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Topology, Symmetry, and Band Theory of Materials

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Glide-Symmetric Z2 Magnetic Topological Crystalline Insulators

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Abstract

We review basic properties of topology and symmetry in band theory and the brief history of topological phases of matter in single-particle systems in the present chapter. First, we explain general properties of Berry phase, Berry connection, and Berry curvature and how these quantities are encoded in band theory described by Bloch electrons. We also explain how they correspond to the Chern number, as an example of topological invariants, associated with Wannier functions localized in real space.

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Notes

  1. 1.

    Namely, it is regarded as a cylinder geometry different from a torus or a sphere we have dealt with.

References

  1. Berry MV (1984) Proceedings of the royal society of London A. Math Phys Sci 392(1802):45. https://doi.org/10.1098/rspa.1984.0023

    Article  Google Scholar 

  2. Vanderbilt D (2018) Berry phases in electronic structure theory: electric polarization, orbital magnetization and topological insulators. Cambridge University Press

    Google Scholar 

  3. Gresch D, Autès G, Yazyev OV, Troyer M, Vanderbilt D, Bernevig BA, Soluyanov AA (2017) Phys Rev B 95:075146. https://doi.org/10.1103/PhysRevB.95.075146

    Article  ADS  Google Scholar 

  4. Joannopoulos JD, Johnson SG, Winn JN, Meade RD (2008) Molding the flow of light

    Google Scholar 

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

    Article  ADS  Google Scholar 

  6. Fang C, Gilbert MJ, Bernevig BA (2012) Phys Rev B 86:115112. https://doi.org/10.1103/PhysRevB.86.115112

    Article  ADS  Google Scholar 

  7. Watanabe H, Po HC, Vishwanath A (2018) Sci Adv 4(8):eaat8685

    Google Scholar 

  8. Po HC, Vishwanath A, Watanabe H (2017) Nat Commun 8(1):50

    ADS  Google Scholar 

  9. Inui T, Tanabe Y, Onodera Y (2012) Group theory and its applications in physics, vol 78. Springer Science & Business Media

    Google Scholar 

  10. Shiozaki K, Sato M, Gomi K (2018) ar**v:1802.06694

  11. Haldane FDM (1988) Phys Rev Lett 61:2015. https://doi.org/10.1103/PhysRevLett.61.2015

    Article  ADS  Google Scholar 

  12. Chang CZ, Zhang J, Feng X, Shen J, Zhang Z, Guo M, Li K, Ou Y, Wei P, Wang LL, Ji ZQ, Feng Y, Ji S, Chen X, Jia J, Dai X, Fang Z, Zhang SC, He K, Wang Y, Lu L, Ma XC, Xue QK (2013) Science 340(6129):167. https://doi.org/10.1126/science.1234414

    Article  ADS  Google Scholar 

  13. Jotzu G, Messer M, Desbuquois R, Lebrat M, Uehlinger T, Greif D, Esslinger T (2014) Nature 515(7526):237

    ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  15. Qi XL, Hughes TL, Zhang SC (2008) Phys Rev B 78:195424. https://doi.org/10.1103/PhysRevB.78.195424

    Article  ADS  Google Scholar 

  16. Laughlin RB (1981) Phys Rev B 23:5632. https://doi.org/10.1103/PhysRevB.23.5632

    Article  ADS  Google Scholar 

  17. Resta R (1992) Ferroelectrics 136(1):51

    Google Scholar 

  18. King-Smith RD, Vanderbilt D (1993) Phys Rev B 47:1651. https://doi.org/10.1103/PhysRevB.47.1651

    Article  ADS  Google Scholar 

  19. Vanderbilt D, King-Smith RD (1993) Phys Rev B 48:4442. https://doi.org/10.1103/PhysRevB.48.4442

    Article  ADS  Google Scholar 

  20. Hasan MZ, Kane CL (2010) Rev Mod Phys 82:3045. https://doi.org/10.1103/RevModPhys.82.3045

    Article  ADS  Google Scholar 

  21. Qi XL, Zhang SC (2011) Rev Mod Phys 83:1057. https://doi.org/10.1103/RevModPhys.83.1057

    Article  ADS  Google Scholar 

  22. Murakami S, Nagaosa N, Zhang SC (2003) Science 301(5638):1348. https://doi.org/10.1126/science.1087128

    Article  ADS  Google Scholar 

  23. Sinova J, Culcer D, Niu Q, Sinitsyn NA, Jungwirth T, MacDonald AH (2004) Phys Rev Lett 92:126603. https://doi.org/10.1103/PhysRevLett.92.126603

    Article  ADS  Google Scholar 

  24. König M, Wiedmann S, Brüne C, Roth A, Buhmann H, Molenkamp LW, Qi XL, Zhang SC (2007) Science 318(5851):766. https://doi.org/10.1126/science.1148047

