Higher-order topological insulators and superconductors

Yan Zhong-Bo

doi:10.7498/aps.68.20191101

Abstract

Very recently, higher-order topological insulators and superconductors have attracted wide attention and aroused the great interest of researchers. Owing to their unconventional bulk-boundary correspondence, higher-order topological insulators and superconductors possess novel boundary modes whose dimensions are always lower than the first-order (or say conventional) topological insulators and superconductors, provided that their bulk dimensions are the same. The essence of higher-order topological phase is the formation of Dirac-mass domain walls on the gapped one-dimensional lower boundary. Roughly speaking, the origins of the formation can be classified as " intrinsic” and " extrinsic” type. For the former case, the formation of domain walls is forced by symmetry, suggesting that the resulting higher-order topological phases can be taken as topological crystalline phases. For this case, the domain walls are quite robust if the corresponding symmetry is preserved. For the latter case, the domain walls are formed simply because the one-dimensional lower boundary modes are gapped in a nontrivial way, however, the nontrivial way is not forced by symmetry. For this case, the domain walls are also stable against perturbations as long as the separations between them are large enough. The domain walls can have various patterns, which indicates that the higher-order topological phases are very rich. In this paper, we first reveal the connection between the higher-order topological phase and the first-order topological phase. Concretely, we show how to realize a higher-order topological phase by breaking some symmetries of a first-order topological phase, as well as stacking lower-dimensional first-order topological systems in an appropriate way. After these, we review the recent progress of theoretical and experimental study of higher-order topological insulators and superconductors. For the higher-order topological insulators, we find that the electronic materials are still laking though a lot of experimental realizations have been achieved. For higher-order topological superconductors, we find that their experimental realization and investigation are still in the very primary stage though quite a lot of relevant theoretical studies have been carried out. In order to comprehensively understand this newly-emerging field there are still many things to be done.

Keywords:

higher-order topological insulator /
higher-order topological superconductor /
bulk-boundary correspondence

Authors and contacts

Yan Zhong-Bo

School of Physics, Sun Yat-sen University, Guangzhou 510275, China

Corresponding author: Yan Zhong-Bo, yanzhb5@mail.sysu.edu.cn

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References

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Cited By

图 1 拓扑物态的边界态示意图　$n = 1$的行对应传统的拓扑物态, 其具有比系统维度低一维的无能隙边界态; $n \geqslant 2$的行对应高阶拓扑物态, 其具有比维度低n维的无能隙边界态

Figure 1. A schematic diagram of the boundary modes of topological matter. The line with $n = 1$ corresponds to conventional topological matters which host gapless modes whose dimensions are one-dimensional lower than the system dimension. The lines with $n \geqslant 2$ correspond to higher-order topological matters which host gapless modes whose dimensions are n-dimensional lower than the system dimension.

DownLoad: Full-Size Img PowerPoint

图 2 从一阶拓扑绝缘体到二阶拓扑绝缘体　(a) 沿x方向取开放边界条件(${L_x} = 100$), 沿y方向取周期边界条件, 参数为$M = B = A = 1$, $\varLambda = 0$, 对应BHZ模型, 能谱反映出无能隙边界态的存在; (b) 插图中红点对应的能量本征态的波函数分布, 参数同(a), 但沿x和y两个方向均取开放边界条件; 红色的深浅对应波函数分布概率的大小, 可以看出对一阶拓扑绝缘体, 波函数分布在整个边界上; (c) 边界条件和参数同(a), 除了此处$\varLambda = 0.5$, 可看出Λ项的出现让边界态打开了能隙; (d) 零模的波函数分布, 参数同(c), 但沿x和y两个方向均取开放边界条件; 从插图中可发现存在四个零模, 这四个零模的波函数局域在四个角上

Figure 2. From first-order topological insulator to second-order topological insulator. (a) Energy spectra for a sample with open boundary condition in the x direction (the system size ${L_x} = 100$) and periodic boundary condition in the y direction. Parameters are $M = B = A = 1$, $\varLambda = 0$, which corresponds to the original BHZ model. The energy spectra reflect the existence of gapless boundary modes. (b) the density profile of a boundary mode. The parameters are the same as in (a), but now open boundary conditions are taken both in the x and y directions. One can see that the density profile of the boundary mode distributes over the whole boundary. (c) the boundary conditions and parameters are the same as in (a), except now $\varLambda = 0.5$. One can see that the presence of the Λ term opens a gap for the boundary modes. (d) the density profiles of zero modes.The parameters are the same as in (c), but now open boundary conditions are taken both in the x and y directions. One can see that there are four zero-energy modes in the inset. Their wave functions are found to be localized around the corners.

