, Nat. Mater. 14, 318 (2015)ADSCrossRefGoogle Scholar
  113. 113.
    B.M. Wojek, M.H. Berntsen, V. Jonsson, A. Szczerbakow, P. Dziawa, B.J. Kowalski, T. Story, O. Tjernberg, Nat. Commun. 6, 8463 (2015)ADSCrossRefGoogle Scholar
  114. 114.
    M. Serbyn, L. Fu, Phys. Rev. B 90, 035402 (2014)ADSCrossRefGoogle Scholar
  115. 115.
    C. Fang, M.J. Gilbert, B.A. Bernevig, Phys. Rev. Lett. 112, 046801 (2014)ADSCrossRefGoogle Scholar
  116. 116.
    F. Zhang, X. Li, J. Feng, C.L. Kane, E.J. Mele, arXiv:1309.7682
  117. 117.
    E. Tang, L. Fu, Nat. Phys. 10, 964 (2014)CrossRefGoogle Scholar
  118. 118.
    J. Shen, Y. Xie, J.J. Cha, Nano Lett. 15, 3827 (2015)ADSCrossRefGoogle Scholar
  119. 119.
    R. Zhong, X. He, J.A. Schneeloch, C. Zhang, T. Liu, I. Pletikosić, T. Yilmaz, B. Sinkovic, Q. Li, W. Ku, T. Valla, J.M. Tranquada, G. Gu, Phys. Rev. B 91, 195321 (2015)ADSCrossRefGoogle Scholar
  120. 120.
    I. Zeljkovic, D.Walkup, B.A. Assaf, K.L. Scipioni, R. Sankar, F. Chou, V. Madhavan, Nat. Nanotech. 10, 849 (2015)Google Scholar
  121. 121.
    M. Kargarian, G.A. Fiete, Phys. Rev. Lett. 110, 156403 (2013)ADSCrossRefGoogle Scholar
  122. 122.
    T.H. Hsieh, J. Liu, L. Fu, Phys. Rev. B 90, 081112 (2014)ADSCrossRefGoogle Scholar
  123. 123.
    H. Weng, J. Zhao, Z. Wang, Z. Fang, X. Dai, Phys. Rev. Lett. 112, 016403 (2014)ADSCrossRefGoogle Scholar
  124. 124.
    M. Ye, J.W. Allen, K. Sun, arXiv:1307.7191
  125. 125.
    A.P. Schnyder, S. Ryu, A. Furusaki, A.W.W. Ludwig, Phys. Rev. B 78, 195125 (2008)ADSCrossRefGoogle Scholar
  126. 126.
    A. Kitaev, AIP Conf. Proc. 1134, 22 (2009)ADSCrossRefGoogle Scholar
  127. 127.
    S. Ryu, A.P. Schnyder, A. Furusaki, A.W. Ludwig, New J. Phys. 12, 065010 (2010)ADSCrossRefGoogle Scholar
  128. 128.
    A. Altland, M.R. Zirnbauer, Phys. Rev. B 55, 1142 (1997)ADSCrossRefGoogle Scholar
  129. 129.
    C.-K. Chiu, J.C.Y. Teo, A.P. Schnyder, S. Ryu, Rev. Mod. Phys. 88, 035005 (2016)ADSCrossRefGoogle Scholar
  130. 130.
    J.C.Y. Teo, C.L. Kane, Phys. Rev. B 82, 115120 (2010)ADSCrossRefGoogle Scholar
  131. 131.
    G.E. Volovik, The Universe in a Helium Droplet (Oxford University Press, Oxford, 2003)zbMATHGoogle Scholar
  132. 132.
    S. Matsuura, P.-Y. Chang, A.P. Schnyder, S. Ryu, New J. Phys. 15, 065001 (2013)ADSMathSciNetCrossRefGoogle Scholar
  133. 133.
    C.-K. Chiu, A.P. Schnyder, Phys. Rev. B 90, 205136 (2014)ADSCrossRefGoogle Scholar
  134. 134.
    K. Shiozaki, M. Sato, Phys. Rev. B 90, 165114 (2014)ADSCrossRefGoogle Scholar
  135. 135.
    F.D.M. Haldane, Phys. Rev. Lett. 93, 206602 (2004)ADSCrossRefGoogle Scholar
  136. 136.
    C.-K. Chiu, H. Yao, S. Ryu, Phys. Rev. B 88, 075142 (2013)ADSCrossRefGoogle Scholar
  137. 137.
    T. Morimoto, A. Furusaki, Phys. Rev. B 88, 125129 (2013)ADSCrossRefGoogle Scholar
  138. 138.