    Article  ADS  Google Scholar 

  25. Bernevig BA, Hughes TL, Zhang SC (2006) Science 314(5806):1757. https://doi.org/10.1126/science.1133734

    Article  ADS  Google Scholar 

  26. Hsieh D, Qian D, Wray L, **a Y, Hor YS, Cava RJ, Hasan MZ (2008) Nature 452(7190):970

    ADS  Google Scholar 

  27. Zhang H, Liu CX, Qi XL, Dai X, Fang Z, Zhang SC (2009) Nat Phys 5(6):438

    Google Scholar 

  28. **a Y, Qian D, Hsieh D, Wray L, Pal A, Lin H, Bansil A, Grauer D, Hor YS, Cava RJ et al (2009) Nat Phys 5(6):398

    Google Scholar 

  29. Chen YL, Analytis JG, Chu JH, Liu ZK, Mo SK, Qi XL, Zhang HJ, Lu DH, Dai X, Fang Z, Zhang SC, Fisher IR, Hussain Z, Shen ZX (2009) Science 325(5937):178. https://doi.org/10.1126/science.1173034

    Article  ADS  Google Scholar 

  30. Fu L, Kane CL (2006) Phys Rev B 74:195312. https://doi.org/10.1103/PhysRevB.74.195312

    Article  ADS  Google Scholar 

  31. Soluyanov AA, Vanderbilt D (2011) Phys Rev B 83:035108. https://doi.org/10.1103/PhysRevB.83.035108

    Article  ADS  Google Scholar 

  32. Soluyanov AA, Vanderbilt D (2011) Phys Rev B 83:235401. https://doi.org/10.1103/PhysRevB.83.235401

    Article  ADS  Google Scholar 

  33. Yu R, Qi XL, Bernevig A, Fang Z, Dai X (2011) Phys Rev B 84:075119. https://doi.org/10.1103/PhysRevB.84.075119

    Article  ADS  Google Scholar 

  34. Fu L (2011) Phys Rev Lett 106:106802. https://doi.org/10.1103/PhysRevLett.106.106802

    Article  ADS  Google Scholar 

  35. Hughes TL, Prodan E, Bernevig BA (2011) Phys Rev B 83:245132. https://doi.org/10.1103/PhysRevB.83.245132

    Article  ADS  Google Scholar 

  36. Fang C, Gilbert MJ, Bernevig BA (2013) Phys Rev B 87:035119. https://doi.org/10.1103/PhysRevB.87.035119

    Article  ADS  Google Scholar 

  37. Alexandradinata A, Fang C, Gilbert MJ, Bernevig BA (2014) Phys Rev Lett 113:116403. https://doi.org/10.1103/PhysRevLett.113.116403

    Article  ADS  Google Scholar 

  38. Hsieh TH, Lin H, Liu J, Duan W, Bansil A, Fu L (2012) Nat Commun 3:982. https://doi.org/10.1038/ncomms1969

    Article  ADS  Google Scholar 

  39. Sun Y, Zhong Z, Shirakawa T, Franchini C, Li D, Li Y, Yunoki S, Chen XQ (2013) Phys Rev B 88:235122. https://doi.org/10.1103/PhysRevB.88.235122

    Article  ADS  Google Scholar 

  40. Kargarian M, Fiete GA (2013) Phys Rev Lett 110:156403. https://doi.org/10.1103/PhysRevLett.110.156403

    Article  ADS  Google Scholar 

  41. Kindermann M (2015) Phys Rev Lett 114:226802. https://doi.org/10.1103/PhysRevLett.114.226802

    Article  ADS  Google Scholar 

  42. Weng H, Zhao J, Wang Z, Fang Z, Dai X (2014) Phys Rev Lett 112:016403. https://doi.org/10.1103/PhysRevLett.112.016403

    Article  ADS  Google Scholar 

  43. Hsieh TH, Liu J, Fu L (2014) Phys Rev B 90:081112. https://doi.org/10.1103/PhysRevB.90.081112

    Article  ADS  Google Scholar 

  44. Slager RJ, Mesaros A, Juricic V, Zaanen J (2013) Nat Phys 9:98. https://doi.org/10.1038/nphys2513

    Article  Google Scholar 

  45. Chiu CK, Yao H, Ryu S (2013) Phys Rev B 88:075142. https://doi.org/10.1103/PhysRevB.88.075142

    Article  ADS  Google Scholar 

  46. Morimoto T, Furusaki A (2013) Phys Rev B 88:125129. https://doi.org/10.1103/PhysRevB.88.125129

    Article  ADS  Google Scholar 

  47. Jadaun P, **ao D, Niu Q, Banerjee SK (2013) Phys Rev B 88:085110. https://doi.org/10.1103/PhysRevB.88.085110

    Article  ADS  Google Scholar 

  48. Shiozaki K, Sato M (2014) Phys Rev B 90:165114. https://doi.org/10.1103/PhysRevB.90.165114

    Article  ADS  Google Scholar 

  49. Chiu CK, Teo JCY, Schnyder AP, Ryu S (2016) Rev Mod Phys 88:035005. https://doi.org/10.1103/RevModPhys.88.035005

    Article  ADS  Google Scholar 

  50. Dziawa P, Kowalski BJ, Dybko K, Buczko R, Szczerbakow A, Szot M, Lusakowska E, Balasubramanian T, Wojek BM, Berntsen MH, Tjernberg O, Story T (2012) Nat Mater 11:1023. https://doi.org/10.1038/nmat3449

