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Majorana States

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Topology in Magnetism

Part of the book series: Springer Series in Solid-State Sciences ((SSSOL,volume 192))

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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.

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Notes

  1. 1.

    The quantum phase difference \(e^{i\phi }\) between the two can be absorbed by a gauge redefinition \(\psi _0\rightarrow e^{i\phi /2}\psi _0\).

  2. 2.

    If there are multiple zero modes, there exists an orthogonal transformation that rotates to a new basis where all zero modes are self-conjugate (12.9). For instance, given a zero mode \(\psi _0\) which is not self-conjugate, one can define the self-conjugate zero modes \(\psi _0+\Xi \psi _0^\dagger \), \(i\psi _0-i\Xi \psi _0^\dagger \). This process can be repeated until all zero modes are self-conjugate.

  3. 3.

    Even if there are accidental mid-gap BdG-states, they are not particle-hole symmetric and their wavefunctions carry unequal weights between electron and hole. Consequently, the electric conductance \(\sigma =(2e^2/h)|S_{eh}|^2\) will be smaller than the quantized value \(2e^2/h\).

  4. 4.

    The orthornormal frame corresponds to a N-dimensional subspace \(\mathcal {V}_\mathbf{k}=\mathrm {span}\{\mathbf{u}_m(\mathbf{k})\}_{m=1,\ldots ,N}\) for each \(\mathbf{k}\). The collection of these subspaces is known as a vector bundle \(\mathcal {V}=\coprod _{\mathbf{k}\in BZ}\mathcal {V}_\mathbf{k}\). The vector bundle is trivial if it decomposes into a cartesian product \(\mathcal {V}\cong BZ\times \mathbb {C}^N\), which is the case if and only if there is a global continuous eigen-frame \(\mathbf{u}_m(\mathbf{k})\). Otherwise, the vector bundle is topological. For the mathematics of vector bundles and their classification, we refer the readers to [70,71,72].

  5. 5.

    Contrary to the SOC wire model, by computing the Chern-Simons invariant (12.33) here, the weak field phase (i.e. the superconducting edge) is topological while the strong field phase (i.e. the magnetic edge) is trivial.

  6. 6.

    In reality the pairing strength plateaus \(\Delta (\mathbf{r})=|\Delta _{\infty }|e^{i\varphi (\mathbf{r})}\) for \(|\mathbf{r}|\gg 0\) far away from the vortex. However, because of the non-trivial winding of the phase, \(\Delta (\mathbf{r})\) cannot be compactified or otherwise it would be discontinuous at \(\infty \). We deform the model and let the pairing strength plateaus up to some long length scale L until it eventually diverges at \(\infty \). In this case after compactifying both real and pairing spaces \(\mathbb {S}^2=\mathbb {R}^2\cup \{\infty \}\), the pairing function \(\Delta :\mathbb {S}^2\rightarrow \mathbb {S}^2\) is continuous.

  7. 7.

    The Nambu doubled BdG theory of the bottom surface can be regularized at large momentum by adding \(\varepsilon k^2\sigma _z\tau _z\), for instance, \(\varepsilon \) could be \(\hbar ^2/2m\). Depending on the relative sign of \(\varepsilon \) and B, the Chern numbers of the \((+,-)\) sectors are now \((\pm 1,0)\) or \((0,\pm 1)\) as \(\hat{\mathbf{h}}_\pm \) cover either the entire Bloch sphere or nothing at all.

  8. 8.

    The chiral Dirac interface fermion between time reversal conjugate magnetic TI surface domains can be illustrated by solving the Bloch Hamiltonian \(H_{\mathrm {surface}}(\mathbf{k}_\Vert )=\lambda (k_x\sigma _x+k_y\sigma _y)+g\mu _BB(x)\sigma _z\) for \(k_x\leftrightarrow -i\partial _x\) and B(x) changes sign across the domain wall along the y-axis. The derivation is similar to that of the Jackiw-Rebbi model (12.76) and is carried out in the context of a BdG model later in (12.103).

