Stiefel-Whitney classes and topological phases in band theory

doi:10.1088/1674-1056/ab4d3b

Abstract We review the recent progress in the study of topological phases in systems with space-time inversion symmetry I_ST. I_ST is an anti-unitary symmetry which is local in momentum space and satisfies I_ST²=1 such as PT in two dimensions (2D) and three dimensions (3D) without spin-orbit coupling and C₂T in 2D with or without spin-orbit coupling, where P, T, C₂ indicate the inversion, time-reversal, and two-fold rotation symmetries, respectively. Under I_ST, the Hamiltonian and the periodic part of the Bloch wave function can be constrained to be real-valued, which makes the Berry curvature and the Chern number vanish. In this class of systems, gapped band structures of real wave functions can be topologically distinguished by the Stiefel-Whitney numbers instead. The first and second Stiefel-Whitney numbers w₁ and w₂, respectively, are the corresponding invariants in 1D and 2D, which are equivalent to the quantized Berry phase and the Z₂ monopole charge, respectively. We first describe the topological phases characterized by the first Stiefel-Whitney number, including 1D topological insulators with quantized charge polarization, 2D Dirac semimetals, and 3D nodal line semimetals. Next we review how the second Stiefel-Whitney class characterizes the 3D nodal line semimetals carrying a Z₂ monopole charge. In particular, we explain how the second Stiefel-Whitney number w₂, the Z₂ monopole charge, and the linking number between nodal lines are related. Finally, we review the properties of 2D and 3D topological insulators characterized by the nontrivial second Stiefel Whitney class.

Keywords: topological semimetal

Received: 30 March 2019
Revised: 30 September 2019
Accepted manuscript online:

PACS:	71.20.-b	(Electron density of states and band structure of crystalline solids)
	73.20.-r	(Electron states at surfaces and interfaces)

Corresponding Authors: Bohm-Jung Yang
E-mail: bjyang@snu.ac.kr

Cite this article:

Junyeong Ahn, Sungjoon Park, Dongwook Kim, Youngkuk Kim, Bohm-Jung Yang Stiefel-Whitney classes and topological phases in band theory 2019 Chin. Phys. B 28 117101