    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 (2014)ADSCrossRefGoogle Scholar
  139. 139.
    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, 294 (2015)ADSCrossRefGoogle Scholar
  140. 140.
    X. Chen, Z.-C. Gu, Z.-X. Liu, X.-G. Wen, Science 338, 1604 (2012)ADSMathSciNetCrossRefGoogle Scholar
  141. 141.
    X.-G. Wen, Phys. Rev. B 89, 035147 (2014)ADSCrossRefGoogle Scholar
  142. 142.
    Y.-Z. You, C. Xu, Phys. Rev. B 90, 245120 (2014)ADSCrossRefGoogle Scholar
  143. 143.
    Z.-C. Gu, X.-G. Wen, Phys. Rev. B 80, 155131 (2009)ADSCrossRefGoogle Scholar
  144. 144.
    T. Senthil, Annu. Rev. Condens. Matter Phys. 6, 299 (2015)ADSCrossRefGoogle Scholar
  145. 145.
    A. Kitaev, Homotopy-theoretic approach to SPT phases in action: Z16 classification of three-dimensional superconductors, in IPAM Workshop “Symmetry and Topology in Quantum Matter” (2015)Google Scholar
  146. 146.
    X. Chen, Z.-C. Gu, X.-G. Wen, Phys. Rev. B 82, 155138 (2010)ADSCrossRefGoogle Scholar
  147. 147.
    X.-G. Wen, Adv. Phys. 44, 405 (1995)ADSCrossRefGoogle Scholar
  148. 148.
    X.-G. Wen, Quantum Field Theory of Many-Body Systems (Oxford University Press, Oxford, 2007)CrossRefGoogle Scholar
  149. 149.
    X. Chen, Z.-C. Gu, Z.-X. Liu, X.-G. Wen, Phys. Rev. B 87, 155114 (2013)ADSCrossRefGoogle Scholar
  150. 150.
    L. Fidkowski, A. Kitaev, Phys. Rev. B 81, 134509 (2010)ADSCrossRefGoogle Scholar
  151. 151.
    L. Fidkowski, A. Kitaev, Phys. Rev. B 83, 075103 (2011)ADSCrossRefGoogle Scholar
  152. 152.
    S. Ryu, S.-C. Zhang, Phys. Rev. B 85, 245132 (2012)ADSCrossRefGoogle Scholar
  153. 153.
    H. Yao, S. Ryu, Phys. Rev. B 88, 064507 (2013)ADSCrossRefGoogle Scholar
  154. 154.
    X.-L. Qi, New J. Phys. 15, 065002 (2013)ADSMathSciNetCrossRefGoogle Scholar
  155. 155.
    L. Fidkowski, X. Chen, A. Vishwanath, Phys. Rev. X 3, 041016 (2013)Google Scholar
  156. 156.
    C. Wang, T. Senthil, Phys. Rev. B 89, 195124 (2014)ADSCrossRefGoogle Scholar
  157. 157.
    Z.-C. Gu, M. Levin, Phys. Rev. B 89, 201113 (2014)ADSCrossRefGoogle Scholar
  158. 158.
    T. Neupert, C. Chamon, C. Mudry, R. Thomale, Phys. Rev. B 90, 205101 (2014)ADSCrossRefGoogle Scholar
  159. 159.
    T. Morimoto, A. Furusaki, C. Mudry, Phys. Rev. B 92, 125104 (2015)ADSCrossRefGoogle Scholar
  160. 160.
    M. Cheng, Z.-C. Gu, Phys. Rev. Lett. 112, 141602 (2014)ADSCrossRefGoogle Scholar
  161. 161.
    M. Levin, Z.-C. Gu, Phys. Rev. B 86, 115109 (2012)ADSCrossRefGoogle Scholar
  162. 162.
    M. Barkeshli, P. Bonderson, M. Cheng, Z. Wang, arXiv:1410.4540
  163. 163.
    A. Kapustin, R. Thorngren, Phys. Rev. Lett. 112, 231602 (2014)ADSCrossRefGoogle Scholar
  164. 164.
    A. Kapustin, arXiv:1403.1467

Copyright information

© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Massachusetts Institute of TechnologyCambridgeUSA

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