  9. 9.

    Here we do not make the distinction between Dirac and Weyl (semi)metals because they both consist of pairs of Weyl fermions with opposite chiralities. We do not pay attention to their spatial symmetry origins and their locations in momentum space.

References

  1. J. Alicea, Rep. Prog. Phys. 75, 076501 (2012). https://doi.org/10.1088/0034-4885/75/7/076501

    Article  ADS  Google Scholar 

  2. C.W.J. Beenakker, Annu. Rev. Conden. Matter Phys. 4, 113 (2013). https://doi.org/10.1146/annurev-conmatphys-030212-184337

    Article  ADS  Google Scholar 

  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.88.035005

    Article  ADS  Google Scholar 

  4. S. Das Sarma, M. Freedman, C. Nayak, Npj Quantum Inf. 1, 15001 (2015). https://doi.org/10.1038/npjqi.2015.1

  5. S.R. Elliott, M. Franz, Rev. Mod. Phys. 87, 137 (2015). https://doi.org/10.1103/RevModPhys.87.137

    Article  ADS  Google Scholar 

  6. M.Z. Hasan, C.L. Kane, Rev. Mod. Phys. 82, 3045 (2010). https://doi.org/10.1103/RevModPhys.82.3045

    Article  ADS  Google Scholar 

  7. M. Leijnse, K. Flensberg, Semicond. Sci. Technol. 27(12), 124003 (2012). https://doi.org/10.1088/0268-1242/27/12/124003

    Article  ADS  Google Scholar 

  8. X.-L. Qi, S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011). https://doi.org/10.1103/RevModPhys.83.1057

    Article  ADS  Google Scholar 

  9. T.D. Stanescu, S. Tewari, J. Phys. Condens. Matter 25, 233201 (2013), http://stacks.iop.org/0953-8984/25/i=23/a=233201

    ADS  Google Scholar 

  10. E. Schrödinger, Phys. Rev. 28, 1049 (1926). https://doi.org/10.1103/PhysRev.28.1049

    Article  ADS  Google Scholar 

  11. A. Einstein, Ann. der Phys. 322(10), 891 (1905). https://doi.org/10.1002/andp.19053221004

    Article  ADS  Google Scholar 

  12. P.A.M. Dirac, Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 117(778), 610 (1928). https://doi.org/10.1098/rspa.1928.0023

    Article  ADS  Google Scholar 

  13. E. Majorana, Il Nuovo Cimento (1924–1942) 14(4), 171 (1937). https://doi.org/10.1007/BF02961314

    Article  ADS  Google Scholar 

  14. A. Kitaev, AIP Conf. Proc. 1134, 22 (2008). https://doi.org/10.1063/1.3149495

    Article  ADS  Google Scholar 

  15. X.-L. Qi, T.L. Hughes, S. Raghu, S.-C. Zhang, Phys. Rev. Lett. 102, 187001 (2009). https://doi.org/10.1103/PhysRevLett.102.187001

    Article  ADS  Google Scholar 

  16. A.P. Schnyder, S. Ryu, A. Furusaki, A.W.W. Ludwig, Phys. Rev. B 78, 195125 (2008). https://doi.org/10.1103/PhysRevB.78.195125

    Article  ADS  Google Scholar 

  17. A.Y. Kitaev, Phys.-Usp. 44, 131 (2001). https://doi.org/10.1070/1063-7869/44/10S/S29

    Article  ADS  Google Scholar 

  18. A. Cook, M. Franz, Phys. Rev. B 84, 201105 (2011). https://doi.org/10.1103/PhysRevB.84.201105

    Article  ADS  Google Scholar 

  19. R.M. Lutchyn, J.D. Sau, S. Das Sarma, Phys. Rev. Lett. 105, 077001 (2010). https://doi.org/10.1103/PhysRevLett.105.077001

    Article  ADS  Google Scholar 

  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.1259327

    Article  ADS  Google Scholar 

  21. Y. Oreg, G. Refael, F. von Oppen, Phys. Rev. Lett. 105, 177002 (2010). https://doi.org/10.1103/PhysRevLett.105.177002