[32]	Konig M, Wiedmann S, Brune C, Roth A, Buhmann H, Molenkamp L W, Qi X L and Zhang S C 2007 Science 318 766
[1]	Hasan M Z and Kane C L 2010 Rev. Mod. Phys. 82 3045
[33]	Kim J, Baik S S, Jung S W, Sohn Y, Ryu S H, Choi H J, Yang B J and Kim K S 2017 Phys. Rev. Lett. 119 226801
[2]	Armitage N, Mele E and Vishwanath A 2018 Rev. Mod. Phys. 90 015001
[34]	Park S and Yang B J 2017 Phys. Rev. B 96 125127
[3]	Murakami S 2007 New Journal of Physics 9 356
[35]	Chen Y, Xie Y, Yang S A, Pan H, Zhang F, Cohen M L and Zhang S 2015 Nano Lett. 15 6974
[4]	Wan X, Turner A M, Vishwanath A and Savrasov S Y 2011 Physical Review B 83 205101
[36]	Weng H, Liang Y, Xu Q, Yu R, Fang Z, Dai X and Kawazoe Y 2015 Phys. Rev. B 92 045108
[5]	Fang C and Fu L 2015 Phys. Rev. B 91 161105
[37]	Xie L S, Schoop L M, Seibel E M, Gibson Q D, Xie W and Cava R J 2015 Apl Mater. 3 083602
[6]	Ahn J and Yang B J 2017 Phys. Rev. Lett. 118 156401
[38]	Chan Y H, Chiu C K, Chou M Y and Schnyder A P 2016 Phys. Rev. B 93 205132
[7]	Ahn J, Kim D, Kim Y and Yang B J 2018 Phys. Rev. Lett. 121 106403
[39]	Kim Y, Wieder B J, Kane C L and Rappe A M 2015 Phys. Rev. Lett. 115 036806
[40]	Yu R, Weng H, Fang Z, Dai X and Hu X 2015 Phys. Rev. Lett. 115 036807
[8]	Bouhon A, Slager R J and Bzdusek T 2019 arXiv:1907.10611
[41]	Fang C, Weng H, Dai X and Fang Z 2016 Chin. Phys. B 25 117106
[9]	Nakahara M 2003 Geometry, Topology and Physics (Philadelphia:CRC Press)
[42]	Fang C, Chen Y, Kee H Y and Fu L 2015 Phys. Rev. B 92 081201
[10]	Hatcher A Vector Bundles and K-Theory (unpublished)
[43]	Morimoto T and Furusaki A 2014 Phys. Rev. B 89 235127
[11]	Hatcher A 2002 Algebraic Topology (Cambridge:Cambridge University Press)
[44]	Zhao Y X and Lu Y 2017 Phys. Rev. Lett. 118 056401
[12]	Shiozaki K, Sato M and Gomi K 2017 Phys. Rev. B 95 235425
[45]	Song Z, Zhang T and Fang C 2018 Phys. Rev. X 8 031069
[13]	Bzdušek T and Sigrist M 2017 Phys. Rev. B 96 155105
[46]	Prasolov V V 2006 Elements of combinatorial and differential topology Vol. 74 (American Mathematical Soc.)
[14]	Ahn J, Park S and Yang B J 2019 Phys. Rev. X 9 021013
[47]	Chooquet-Bruhat Y 2000 Analysis, Manifolds, and Physics-Part II (Amsterdam:Elsevier)
[15]	Po H C, Watanabe H and Vishwanath A 2018 Phys. Rev. Lett. 121 126402
[48]	Yu R, Qi X L, Bernevig B A, Fang Z and Dai X 2011 Phys. Rev. B 84 075119
[16]	Cano J, Bradlyn B, Wang Z, Elcoro L, Vergniory M G, Felser C, Aroyo M I and Bernevig B A 2018 Phys. Rev. Lett. 120 266401
[49]	Alexandradinata A, Dai X and Bernevig B A 2014 Phys. Rev. B 89 155114
[50]	Alexandradinata A, Wang Z and Bernevig B A 2016 Phys. Rev. X 6 021008
[17]	Bouhon A, Black-Schaffer A M and Slager R J 2018 arXiv:1804.09719
[51]	Soluyanov A A and Vanderbilt D 2012 Physical Review B 85 115415
[18]	Wang Z, Wieder B J, Li J, Yan B and Bernevig B A 2019 Phys. Rev. Lett. 123 186401
[52]	Nomura T, Habe T, Sakamoto R and Koshino M 2018 Physical Review Materials 2 054204
[19]	Bradlyn B, Wang Z, Cano J and Bernevig B A 2019 Phys. Rev. B 99 045140
[53]	Ramamurthy S T and Hughes T L 2015 Physical Review B 92 085105
[20]	Song Z, Wang Z, Shi W, Li G, Fang C and Bernevig B A 2019 Phys. Rev. Lett. 123 036401
[54]	Zhao Y X, Schnyder A P and Wang Z D 2016 Phys. Rev. Lett. 116 156402
[21]	Po H C, Zou L, Senthil T and Vishwanath A 2019 Phys. Rev. B 99 195455
[22]	Liu S, Vishwanath A and Khalaf E 2019 Phys. Rev. X 9 031003
[55]	Wieder B J and Bernevig B A 2018 arXiv:1810.02373
[23]	Ahn J and Yang B J 2019 Phys. Rev. B 99 235125
[56]	Zhang F, Kane C L and Mele E J 2013 Phys. Rev. Lett. 110 046404
[24]	Kirby R and Taylor L 1991 Pin structures on low-dimensional manifolds (London Mathematical Society Lecture Note Series Vol. 2) (Cambridge:Cambridge University Press)
[25]	Su W, Schrieffer J and Heeger A J 1979 Physical review letters 42 1698
[57]	Fang C and Fu L 2017 arXiv:1709.01929