    Article  ADS  Google Scholar 

  51. Tanaka Y, Ren Z, Sato T, Nakayama K, Souma S, Takahashi T, Segawa K, Ando Y (2012) Nat Phys 8:800. https://doi.org/10.1038/nphys2442

    Article  Google Scholar 

  52. Xu SY, Liu C, Alidoust N, Neupane M, Qian D, Belopolski I, Denlinger J, Wang Y, Lin H, Wray L, Landolt G, Slomski B, Dil J, Marcinkova A, Morosan E, Gibson Q, Sankar R, Chou F, Cava R, Bansil A, Hasan M (2012) Nat Commun 3:1192. https://doi.org/10.1038/ncomms2191

    Article  ADS  Google Scholar 

  53. Okada Y, Serbyn M, Lin H, Walkup D, Zhou W, Dhital C, Neupane M, Xu S, Wang YJ, Sankar R et al (2013) Science 341(6153):1496

    ADS  Google Scholar 

  54. Liang T, Gibson Q, **ong J, Hirschberger M, Koduvayur SP, Cava R, Ong N (2013) ar**v:1307.4022

  55. Gyenis A, Drozdov IK, Nadj-Perge S, Jeong OB, Seo J, Pletikosić I, Valla T, Gu GD, Yazdani A (2013) Phys Rev B 88:125414. https://doi.org/10.1103/PhysRevB.88.125414

    Article  ADS  Google Scholar 

  56. Liu J, Hsieh TH, Wei P, Duan W, Moodera J, Fu L (2014) Nat Mater 13:178. https://doi.org/10.1038/nmat3828

    Article  ADS  Google Scholar 

  57. Liu J, Duan W, Fu L (2013) Phys Rev B 88:241303. https://doi.org/10.1103/PhysRevB.88.241303

    Article  ADS  Google Scholar 

  58. Fang C, Gilbert MJ, Xu SY, Bernevig BA, Hasan MZ (2013) Phys Rev B 88:125141. https://doi.org/10.1103/PhysRevB.88.125141

    Article  ADS  Google Scholar 

  59. Ozawa H, Yamakage A, Sato M, Tanaka Y (2014) Phys Rev B 90:045309. https://doi.org/10.1103/PhysRevB.90.045309

    Article  ADS  Google Scholar 

  60. Wrasse EO, Schmidt TM (2014) Nano Lett 14(10):5717

    ADS  Google Scholar 

  61. Liu J, Qian X, Fu L (2015) Nano Lett 15(4):2657

    ADS  Google Scholar 

  62. Teo JCY, Fu L, Kane CL (2008) Phys Rev B 78:045426. https://doi.org/10.1103/PhysRevB.78.045426

    Article  ADS  Google Scholar 

  63. Parameswaran SA, Turner AM, Arovas DP, Vishwanath A (2013) Nat Phys 9(5):299

    Google Scholar 

  64. Varjas D, de Juan F, Lu YM (2015) Phys Rev B 92:195116. https://doi.org/10.1103/PhysRevB.92.195116

    Article  ADS  Google Scholar 

  65. Watanabe H, Po HC, Vishwanath A, Zaletel M (2015) Proc Natl Acad Sci USA 112(47):14551

    ADS  Google Scholar 

  66. Po HC, Watanabe H, Zaletel MP, Vishwanath A (2016) Sci Adv 2(4):e1501782

    ADS  Google Scholar 

  67. Dong XY, Liu CX (2016) Phys Rev B 93:045429. https://doi.org/10.1103/PhysRevB.93.045429

    Article  ADS  Google Scholar 

  68. Shiozaki K, Sato M, Gomi K (2016) Phys Rev B 93:195413. https://doi.org/10.1103/PhysRevB.93.195413

    Article  ADS  Google Scholar 

  69. Liu CX, Zhang RX, VanLeeuwen BK (2014) Phys Rev B 90:085304. https://doi.org/10.1103/PhysRevB.90.085304

    Article  ADS  Google Scholar 

  70. Fang C, Fu L (2015) Phys Rev B 91:161105(R). https://doi.org/10.1103/PhysRevB.91.161105

    Article  ADS  Google Scholar 

  71. Shiozaki K, Sato M, Gomi K (2015) Phys Rev B 91:155120. https://doi.org/10.1103/PhysRevB.91.155120

    Article  ADS  Google Scholar 

  72. Lu L, Fang C, Fu L, Johnson SG, Joannopoulos JD, Soljačić M (2016) Nat Phys 12(4):337

    Google Scholar 

  73. Wang Z, Alexandradinata A, Cava RJ, Bernevig BA (2016) Nature 532(7598):189

    ADS  Google Scholar 

  74. Alexandradinata A, Wang Z, Bernevig BA (2016) Phys Rev X 6:021008. https://doi.org/10.1103/PhysRevX.6.021008

    Article  Google Scholar 

  75. Ma J, Yi C, Lv B, Wang Z, Nie S, Wang L, Kong L, Huang Y, Richard P, Zhang P, Yaji K, Kuroda K, Shin S, Weng H, Bernevig BA, Shi Y, Qian T, Ding H (2017) Sci Adv 3(5). https://doi.org/10.1126/sciadv.1602415