    Article  ADS  Google Scholar 

  22. V. Gurarie, L. Radzihovsky, Phys. Rev. B 75, 212509 (2007b). https://doi.org/10.1103/PhysRevB.75.212509

    Article  ADS  Google Scholar 

  23. D.A. Ivanov, Phys. Rev. Lett. 86, 268 (2001). https://doi.org/10.1103/PhysRevLett.86.268

    Article  ADS  Google Scholar 

  24. A. Kitaev, Ann. Phys. 321(1), 2 (2006). https://doi.org/10.1016/j.aop.2005.10.005

    Article  ADS  MathSciNet  Google Scholar 

  25. N. Read, D. Green, Phys. Rev. B 61, 10267 (2000). https://doi.org/10.1103/PhysRevB.61.10267

    Article  ADS  Google Scholar 

  26. G.E. Volovik, Pisma Zh. Eksp. Teor. Fiz. 70, 601 (1999). https://doi.org/10.1134/1.568223

    Article  ADS  Google Scholar 

  27. G.E. Volovik, The Universe in a Helium Droplet (Oxford University Press, 2003)

    Google Scholar 

  28. C.-K. Chiu, M.J. Gilbert, T.L. Hughes, Phys. Rev. B 84(14), 144507 (2011). https://doi.org/10.1103/PhysRevB.84.144507

    Article  ADS  Google Scholar 

  29. L. Fu, C.L. Kane, Phys. Rev. Lett. 100, 096407 (2008). https://doi.org/10.1103/PhysRevLett.100.096407

    Article  ADS  Google Scholar 

  30. L. Fu, C.L. Kane, Phys. Rev. B 79, 161408(R) (2009a). https://doi.org/10.1103/PhysRevB.79.161408

    Article  ADS  Google Scholar 

  31. P. Hosur, P. Ghaemi, R.S.K. Mong, A. Vishwanath, Phys. Rev. Lett. 107(9), 097001 (2011). https://doi.org/10.1103/PhysRevLett.107.097001

    Article  ADS  Google Scholar 

  32. J.C.Y. Teo, C.L. Kane, Phys. Rev. Lett. 104, 046401 (2010a). https://doi.org/10.1103/PhysRevLett.104.046401

    Article  ADS  Google Scholar 

  33. J.C.Y. Teo, C.L. Kane, Phys. Rev. B 82, 115120 (2010b). https://doi.org/10.1103/PhysRevB.82.115120

    Article  ADS  Google Scholar 

  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.112.217001

    Article  ADS  Google Scholar 

  35. P.G. De Gennes, Superconductivity Of Metals And Alloys (Westview Press, 1999)

    Google Scholar 

  36. A.J. Leggett, Quantum Liquids: Bose Condensation and Cooper Pairing in Condensed-Matter Systems (Oxford University Press, 2006)

    Google Scholar 

  37. A. Altland, M.R. Zirnbauer, Phys. Rev. B 55, 1142 (1997). https://doi.org/10.1103/PhysRevB.55.1142

    Article  ADS  Google Scholar 

  38. F. Wilczek, Nat. Phys. 5, 614 (2009). https://doi.org/10.1038/nphys1380

    Article  Google Scholar 

  39. M. Freedman, A. Kitaev, M. Larsen, Z. Wang, Bull. Am. Math. Soc. 40, 31 (2002). https://doi.org/10.1090/S0273-0979-02-00964-3