DownLoad: Full-Size Img PowerPoint

图 3 从一维一阶拓扑绝缘体到二维二阶拓扑绝缘体　(a) 一维SSH链的示意图; (b) 利用一维SSH链构造二维二阶拓扑绝缘体, 每个单位元胞中有一个${\text{π}}$磁通

Figure 3. Constructing two-dimensional second-order topological insulator by using one-dimensional topological insulator: (a) A schematic diagram of the SSH chain; (b) using the one-dimensional SSH chains to construct a two-dimensional second-order topological insulator, within each small square, there is a ${\text{π}}$-flux.

DownLoad: Full-Size Img PowerPoint

[1]	Kosterlitz J M, Thouless D J 1973 J. Phys. C 6 1181 Google Scholar
[2]	Klitzing K V, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494 Google Scholar
[3]	Tsui D C, Stormer H L, Gossard A C 1982 Phys. Rev. Lett. 48 1559 Google Scholar
[4]	Wen X G 2017 Rev. Mod. Phys. 89 041004
[5]	Thouless D J, Kohmoto M, Nightingale M P, den Nijs M 1982 Phys. Rev. Lett. 49 405 Google Scholar
[6]	Klitzing K V 2019 Phys. Rev. Lett. 122 200001 Google Scholar
[7]	Wen X G 2004 Quantum Field Theory of Many-body Systems: from the Origin of Sound to an Origin of Light and Electrons (1st Ed.) (New York: Oxford University Press) pp5–9
[8]	Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 226801 Google Scholar
[9]	Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 146802 Google Scholar
[10]	Bernevig B A, Zhang S C 2006 Phys. Rev. Lett. 96 106802 Google Scholar
[11]	Bernevig B A, Hughes T L, Zhang S C 2006 Science 314 1757 Google Scholar
[12]	Koing M, Wiedmann S, Brune C, Roth A, Buhmann H, Molenkamp L W, Qi X L, Zhang S C 2007 Science 318 766 Google Scholar
[13]	Fu L, Kane C L, Mele E J 2007 Phys. Rev. Lett. 98 106803 Google Scholar
[14]	Moore J E, Balents L 2007 Phys. Rev. B 75 121306 Google Scholar
[15]	Roy R 2009 Phys. Rev. B 79 195322 Google Scholar
[16]	Kitaev A Y 2001 Physics-Uspekhi 44 131 Google Scholar
[17]	Ivanov D A 2001 Phys. Rev. Lett. 86 268 Google Scholar
[18]	Kitaev A Y 2003 Ann. Phys. 303 2 Google Scholar
[19]	Nayak C, Simon S H, Stern A, Freedman M, Sarma S D 2008 Rev. Mod. Phys. 80 1083
[20]	Sarma S D, Freedman M, Nayak C 2015 npj Quantum Information 1 15001 Google Scholar
[21]	Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057 Google Scholar
[22]	Alicea J 2012 Rep. Prog. Phys. 75 076501 Google Scholar
[23]	Beenakker C W J 2013 Annu. Rev. Condens. Matter Phys. 4 113 Google Scholar
[24]	Stanescu T D, Tewari S 2013 J. Phys: Condens. Matter 25 233201 Google Scholar
[25]	Schnyder A P, Ryu S, Furusaki A, Ludwig A W W 2008 Phys. Rev. B 78 195125 Google Scholar
[26]	Kitaev A Y 2009 AIP Conference Proceedings 1134 22
[27]	Fu L 2011 Phys. Rev. Lett. 106 106802 Google Scholar
[28]	Zhang T T, Jiang Y, Song Z D, Huang H, He Y Q, Fang Z, Weng H M, Fang C 2019 Nature 566 475 Google Scholar
[29]	Vergniory M G, Elcoro L, Felser C, Regnault N, Bernevig B A, Wang Z J 2019 Nature 566 480
[30]	Feng T, Po H C, Vishwanath A, Wan X G 2019 Nature 566 486 Google Scholar
[31]	Benalcazar W A, Bernevig B A, Hughes T L 2017 Science 357 61 Google Scholar