  76. Ezawa M (2016) Phys Rev B 94:155148. https://doi.org/10.1103/PhysRevB.94.155148

    Article  ADS  Google Scholar 

  77. Chang PY, Erten O, Coleman P (2017) Nat Phys 13(8):794

    Google Scholar 

  78. Zhang D, Shi M, Zhu T, **ng D, Zhang H, Wang J (2019) Phys Rev Lett 122:206401. https://doi.org/10.1103/PhysRevLett.122.206401

    Article  ADS  Google Scholar 

  79. Li J, Li Y, Du S, Wang Z, Gu BL, Zhang SC, He K, Duan W, Xu Y (2019) Sci Adv 5(6). https://doi.org/10.1126/sciadv.aaw5685

  80. Vidal RC, Bentmann H, Peixoto TRF, Zeugner A, Moser S, Min CH, Schatz S, Kißner K, Ünzelmann M, Fornari CI, Vasili HB, Valvidares M, Sakamoto K, Mondal D, Fujii J, Vobornik I, Jung S, Cacho C, Kim TK, Koch RJ, Jozwiak C, Bostwick A, Denlinger JD, Rotenberg E, Buck J, Hoesch M, Diekmann F, Rohlf S, Kalläne M, Rossnagel K, Otrokov MM, Chulkov EV, Ruck M, Isaeva A, Reinert F (2019) Phys Rev B 100:121104. https://doi.org/10.1103/PhysRevB.100.121104

    Article  ADS  Google Scholar 

  81. Wu J, Liu F, Sasase M, Ienaga K, Obata Y, Yukawa R, Horiba K, Kumigashira H, Okuma S, Inoshita T, Hosono H (2019) Sci Adv 5(11). https://doi.org/10.1126/sciadv.aax9989

  82. Li H, Gao SY, Duan SF, Xu YF, Zhu KJ, Tian SJ, Gao JC, Fan WH, Rao ZC, Huang JR, Li JJ, Yan DY, Liu ZT, Liu WL, Huang YB, Li YL, Liu Y, Zhang GB, Zhang P, Kondo T, Shin S, Lei HC, Shi YG, Zhang WT, Weng HM, Qian T, Ding H (2019) Phys Rev X 9:041039. https://doi.org/10.1103/PhysRevX.9.041039

    Article  Google Scholar 

  83. Hao YJ, Liu P, Feng Y, Ma XM, Schwier EF, Arita M, Kumar S, Hu C, Lu R, Zeng M, Wang Y, Hao Z, Sun HY, Zhang K, Mei J, Ni N, Wu L, Shimada K, Chen C, Liu Q, Liu C (2019) Phys Rev X 9:041038. https://doi.org/10.1103/PhysRevX.9.041038

    Article  Google Scholar 

  84. Chen YJ, Xu LX, Li JH, Li YW, Wang HY, Zhang CF, Li H, Wu Y, Liang AJ, Chen C, Jung SW, Cacho C, Mao YH, Liu S, Wang MX, Guo YF, Xu Y, Liu ZK, Yang LX, Chen YL (2019) Phys Rev X 9:041040. https://doi.org/10.1103/PhysRevX.9.041040

    Article  Google Scholar 

  85. Benalcazar WA, Bernevig BA, Hughes TL (2017) Science 357(6346):61

    ADS  Google Scholar 

  86. Benalcazar WA, Bernevig BA, Hughes TL (2017) Phys Rev B 96:245115. https://doi.org/10.1103/PhysRevB.96.245115

    Article  ADS  Google Scholar 

  87. Song Z, Fang Z, Fang C (2017) Phys Rev Lett 119:246402. https://doi.org/10.1103/PhysRevLett.119.246402

    Article  ADS  Google Scholar 

  88. Langbehn J, Peng Y, Trifunovic L, von Oppen F, Brouwer PW (2017) Phys Rev Lett 119:246401. https://doi.org/10.1103/PhysRevLett.119.246401

    Article  ADS  Google Scholar 

  89. Schindler F, Cook AM, Vergniory MG, Wang Z, Parkin SS, Bernevig BA, Neupert T (2018) Sci Adv 4(6):eaat0346

    Google Scholar 

  90. Khalaf E (2018) Phys Rev B 97:205136. https://doi.org/10.1103/PhysRevB.97.205136

    Article  ADS  Google Scholar 

  91. Khalaf E, Po HC, Vishwanath A, Watanabe H (2018) Phys Rev X 8:031070. https://doi.org/10.1103/PhysRevX.8.031070

    Article  Google Scholar 

  92. Matsugatani A, Watanabe H (2018) Phys Rev B 98:205129. https://doi.org/10.1103/PhysRevB.98.205129

    Article  ADS  Google Scholar 

  93. Ahn J, Yang BJ (2019) Phys Rev B 99:235125. https://doi.org/10.1103/PhysRevB.99.235125

    Article  ADS  MathSciNet  Google Scholar 

  94. Zhang F, Kane CL, Mele EJ (2013) Phys Rev Lett 110:046404. https://doi.org/10.1103/PhysRevLett.110.046404

    Article  ADS  Google Scholar 

  95. Schindler F, Wang Z, Vergniory MG, Cook AM, Murani A, Sengupta S, Kasumov AY, Deblock R, Jeon S, Drozdov I et al (2018) Nat Phys 14(9):918