    Article  Google Scholar 

  40. A. Kitaev, Ann. Phys. 303, 2 (2003). https://doi.org/10.1016/S0003-4916(02)00018-0

    Article  ADS  Google Scholar 

  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.80.1083

    Article  ADS  Google Scholar 

  42. J. Preskill, Topological Quantum Computation, http://www.theory.caltech.edu/~preskill/ph219/topological.pdf (2004)

  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_31

    Chapter  Google Scholar 

  44. Z. Wang, Topological Quantum Computation (American Mathematics Society, 2010)

    Google Scholar 

  45. W.P. Su, J.R. Schrieffer, A.J. Heeger, Phys. Rev. B 22, 2099 (1980). https://doi.org/10.1103/PhysRevB.22.2099

    Article  ADS  Google Scholar 

  46. C.L. Kane, E.J. Mele, Phys. Rev. Lett. 95, 226801 (2005a). https://doi.org/10.1103/PhysRevLett.95.226801

    Article  ADS  Google Scholar 

  47. C.L. Kane, E.J. Mele, Phys. Rev. Lett. 95, 146802 (2005b). https://doi.org/10.1103/PhysRevLett.95.146802

    Article  ADS  Google Scholar 

  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.112.026602

    Article  ADS  Google Scholar 

  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.1148047

    Article  ADS  Google Scholar 

  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.100.236601

    Article  ADS  Google Scholar 

  51. V. Mourik, K. Zuo, S. Frolov, S. Plissard, E. Bakkers, L. Kouwenhoven, Science 336, 1003 (2012). https://doi.org/10.1126/science.1222360

    Article  ADS  Google Scholar 

  52. A.R. Akhmerov, J. Nilsson, C.W.J. Beenakker, Phys. Rev. Lett. 102, (2009). https://doi.org/10.1103/PhysRevLett.102.216404

  53. A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, H. Shtrikman, Nat. Phys. 8, 887 (2012). https://doi.org/10.1038/nphys2479

    Article  Google Scholar 

  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/nl303758w

    Article  ADS  Google Scholar 

  55. L. Fu, C.L. Kane, Phys. Rev. Lett. 102, 216403 (2009b). https://doi.org/10.1103/PhysRevLett.102.216403

    Article  ADS  Google Scholar 

  56. K.T. Law, P.A. Lee, T.K. Ng, Phys. Rev. Lett. 103, 237001 (2009). https://doi.org/10.1103/PhysRevLett.103.237001

    Article  ADS  Google Scholar 

  57. L.P. Rokhinson, X. Liu, J. K. Furdyna, Nat. Phys. 8, 795 (2012). https://doi.org/10.1038/nphys2429

    Article  ADS  Google Scholar 

  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. L. Fu, Phys. Rev. Lett. 104, 056402 (2010). https://doi.org/10.1103/PhysRevLett.104.056402

    Article  ADS  Google Scholar 

  60. B. van Heck, R.M. Lutchyn, L.I. Glazman, Phys. Rev. B 93, 235431 (2016). https://doi.org/10.1103/PhysRevB.93.235431

    Article  ADS  Google Scholar 

  61. R. Hützen, A. Zazunov, B. Braunecker, A.L. Yeyati, R. Egger, Phys. Rev. Lett. 109, 166403 (2012). https://doi.org/10.1103/PhysRevLett.109.166403

    Article  ADS  Google Scholar 

  62. J.D. Sau, B. Swingle, S. Tewari, Phys. Rev. B 92, 020511 (2015). https://doi.org/10.1103/PhysRevB.92.020511

    Article  ADS  Google Scholar 

  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/nature17162

    Article  ADS  Google Scholar 

  64. G.E. Volovik, JETP Lett. 55, 368 (1992), http://www.jetpletters.ac.ru/ps/1273/article_19263.shtml

  65. A. Cappelli, M. Huerta, G.R. Zemba, Nucl. Phys. B 636(3), 568 (2002). https://doi.org/10.1016/S0550-3213(02)00340-1

    Article  ADS  Google Scholar 

  66. C.L. Kane, M.P.A. Fisher, Phys. Rev. B 55, 15832 (1997). https://doi.org/10.1103/PhysRevB.55.15832

    Article  ADS  Google Scholar 

  67. V. Gurarie, L. Radzihovsky, Ann. Phys. 322(1), 2 (2007). https://doi.org/10.1016/j.aop.2006.10.009

    Article  ADS  MathSciNet  Google Scholar 

  68. M. Sigrist, K. Ueda, Rev. Mod. Phys. 63, 239 (1991). https://doi.org/10.1103/RevModPhys.63.239

    Article  ADS  Google Scholar 

  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/10.5772/48579

    Google Scholar 

  70. M. Atiyah, K-Theory (Westview Press, 1994)

    Google Scholar 

  71. H.B. Lawson, M.-L. Michelsohn, Spin Geometry (Princeton University Press, 1990)

    Google Scholar 

  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. J.D. Jackson, Classical Electrodynamics 3rd edn. (Wiley, New York, 1999)