[32]	Schindler F, Cook A M, Vergniory M G, Wang Z J, Parkin S S P, Bernevig B A, Neupert T 2018 Science Advances 4 eaat0346 Google Scholar
[33]	Song Z D, Fang Z, Fang C 2017 Phys. Rev. Lett. 119 246402
[34]	Langbehn J, Peng Y, Trifunovic L, Oppen F V, Brouwer P W 2017 Phys. Rev. Lett. 119 246401
[35]	Benalcazar W A, Bernevig B A, Hughes T L 2017 Phys. Rev. B 96 245115 Google Scholar
[36]	Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045 Google Scholar
[37]	Fu L, Kane C L 2007 Phys. Rev. B 76 045302 Google Scholar
[38]	Yan Z B, Song F, Wang Z 2018 Phys. Rev. Lett. 121 096803 Google Scholar
[39]	Jackiw R, Rebbi C 1976 Phys. Rev. D 13 3398 Google Scholar
[40]	Su W P, Schrieffer J R, Heeger A J 1979 Phys. Rev. Lett. 42 1698 Google Scholar
[41]	Matsugatani A, Watanabe H 2018 Phys. Rev. B 98 205129 Google Scholar
[42]	Sitte M, Rosch A, Altman E, Fritz L 2012 Phys. Rev. Lett. 108 126807 Google Scholar
[43]	Zhang F, Kane C L, Mele E J 2013 Phys. Rev. Lett. 110 046404 Google Scholar
[44]	Khalaf E 2018 Phys. Rev. B 97 205136 Google Scholar
[45]	Geier M, Trifunovic L, Hoskam M, Brouwer P W 2018 Phys. Rev. B 97 205135 Google Scholar
[46]	Franca S, van den Brink J, Fulga I C 2018 Phys. Rev. B 98 201114 Google Scholar
[47]	Ezawa M 2018 Phys. Rev. Lett. 120 026801 Google Scholar
[48]	Miert G V, Ortix C 2018 Phys. Rev. B 98 081110 Google Scholar
[49]	Ezawa M 2018 Phys. Rev. B 98 045125 Google Scholar
[50]	Calugaru D, Juricic V, Roy B 2019 Phys. Rev. B 99 041301 Google Scholar
[51]	Trifunovic L, Brouwer P W 2019 Phys. Rev. X 9 011012
[52]	Ahn J, Yang B J 2018 arXiv: 1810.05363 [cond-mat.mes-hall]
[53]	Kudo K, Yoshida T, Hatsugai Y 2019 arXiv: 1905.03484 [cond-mat.str-el]
[54]	Peterson C W, Benalcazar W A, Hughes T L, Bahl G 2018 Nature 555 346 Google Scholar
[55]	Serra-Garcia M, Peri V, Susstrunk R, Bilal O R, Larsen T, Villanueva L G, Huber S D 2018 Nature 555 342 Google Scholar
[56]	Imhof S, Berger C, Bayer F, Brehm J, Molenkamp L W, Kiessling T, Schindler F, Lee C H, Greiter M, Neupert T, Thomale R 2018 Nat. Phys. 14 925 Google Scholar
[57]	Xue H R, Yang Y H, Gao F, Chong Y D, Zhang B L 2019 Nature Materials 18 108 Google Scholar
[58]	Ni X, Weiner M, Alu A, Khanikaev A B 2019 Nature Materials 18 113 Google Scholar
[59]	Zhang X J, Wang H X, Lin Z K, Tian Y, Xie B Y, Lu M H, Chen Y F, Jiang J H 2019 Nat. Phys. 15 582 Google Scholar
[60]	Xie B Y, Wang H F, Wang H X, Zhu X Y, Jiang J H, Lu M H, Chen Y F 2018 Phys. Rev. B 98 205147 Google Scholar
[61]	Xie B Y, Su G X, Wang H F, Su H, Shen X P, Zhan P, Lu M H, Wang Z L, Chen Y F 2019 Phys. Rev. Lett. 122 233903 Google Scholar
[62]	Chen X D, Deng W M, Shi F L, Zhao F L, Chen M, Dong J W 2019 Phys. Rev. Lett. 122 233902 Google Scholar
[63]	Fan H Y, Xia B Z, Tong L, Zheng S J, Yu D J 2019 Phys. Rev. Lett. 122 204301 Google Scholar