    Google Scholar 

  96. Xu Y, Song Z, Wang Z, Weng H, Dai X (2019) Phys Rev Lett 122:256402. https://doi.org/10.1103/PhysRevLett.122.256402

    Article  ADS  Google Scholar 

  97. Wang Z, Wieder BJ, Li J, Yan B, Bernevig BA (2019) Phys Rev Lett 123:186401. https://doi.org/10.1103/PhysRevLett.123.186401

    Article  ADS  Google Scholar 

  98. Ezawa M (2019) Sci Rep 9(1):5286

    ADS  Google Scholar 

  99. Yue C, Xu Y, Song Z, Weng H, Lu YM, Fang C, Dai X (2019) Nat Phys 15(6):577

    Google Scholar 

  100. Freed DS, Moore GW (2013) Annales Henri Poincaré, vol 14. Springer, pp 1927–2023

    Google Scholar 

  101. Read N (2017) Phys Rev B 95:115309. https://doi.org/10.1103/PhysRevB.95.115309

    Article  ADS  Google Scholar 

  102. Shiozaki K, Sato M, Gomi K (2017) Phys Rev B 95:235425. https://doi.org/10.1103/PhysRevB.95.235425

    Article  ADS  Google Scholar 

  103. Shiozaki K, **ong CZ, Gomi K (2018) ar**v:1810.00801

  104. Kruthoff J, de Boer J, van Wezel J, Kane CL, Slager RJ (2017) Phys Rev X 7:041069. https://doi.org/10.1103/PhysRevX.7.041069

    Article  Google Scholar 

  105. Bradlyn B, Elcoro L, Cano J, Vergniory M, Wang Z, Felser C, Aroyo M, Bernevig BA (2017) Nature 547(7663):298

    ADS  Google Scholar 

  106. Herring C (1937) Phys Rev 52:365. https://doi.org/10.1103/PhysRev.52.365

    Article  ADS  Google Scholar 

  107. Murakami S (2007) New J Phys 9(9):356

    Google Scholar 

  108. Murakami S, Kuga SI (2008) Phys Rev B 78:165313. https://doi.org/10.1103/PhysRevB.78.165313

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

    Article  ADS  Google Scholar 

  110. Wan X, Turner AM, Vishwanath A, Savrasov SY (2011) Phys Rev B 83:205101. https://doi.org/10.1103/PhysRevB.83.205101

    Article  ADS  Google Scholar 

  111. Volovik GE (2003) The universe in a helium droplet, vol 117. Oxford University Press on Demand

    Google Scholar 

  112. Yang KY, Lu YM, Ran Y (2011) Phys Rev B 84:075129. https://doi.org/10.1103/PhysRevB.84.075129

    Article  ADS  Google Scholar 

  113. Witczak-Krempa W, Kim YB (2012) Phys Rev B 85:045124. https://doi.org/10.1103/PhysRevB.85.045124

    Article  ADS  Google Scholar 

  114. Okugawa R, Murakami S (2014) Phys Rev B 89:235315. https://doi.org/10.1103/PhysRevB.89.235315

    Article  ADS  Google Scholar 

  115. Liu J, Vanderbilt D (2014) Phys Rev B 90:155316. https://doi.org/10.1103/PhysRevB.90.155316

    Article  ADS  Google Scholar 

  116. Soluyanov AA, Gresch D, Wang Z, Wu Q, Troyer M, Dai X, Bernevig BA (2015) Nature 527(7579):495

    ADS  Google Scholar 

  117. Sun Y, Wu SC, Ali MN, Felser C, Yan B (2015) Phys Rev B 92:161107. https://doi.org/10.1103/PhysRevB.92.161107

    Article  ADS  Google Scholar 

  118. Hirayama M, Okugawa R, Ishibashi S, Murakami S, Miyake T (2015) Phys Rev Lett 114:206401. https://doi.org/10.1103/PhysRevLett.114.206401

    Article  ADS  Google Scholar 

  119. Huang SM, Xu SY, Belopolski I, Lee CC, Chang G, Chang TR, Wang B, Alidoust N, Bian G, Neupane M, Sanchez D, Zheng H, Jeng HT, Bansil A, Neupert T, Lin H, Hasan MZ (2016) Proc Natl Acad Sci 113(5):1180. https://doi.org/10.1073/pnas.1514581113

    Article  ADS  Google Scholar 

  120. Lv BQ, Weng HM, Fu BB, Wang XP, Miao H, Ma J, Richard P, Huang XC, Zhao LX, Chen GF, Fang Z, Dai X, Qian T, Ding H (2015) Phys Rev X 5:031013. https://doi.org/10.1103/PhysRevX.5.031013