    Google Scholar 

  74. P. Di Francesco, P. Mathieu, D. Senechal, Conformal Field Theory (Springer, New York, 1999)

    Google Scholar 

  75. M.F. Atiyah, V.K. Patodi, I.M. Singer, Math. Proc. Camb. Philos. Soc. 77(1), 43 (1975a). https://doi.org/10.1017/S0305004100049410

    Article  Google Scholar 

  76. M.F. Atiyah, V.K. Patodi, I.M. Singer, Math. Proc. Camb. Philos. Soc. 78(3), 405 (1975b). https://doi.org/10.1017/S0305004100051872

    Article  Google Scholar 

  77. M.F. Atiyah, V.K. Patodi, I.M. Singer, Math. Proc. Camb. Philos. Soc. 79(1), 71 (1976). https://doi.org/10.1017/S0305004100052105

    Article  Google Scholar 

  78. B.I. Halperin, Phys. Rev. B 25, 2185 (1982). https://doi.org/10.1103/PhysRevB.25.2185

    Article  ADS  Google Scholar 

  79. Y. Hatsugai, Phys. Rev. Lett. 71, 3697 (1993). https://doi.org/10.1103/PhysRevLett.71.3697

    Article  ADS  MathSciNet  Google Scholar 

  80. R.B. Laughlin, Phys. Rev. B 23, 5632 (1981). https://doi.org/10.1103/PhysRevB.23.5632

    Article  ADS  Google Scholar 

  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. D.J. Thouless, M. Kohmoto, M.P. Nightingale, M. den Nijs, Phys. Rev. Lett. 49, 405 (1982). https://doi.org/10.1103/PhysRevLett.49.405

    Article  ADS  Google Scholar 

  83. J. Bellissard, A. van Elst, H. Schulz-Baldes, J. Math. Phys. 35(10), 5373 (1994). https://doi.org/10.1063/1.530758

    Article  ADS  MathSciNet  Google Scholar 

  84. T.A. Loring, M.B. Hastings, EPL (Europhysics Lett.) 92(6), 67004 (2010), http://stacks.iop.org/0295-5075/92/i=6/a=67004

    Article  ADS  Google Scholar 

  85. E. Prodan, H. Schulz-Baldes, J. Funct. Anal. 271(5), 1150 (2016). https://doi.org/10.1016/j.jfa.2016.06.001

    Article  MathSciNet  Google Scholar 

  86. P.W. Anderson, P. Morel, Phys. Rev. 123, 1911 (1961). https://doi.org/10.1103/PhysRev.123.1911

    Article  ADS  MathSciNet  Google Scholar 

  87. A.J. Leggett, Rev. Mod. Phys. 47, 331 (1975). https://doi.org/10.1103/RevModPhys.47.331

    Article  ADS  Google Scholar 

  88. G.E. Volovik, JETP Lett. 93(2), 66 (2011). https://doi.org/10.1134/S0021364011020147

    Article  ADS  Google Scholar 

  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/10.1038/29038

    Article  ADS  Google Scholar 

  90. K.D. Nelson, Z.Q. Mao, Y. Maeno, Y. Liu, Science 306(5699), 1151 (2004). https://doi.org/10.1126/science.1103881

    Article  ADS  Google Scholar 

  91. T.M. Rice, M. Sigrist, J. Phys.: Condens. Matter 7, L643 (1995). https://doi.org/10.1088/0953-8984/7/47/002

    Article  ADS  Google Scholar 

  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.97.167002

    Article  ADS  Google Scholar 

  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.76.014526

    Article  ADS  Google Scholar 

  94. Y. Maeno, T.M. Rice, M. Sigrist, Phys. Today 54, 42 (2001). https://doi.org/10.1063/1.1349611

    Article  ADS  Google Scholar 

  95. S. Raghu, A. Kapitulnik, S.A. Kivelson, Phys. Rev. Lett. 105, 136401 (2010). https://doi.org/10.1103/PhysRevLett.105.136401