[64]	Schindler F, Wang Z J, Vergniory M G, Cook A M, Murani A, Sengupta S, Kasumov A Y, Deblock R, Jeon S, Drozdov I, Bouchiat H, Gueron S, Yazdani A, Bernevig B A, Nepert T 2018 Nat. Phys. 14 918 Google Scholar
[65]	Yue C M, Xu Y F, Song Z D, Weng H M, Lu Y M, Fang C, Dai X 2019 Nat. Phys. 15 577 Google Scholar
[66]	Xu Y F, Song Z D, Wang Z J, Weng H M, Dai X 2019 Phys. Rev. Lett. 122 256402 Google Scholar
[67]	Wang Z J, Wieder B J, Li J, Yan B H, Bernevig B A 2018 arXiv: 1806.11116 [cond-mat.mltr-sci]
[68]	Sheng X L, Chen C, Liu H Y, Chen Z Y, Zhao Y X, Yu Z M, Yang S Y A 2019 arXiv: 1904.09985 [cond-mat.mes-hall]
[69]	Lee E, Kim R, Ahn J, Yang B J 2019 arXiv: 1904.11452 [cond-mat.mltr-sci]
[70]	Kempkes S N, Slot M R, van den Broeke J J, Capiod P, Benalcazar W A, Vanmaekelbergh D, Bercioux D, Swart I, Smith C M 2019 arXiv: 1905.06053 [cond-mat.mes-hall]
[71]	Zhu X Y 2018 Phys. Rev. B 97 205134 Google Scholar
[72]	Qi X L, Hughes T L, Raghu S, Zhang S C 2009 Phys. Rev. Lett. 102 187001 Google Scholar
[73]	Wang Q Y, Liu C C, Lu Y M, Zhang F 2018 Phys. Rev. Lett. 121 186801 Google Scholar
[74]	Zhang R X, Cole W S, Sarma S D 2019 Phys. Rev. Lett. 122 187001 Google Scholar
[75]	Liu T, He J J, Nori F 2018 Phys. Rev. B 98 245413 Google Scholar
[76]	Wu Y J, Hou J P, Li Y M, Luo X W, Zhang C W 2019 arXiv: 1905.08896 [cond-mat.mes-hall]
[77]	Pan X H, Yang K J, Chen L, Xu G, Liu C X, Liu X 2018 arXiv: 1812.10989 [cond-mat.mes-hall]
[78]	Zhang R X, Cole W S, Wu X X, Sarma D S 2019 arXiv: 1905.10647 [cond-mat.supr-con]
[79]	Wu X X, Liu X, Thomale R, Liu C X 2019 arXiv: 1905.10648 [cond-mat.supr-con]
[80]	Peng Y, Xu Y 2019 Phys. Rev. B 99 195431 Google Scholar
[81]	Yan Z B 2019 arXiv: 1907.02070 [cond-mat.mes-hall]
[82]	Hsu C H, Stano P, Klinovaja J, Loss D 2018 Phys. Rev. Lett. 121 196801 Google Scholar
[83]	Queiroz R, Stern A 2019 Phys. Rev. Lett. 123 036802 Google Scholar
[84]	Wang Y X, Lin M, Hughes T L 2018 Phys. Rev. B 98 165144 Google Scholar
[85]	Wu Z G, Yan Z B, Huang W 2019 Phys. Rev. B 99 020508 Google Scholar
[86]	Zhu X Y 2019 Phys. Rev. Lett. 122 236401 Google Scholar
[87]	Shapourian H, Wang Y X, Ryu S 2018 Phys. Rev. B 97 094508 Google Scholar
[88]	Yan Z B 2019 arXiv: 1905.11411 [cond-mat.supr-con]
[89]	Ahn J, Yang B J 2019 arXiv: 1906.02709 [cond-mat.mes-hall]
[90]	Gray M J, Freudenstein J, Zhao Y S F, Oconnor R, Jenkins S, Kumar N, Hoek M, Koper A, Taniguchi T, Watanabe K, Zhong R D, Gu G D, Burch K S 2019 Nano Lett. 19 4890 Google Scholar
[91]	Zhang P, Yaji K, Hashimoto T, Ota Y, Kondo T, Okazaki K, Wang Z J, Wen J S, Gu G D, Ding H, Shin S 2018 Science 360 182 Google Scholar
[92]	Jack B, Xie Y L, Li J, Jeon S, Bernevig B A, Yazdani A 2019 Science 364 1255 Google Scholar
[93]	Lin M, Hughes T L 2018 Phys. Rev. B 98 241103 Google Scholar
[94]	Ghorashi S A A, Hu X, Hughes T L, Rossi E 2019 Phys. Rev. B 100 020509 Google Scholar

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Vol. 68, Issue 22 - 2019-11-20

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