    Article  Google Scholar 

  121. Xu SY, Belopolski I, Alidoust N, Neupane M, Bian G, Zhang C, Sankar R, Chang G, Yuan Z, Lee CC, Huang SM, Zheng H, Ma J, Sanchez DS, Wang B, Bansil A, Chou F, Shibayev PP, Lin H, Jia S, Hasan MZ (2015) Science 349(6248):613. https://doi.org/10.1126/science.aaa9297

    Article  ADS  Google Scholar 

  122. Lv B, Xu N, Weng H, Ma J, Richard P, Huang X, Zhao L, Chen G, Matt C, Bisti F et al (2015) Nat Phys 11(9):724

    Google Scholar 

  123. Yang L, Liu Z, Sun Y, Peng H, Yang H, Zhang T, Zhou B, Zhang Y, Guo Y, Rahn M et al (2015) Nat Phys 11(9):728

    Google Scholar 

  124. Xu SY, Alidoust N, Belopolski I, Yuan Z, Bian G, Chang TR, Zheng H, Strocov VN, Sanchez DS, Chang G et al (2015) Nat Phys 11(9):748

    Google Scholar 

  125. Weng H, Fang C, Fang Z, Bernevig BA, Dai X (2015) Phys Rev X 5:011029. https://doi.org/10.1103/PhysRevX.5.011029

    Article  Google Scholar 

  126. Huang SM, Xu SY, Belopolski I, Lee CC, Chang G, Wang B, Alidoust N, Bian G, Neupane M, Zhang C et al (2015) Nat Commun 6:7373

    ADS  Google Scholar 

  127. Xu G, Weng H, Wang Z, Dai X, Fang Z (2011) Phys Rev Lett 107:186806. https://doi.org/10.1103/PhysRevLett.107.186806

    Article  ADS  Google Scholar 

  128. Wang Z, Vergniory MG, Kushwaha S, Hirschberger M, Chulkov EV, Ernst A, Ong NP, Cava RJ, Bernevig BA (2016) Phys Rev Lett 117:236401. https://doi.org/10.1103/PhysRevLett.117.236401

    Article  ADS  Google Scholar 

  129. Liu DF, Liang AJ, Liu EK, Xu QN, Li YW, Chen C, Pei D, Shi WJ, Mo SK, Dudin P, Kim T, Cacho C, Li G, Sun Y, Yang LX, Liu ZK, Parkin SSP, Felser C, Chen YL (2019) Science 365(6459):1282. https://doi.org/10.1126/science.aav2873

    Article  ADS  Google Scholar 

  130. Young SM, Zaheer S, Teo JCY, Kane CL, Mele EJ, Rappe AM (2012) Phys Rev Lett 108:140405. https://doi.org/10.1103/PhysRevLett.108.140405

    Article  ADS  Google Scholar 

  131. Yang BJ, Nagaosa N (2014) Nat Commun 5:4898

    ADS  Google Scholar 

  132. Wang Z, Sun Y, Chen XQ, Franchini C, Xu G, Weng H, Dai X, Fang Z (2012) Phys Rev B 85:195320. https://doi.org/10.1103/PhysRevB.85.195320

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  134. Liu ZK, Zhou B, Zhang Y, Wang ZJ, Weng HM, Prabhakaran D, Mo SK, Shen ZX, Fang Z, Dai X, Hussain Z, Chen YL (2014) Science 343(6173):864. https://doi.org/10.1126/science.1245085

    Article  ADS  Google Scholar 

  135. Neupane M, Xu SY, Sankar R, Alidoust N, Bian G, Liu C, Belopolski I, Chang TR, Jeng HT, Lin H et al (2014) Nat Commun 5:3786

    ADS  Google Scholar 

  136. Borisenko S, Gibson Q, Evtushinsky D, Zabolotnyy V, Büchner B, Cava RJ (2014) Phys Rev Lett 113:027603. https://doi.org/10.1103/PhysRevLett.113.027603

    Article  ADS  Google Scholar 

  137. Weng H, Liang Y, Xu Q, Yu R, Fang Z, Dai X, Kawazoe Y (2015) Phys Rev B 92:045108. https://doi.org/10.1103/PhysRevB.92.045108

    Article  ADS  Google Scholar 

  138. Chen Y, **e Y, Yang SA, Pan H, Zhang F, Cohen ML, Zhang S (2015) Nano Lett 15(10):6974

    ADS  Google Scholar 

  139. **e LS, Schoop LM, Seibel EM, Gibson QD, **e W, Cava RJ (2015) APL Mater 3(8):083602