    Article  ADS  Google Scholar 

  96. F.D.M. Haldane, Phys. Rev. Lett. 61, 2015 (1988). https://doi.org/10.1103/PhysRevLett.61.2015

    Article  ADS  Google Scholar 

  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-031115-011417

    Article  ADS  Google Scholar 

  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.aag2792

    Article  ADS  MathSciNet  Google Scholar 

  99. A. Ii, K. Yada, M. Sato, Y. Tanaka, Phys. Rev. B 83, 224524 (2011). https://doi.org/10.1103/PhysRevB.83.224524

    Article  ADS  Google Scholar 

  100. X.-L. Qi, T.L. Hughes, S.-C. Zhang, Phys. Rev. B 82, 184516 (2010). https://doi.org/10.1103/PhysRevB.82.184516

    Article  ADS  Google Scholar 

  101. J. Wang, Q. Zhou, B. Lian, S.-C. Zhang, Phys. Rev. B 92, 064520 (2015). https://doi.org/10.1103/PhysRevB.92.064520

    Article  ADS  Google Scholar 

  102. M. Stone, R. Roy, Phys. Rev. B 69, 184511 (2004). https://doi.org/10.1103/PhysRevB.69.184511

    Article  ADS  Google Scholar 

  103. C. Caroli, P.G. de Gennes, J. Matricon, Phys. Lett. 9, 307 (1964). https://doi.org/10.1016/0031-9163(64)90375-0

    Article  ADS  Google Scholar 

  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. 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. E. Fradkin, Field Theories of Condensed Matter Physics, 2nd edn. (Cambridge University Press, 2013)

    Google Scholar 

  107. N. Read, G. Moore, Nucl. Phys. B 360, 362 (1991). https://doi.org/10.1016/0550-3213(91)90407-O

    Article  ADS  Google Scholar 

  108. P.M. Chaikin, T.C. Lubensky, Principles of Condensed Matter Physics (Cambridge University Press, 2000)

    Google Scholar 

  109. D.R. Nelson, Defects and Geometry in Condensed Matter Physics (Cambridge University Press, 2002)

    Google Scholar 

  110. D. Asahi, N. Nagaosa, Phys. Rev. B 86, 100504(R) (2012). https://doi.org/10.1103/PhysRevB.86.100504

    Article  ADS  Google Scholar 

  111. T.L. Hughes, H. Yao, X.-L. Qi, Phys. Rev. B 90, 235123 (2014). https://doi.org/10.1103/PhysRevB.90.235123

    Article  ADS  Google Scholar 

  112. Ran, Y. (2010), arXiv:1006.5454 [cond-mat.str-el]

  113. J.C. Teo, T.L. Hughes, Annu. Rev. Condens. Matter Phys. 8(1), 211 (2017). https://doi.org/10.1146/annurev-conmatphys-031016-025154

    Article  ADS  Google Scholar 

  114. W.A. Benalcazar, J.C.Y. Teo, T.L. Hughes, Phys. Rev. B 89, 224503 (2014). https://doi.org/10.1103/PhysRevB.89.224503