[26]	Serbyn M and Fu L 2014 Phys. Rev. B 90 035402
[58]	Khalaf E 2018 Phys. Rev. B 97 205136
[27]	Hsieh T H, Lin H, Liu J, Duan W, Bansil A and Fu L 2013 Nat. Commun. 4 1901
[59]	Kooi S H, van Miert G and Ortix C 2018 Phys. Rev. B 98 245102
[28]	Fang C, Gilbert M J, Xu S Y, Bernevig B A and Hasan M Z 2013 Phys. Rev. B 88 125141
[60]	Varnava N and Vanderbilt D 2018 Phys. Rev. B 98 245117
[29]	Young S M and Kane C L 2015 Phys. Rev. Lett. 115 126803
[61]	van Miert G and Ortix C 2018 Phys. Rev. B 98 081110
[30]	Wieder B and Kane C L 2016 Phys. Rev. B 94 155108
[62]	Yue C, Xu Y, Song Z, Lu Y M, Weng H, Fang C and Dai X 2019 Nat. Phys. 15 577
[31]	Bernevig B A, Hughes T L and Zhang S C 2006 Science 314 1757
[63]	Schindler F, Cook A M, Vergniory M G, Wang Z, Parkin S S P, Bernevig B A and Neupert T 2018 Sci. Adv. 4 eaat0346
[32]	Konig M, Wiedmann S, Brune C, Roth A, Buhmann H, Molenkamp L W, Qi X L and Zhang S C 2007 Science 318 766
[64]	Ezawa M 2018 Phys. Rev. B 97 241402
[33]	Kim J, Baik S S, Jung S W, Sohn Y, Ryu S H, Choi H J, Yang B J and Kim K S 2017 Phys. Rev. Lett. 119 226801
[65]	Ezawa M 2018 Phys. Rev. B 97 155305
[34]	Park S and Yang B J 2017 Phys. Rev. B 96 125127
[66]	Qi X L, Hughes T L and Zhang S C 2008 Phys. Rev. B 78 195424
[35]	Chen Y, Xie Y, Yang S A, Pan H, Zhang F, Cohen M L and Zhang S 2015 Nano Lett. 15 6974
[67]	Wang Z, Qi X L and Zhang S C 2010 New. J. Phys. 12 065007
[36]	Weng H, Liang Y, Xu Q, Yu R, Fang Z, Dai X and Kawazoe Y 2015 Phys. Rev. B 92 045108
[68]	Wu Q, Soluyanov A A and bzdušek T 2019 Science 365 1273
[37]	Xie L S, Schoop L M, Seibel E M, Gibson Q D, Xie W and Cava R J 2015 Apl Mater. 3 083602
[38]	Chan Y H, Chiu C K, Chou M Y and Schnyder A P 2016 Phys. Rev. B 93 205132
[39]	Kim Y, Wieder B J, Kane C L and Rappe A M 2015 Phys. Rev. Lett. 115 036806
[40]	Yu R, Weng H, Fang Z, Dai X and Hu X 2015 Phys. Rev. Lett. 115 036807
[41]	Fang C, Weng H, Dai X and Fang Z 2016 Chin. Phys. B 25 117106
[42]	Fang C, Chen Y, Kee H Y and Fu L 2015 Phys. Rev. B 92 081201
[43]	Morimoto T and Furusaki A 2014 Phys. Rev. B 89 235127
[44]	Zhao Y X and Lu Y 2017 Phys. Rev. Lett. 118 056401
[45]	Song Z, Zhang T and Fang C 2018 Phys. Rev. X 8 031069
[46]	Prasolov V V 2006 Elements of combinatorial and differential topology Vol. 74 (American Mathematical Soc.)
[47]	Chooquet-Bruhat Y 2000 Analysis, Manifolds, and Physics-Part II (Amsterdam:Elsevier)
[48]	Yu R, Qi X L, Bernevig B A, Fang Z and Dai X 2011 Phys. Rev. B 84 075119
[49]	Alexandradinata A, Dai X and Bernevig B A 2014 Phys. Rev. B 89 155114
[50]	Alexandradinata A, Wang Z and Bernevig B A 2016 Phys. Rev. X 6 021008
[51]	Soluyanov A A and Vanderbilt D 2012 Physical Review B 85 115415
[52]	Nomura T, Habe T, Sakamoto R and Koshino M 2018 Physical Review Materials 2 054204
[53]	Ramamurthy S T and Hughes T L 2015 Physical Review B 92 085105
[54]	Zhao Y X, Schnyder A P and Wang Z D 2016 Phys. Rev. Lett. 116 156402
[55]	Wieder B J and Bernevig B A 2018 arXiv:1810.02373
[56]	Zhang F, Kane C L and Mele E J 2013 Phys. Rev. Lett. 110 046404
[57]	Fang C and Fu L 2017 arXiv:1709.01929
[58]	Khalaf E 2018 Phys. Rev. B 97 205136
[59]	Kooi S H, van Miert G and Ortix C 2018 Phys. Rev. B 98 245102
[60]	Varnava N and Vanderbilt D 2018 Phys. Rev. B 98 245117
[61]	van Miert G and Ortix C 2018 Phys. Rev. B 98 081110
[62]	Yue C, Xu Y, Song Z, Lu Y M, Weng H, Fang C and Dai X 2019 Nat. Phys. 15 577
[63]	Schindler F, Cook A M, Vergniory M G, Wang Z, Parkin S S P, Bernevig B A and Neupert T 2018 Sci. Adv. 4 eaat0346
[64]	Ezawa M 2018 Phys. Rev. B 97 241402
[65]	Ezawa M 2018 Phys. Rev. B 97 155305
[66]	Qi X L, Hughes T L and Zhang S C 2008 Phys. Rev. B 78 195424
[67]	Wang Z, Qi X L and Zhang S C 2010 New. J. Phys. 12 065007
[68]	Wu Q, Soluyanov A A and bzdušek T 2019 Science 365 1273

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