    ADS  Google Scholar 

  140. Chan YH, Chiu CK, Chou MY, Schnyder AP (2016) Phys Rev B 93:205132. https://doi.org/10.1103/PhysRevB.93.205132

    Article  ADS  Google Scholar 

  141. Kim Y, Wieder BJ, Kane CL, Rappe AM (2015) Phys Rev Lett 115:036806. https://doi.org/10.1103/PhysRevLett.115.036806

    Article  ADS  Google Scholar 

  142. Yu R, Weng H, Fang Z, Dai X, Hu X (2015) Phys Rev Lett 115:036807. https://doi.org/10.1103/PhysRevLett.115.036807

    Article  ADS  Google Scholar 

  143. Schoop LM, Ali MN, Straßer C, Topp A, Varykhalov A, Marchenko D, Duppel V, Parkin SS, Lotsch BV, Ast CR (2016) Nat Commun 7:11696

    ADS  Google Scholar 

  144. Neupane M, Belopolski I, Hosen MM, Sanchez DS, Sankar R, Szlawska M, Xu SY, Dimitri K, Dhakal N, Maldonado P, Oppeneer PM, Kaczorowski D, Chou F, Hasan MZ, Durakiewicz T (2016) Phys Rev B 93:201104. https://doi.org/10.1103/PhysRevB.93.201104

    Article  ADS  Google Scholar 

  145. Hu J, Tang Z, Liu J, Liu X, Zhu Y, Graf D, Myhro K, Tran S, Lau CN, Wei J, Mao Z (2016) Phys Rev Lett 117:016602. https://doi.org/10.1103/PhysRevLett.117.016602

    Article  ADS  Google Scholar 

  146. Huang H, Liu J, Vanderbilt D, Duan W (2016) Phys Rev B 93:201114. https://doi.org/10.1103/PhysRevB.93.201114

    Article  ADS  Google Scholar 

  147. Li R, Ma H, Cheng X, Wang S, Li D, Zhang Z, Li Y, Chen XQ (2016) Phys Rev Lett 117:096401. https://doi.org/10.1103/PhysRevLett.117.096401

    Article  ADS  Google Scholar 

  148. Hirayama M, Okugawa R, Miyake T, Murakami S (2017) Nat Commun 8:14022

    ADS  Google Scholar 

  149. Haldane FDM, Raghu S (2008) Phys Rev Lett 100:013904. https://doi.org/10.1103/PhysRevLett.100.013904

    Article  ADS  Google Scholar 

  150. Raghu S, Haldane FDM (2008) Phys Rev A 78:033834. https://doi.org/10.1103/PhysRevA.78.033834

    Article  ADS  Google Scholar 

  151. Wang Z, Chong YD, Joannopoulos JD, Soljačić M (2008) Phys Rev Lett 100:013905. https://doi.org/10.1103/PhysRevLett.100.013905

    Article  ADS  Google Scholar 

  152. Wang Z, Chong Y, Joannopoulos JD, Soljačić M (2009) Nature 461(7265):772

    ADS  Google Scholar 

  153. Hafezi M, Demler EA, Lukin MD, Taylor JM (2011) Nat Phys 7(11):907

    Google Scholar 

  154. Khanikaev AB, Mousavi SH, Tse WK, Kargarian M, MacDonald AH, Shvets G (2013) Nat Mater 12(3):233

    ADS  Google Scholar 

  155. Hafezi M, Mittal S, Fan J, Migdall A, Taylor J (2013) Nat Photonics 7(12):1001

    ADS  Google Scholar 

  156. Mittal S, Fan J, Faez S, Migdall A, Taylor JM, Hafezi M (2014) Phys Rev Lett 113:087403. https://doi.org/10.1103/PhysRevLett.113.087403

    Article  ADS  Google Scholar 

  157. Ma T, Khanikaev AB, Mousavi SH, Shvets G (2015) Phys Rev Lett 114:127401. https://doi.org/10.1103/PhysRevLett.114.127401

    Article  ADS  Google Scholar 

  158. Mittal S, Ganeshan S, Fan J, Vaezi A, Hafezi M (2016) Nat Photonics 10(3):180

    ADS  Google Scholar 

  159. Slobozhanyuk A, Mousavi SH, Ni X, Smirnova D, Kivshar YS, Khanikaev AB (2017) Nat Photonics 11(2):130

    ADS  Google Scholar 

  160. Yang Y, Gao Z, Xue H, Zhang L, He M, Yang Z, Singh R, Chong Y, Zhang B, Chen H (2019) Nature 565(7741):622

    ADS  Google Scholar 

  161. Lu L, Fu L, Joannopoulos JD, Soljačić M (2013) Nat Photonics 7(4):294

    ADS  Google Scholar 

  162. Ochiai T (2017) Phys Rev A 96:043842. https://doi.org/10.1103/PhysRevA.96.043842

    Article  ADS  Google Scholar 

  163. Lu L, Gao H, Wang Z (2018) Nat Commun 9(1):1

    ADS  Google Scholar 

  164. Onose Y, Ideue T, Katsura H, Shiomi Y, Nagaosa N, Tokura Y (2010) Science 329(5989):297. https://doi.org/10.1126/science.1188260