    Article  ADS  Google Scholar 

  115. J.C.Y. Teo, T.L. Hughes, Phys. Rev. Lett. 111, 047006 (2013). https://doi.org/10.1103/PhysRevLett.111.047006

    Article  ADS  Google Scholar 

  116. S.B. Chung, H. Bluhm, E.-A. Kim, Phys. Rev. Lett. 99, 197002 (2007). https://doi.org/10.1103/PhysRevLett.99.197002

    Article  ADS  Google Scholar 

  117. S. Das Sarma, C. Nayak, S. Tewari, Phys. Rev. B 73, 220502 (2006). https://doi.org/10.1103/PhysRevB.73.220502

    Article  Google Scholar 

  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.1193839

    Article  ADS  Google Scholar 

  119. M.M. Salomaa, G.E. Volovik, Phys. Rev. Lett. 55, 1184 (1985). https://doi.org/10.1103/PhysRevLett.55.1184

    Article  ADS  Google Scholar 

  120. S.B. Chung, S.-C. Zhang, Phys. Rev. Lett. 103, 235301 (2009). https://doi.org/10.1103/PhysRevLett.103.235301

    Article  ADS  Google Scholar 

  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.103.155301

    Article  ADS  Google Scholar 

  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.80.013602

    Article  ADS  Google Scholar 

  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. 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.78.214516

    Article  ADS  Google Scholar 

  125. L. Fu, E. Berg, Phys. Rev. Lett. 105, 097001 (2010). https://doi.org/10.1103/PhysRevLett.105.097001

    Article  ADS  Google Scholar 

  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.104.057001

    Article  ADS  Google Scholar 

  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.107.217001

    Article  ADS  Google Scholar 

  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/nphys1762

    Article  Google Scholar 

  129. B.A. Bernevig, T.L. Hughes, S.-C. Zhang, Science 314(5806), 1757 (2006). https://doi.org/10.1126/science.1133734

    Article  ADS  Google Scholar 

  130. J. Nilsson, A.R. Akhmerov, C.W.J. Beenakker, Phys. Rev. Lett. 101, 120403 (2008). https://doi.org/10.1103/PhysRevLett.101.120403

    Article  ADS  Google Scholar 

  131. R. Jackiw, C. Rebbi, Phys. Rev. D 13, 3398 (1976). https://doi.org/10.1103/PhysRevD.13.3398

    Article  ADS  MathSciNet  Google Scholar 

  132. J.C. Gallop, SQUIDs, the Josephson Effects and Superconducting Electronics (CRC Press, 1991)

    Google Scholar 

  133. D.J. Thouless, Phys. Rev. B 27, 6083 (1983). https://doi.org/10.1103/PhysRevB.27.6083

    Article  ADS  MathSciNet  Google Scholar 

  134. L. Fu, C.L. Kane, E.J. Mele, Phys. Rev. Lett. 98, 106803 (2007). https://doi.org/10.1103/PhysRevLett.98.106803

    Article  ADS  Google Scholar 

  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/nature06843

    Article  ADS  Google Scholar 

  136. J.E. Moore, L. Balents, Phys. Rev. B 75, 121306(R) (2007). https://doi.org/10.1103/PhysRevB.75.121306

    Article  ADS  Google Scholar 

  137. X.-L. Qi, T.L. Hughes, S.-C. Zhang, Phys. Rev. B 78, 195424 (2008). https://doi.org/10.1103/PhysRevB.78.195424

    Article  ADS  Google Scholar 

  138. R. Roy, Phys. Rev. B 79, 195322 (2009). https://doi.org/10.1103/PhysRevB.79.195322

    Article  ADS  Google Scholar 

  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/nphys1274

    Article  Google Scholar 

  140. R. Jackiw, P. Rossi, Nucl. Phys. B 190, 681 (1981). https://doi.org/10.1016/0550-3213(81)90044-4

    Article  ADS  Google Scholar 

  141. M.F. Atiyah, G.B. Segal, Ann. Math. 87, 531 (1968). https://doi.org/10.2307/1970716

    Article  MathSciNet  Google Scholar 

  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. M.F. Atiyah, I.M. Singer, Ann. Math. 87, 484 (1968a). https://doi.org/10.2307/1970715