    Article  ADS  Google Scholar 

  165. Mook A, Henk J, Mertig I (2014) Phys Rev B 89:134409. https://doi.org/10.1103/PhysRevB.89.134409

    Article  ADS  Google Scholar 

  166. Lee H, Han JH, Lee PA (2015) Phys Rev B 91:125413. https://doi.org/10.1103/PhysRevB.91.125413

    Article  ADS  Google Scholar 

  167. Hirschberger M, Chisnell R, Lee YS, Ong NP (2015) Phys Rev Lett 115:106603. https://doi.org/10.1103/PhysRevLett.115.106603

    Article  ADS  Google Scholar 

  168. Chisnell R, Helton JS, Freedman DE, Singh DK, Bewley RI, Nocera DG, Lee YS (2015) Phys Rev Lett 115:147201. https://doi.org/10.1103/PhysRevLett.115.147201

    Article  ADS  Google Scholar 

  169. Nakata K, Kim SK, Klinovaja J, Loss D (2017) Phys Rev B 96:224414. https://doi.org/10.1103/PhysRevB.96.224414

    Article  ADS  Google Scholar 

  170. Li FY, Li YD, Kim YB, Balents L, Yu Y, Chen G (2016) Nat Commun 7:12691

    ADS  Google Scholar 

  171. Mook A, Henk J, Mertig I (2016) Phys Rev Lett 117:157204. https://doi.org/10.1103/PhysRevLett.117.157204

    Article  ADS  Google Scholar 

  172. Mook A, Henk J, Mertig I (2017) Phys Rev B 95:014418. https://doi.org/10.1103/PhysRevB.95.014418

    Article  ADS  Google Scholar 

  173. Li K, Li C, Hu J, Li Y, Fang C (2017) Phys Rev Lett 119:247202. https://doi.org/10.1103/PhysRevLett.119.247202

    Article  ADS  Google Scholar 

  174. Yao W, Li C, Wang L, Xue S, Dan Y, Iida K, Kamazawa K, Li K, Fang C, Li Y (2018) Nat Phys 14(10):1011

    Google Scholar 

  175. Zhang L, Ren J, Wang JS, Li B (2010) Phys Rev Lett 105:225901. https://doi.org/10.1103/PhysRevLett.105.225901

    Article  ADS  Google Scholar 

  176. Kane C, Lubensky T (2014) Nat Phys 10(1):39

    Google Scholar 

  177. Yang Z, Gao F, Shi X, Lin X, Gao Z, Chong Y, Zhang B (2015) Phys Rev Lett 114:114301. https://doi.org/10.1103/PhysRevLett.114.114301

    Article  ADS  Google Scholar 

  178. Wang P, Lu L, Bertoldi K (2015) Phys Rev Lett 115:104302. https://doi.org/10.1103/PhysRevLett.115.104302

    Article  ADS  Google Scholar 

  179. **ao M, Chen WJ, He WY, Chan CT (2015) Nat Phys 11(11):920

    Google Scholar 

  180. Süsstrunk R, Huber SD (2015) Science 349(6243):47

    ADS  Google Scholar 

  181. Mousavi SH, Khanikaev AB, Wang Z (2015) Nat Commun 6:8682

    ADS  Google Scholar 

  182. Nash LM, Kleckner D, Read A, Vitelli V, Turner AM, Irvine WT (2015) Proc Natl Acad Sci 112(47):14495

    ADS  Google Scholar 

  183. Huber SD (2016) Nat Phys 12(7):621

    Google Scholar 

  184. Rocklin DZ, Chen BG, Falk M, Vitelli V, Lubensky TC (2016) Phys Rev Lett 116:135503. https://doi.org/10.1103/PhysRevLett.116.135503

    Article  ADS  Google Scholar 

  185. Süsstrunk R, Huber SD (2016) Proc Natl Acad Sci 113(33):E4767

    ADS  Google Scholar 

  186. Liu Y, Xu Y, Zhang SC, Duan W (2017) Phys Rev B 96:064106. https://doi.org/10.1103/PhysRevB.96.064106

    Article  ADS  Google Scholar 

  187. Zhang T, Song Z, Alexandradinata A, Weng H, Fang C, Lu L, Fang Z (2018) Phys Rev Lett 120:016401. https://doi.org/10.1103/PhysRevLett.120.016401

    Article  ADS  Google Scholar 

  188. Weick G, Woollacott C, Barnes WL, Hess O, Mariani E (2013) Phys Rev Lett 110:106801. https://doi.org/10.1103/PhysRevLett.110.106801

    Article  ADS  Google Scholar 

  189. ** D, Lu L, Wang Z, Fang C, Joannopoulos JD, Soljačić M, Fu L, Fang NX (2016) Nat Commun 7:13486

    ADS  Google Scholar 

  190. ** D, Christensen T, Soljačić M, Fang NX, Lu L, Zhang X (2017) Phys Rev Lett 118:245301. https://doi.org/10.1103/PhysRevLett.118.245301

    Article  ADS  Google Scholar 

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    Dr. Heejae Kim

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Kim, H. (2022). Topology, Symmetry, and Band Theory of Materials. In: Glide-Symmetric Z2 Magnetic Topological Crystalline Insulators. Springer Theses. Springer, Singapore. https://doi.org/10.1007/978-981-16-9077-8_2

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