    Article  MathSciNet  Google Scholar 

  144. M.F. Atiyah, I.M. Singer, Ann. Math. 87, 546 (1968b). https://doi.org/10.2307/1970717

    Article  MathSciNet  Google Scholar 

  145. M.F. Atiyah, I.M. Singer, Ann. Math. 93, 119 (1971a). https://doi.org/10.2307/1970756

    Article  MathSciNet  Google Scholar 

  146. M.F. Atiyah, I.M. Singer, Ann. Math. 93, 139 (1971b). https://doi.org/10.2307/1970757

    Article  MathSciNet  Google Scholar 

  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.83.115132

    Article  ADS  Google Scholar 

  148. X.-L. Qi, E. Witten, S.-C. Zhang, Phys. Rev. B 87, 134519 (2013). https://doi.org/10.1103/PhysRevB.87.134519

    Article  ADS  Google Scholar 

  149. A.A. Burkov, L. Balents, Phys. Rev. Lett. 107, 127205 (2011). https://doi.org/10.1103/PhysRevLett.107.127205

    Article  ADS  Google Scholar 

  150. A.A. Burkov, M.D. Hook, L. Balents, Phys. Rev. B 84, 235126 (2011). https://doi.org/10.1103/PhysRevB.84.235126

    Article  ADS  Google Scholar 

  151. S. Murakami, New J. Phys. 9, 356 (2007). https://doi.org/10.1088/1367-2630/9/9/356

    Article  ADS  Google Scholar 

  152. O. Vafek, A. Vishwanath, Annu. Rev. Condens. Matter Phys. 5(1), 83 (2014). https://doi.org/10.1146/annurev-conmatphys-031113-133841

    Article  ADS  Google Scholar 

  153. X. Wan, A.M. Turner, A. Vishwanath, S.Y. Savrasov, Phys. Rev. B 83, 205101 (2011). https://doi.org/10.1103/PhysRevB.83.205101

    Article  ADS  Google Scholar 

  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.85.195320

    Article  ADS  Google Scholar 

  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.108.140405

    Article  ADS  Google Scholar 

  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.113.027603

    Article  ADS  Google Scholar 

  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.5.031023

    Article  Google Scholar 

  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/nmat4023

    Article  ADS  Google Scholar 

  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/nmat4143

    Article  ADS  Google Scholar 

  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/nmat3990

    Article  ADS  Google Scholar 

  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.1245085

    Article  ADS  Google Scholar 

  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.5.031013

    Article  Google Scholar 

  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/nphys3426

    Article  Google Scholar 

  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/ncomms4786

    Article  Google Scholar 

  165. Z. Wang, H. Weng, Q. Wu, X. Dai, Z. Fang, Phys. Rev. B 88, 125427 (2013). https://doi.org/10.1103/PhysRevB.88.125427

    Article  ADS  Google Scholar 

  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.aaa9297

    Article  ADS  Google Scholar 

  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.1256742

    Article  ADS  Google Scholar 

  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/ncomms10735

    Article  ADS  Google Scholar 

  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/nmat4455

    Article  ADS  Google Scholar 

  170. S. Kobayashi, M. Sato, Phys. Rev. Lett. 115, 187001 (2015). https://doi.org/10.1103/PhysRevLett.115.187001

    Article  ADS  Google Scholar 

  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. M.-J. Park, J.C.Y. Teo, M.J. Gilbert, To appear soon (2017)

    Google Scholar 

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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.

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  1. Department of Physics, University of Virginia, Charlottesville, VA, USA

    Jeffrey C. Y. Teo

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  1. University of New Hampshire, Durham, NH, USA

    Jiadong Zang

  2. Unité Mixte de Physique CNRS/Thales, Palaiseau, France

    Vincent Cros

  3. Materials Science Division, Argonne National Laboratory, Lemont, IL, USA

    Axel Hoffmann

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Teo, J.C.Y. (2018). Majorana States. In: Zang, J., Cros, V., Hoffmann, A. (eds) Topology in Magnetism. Springer Series in Solid-State Sciences, vol 192. Springer, Cham. https://doi.org/10.1007/978-3-319-97334-0_12

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