狄拉克量子材料中的输运理论进展

王焕文; 付博; 沈顺清

doi:10.7498/aps.72.20230672

摘要

狄拉克量子材料具有独特的电子结构, 可以用无质量和有质量的狄拉克方程描述. 从奇异的量子流体到晶体材料的多种系统均已发现了狄拉克量子材料. 由于其拓扑非平庸的能带结构, 狄拉克量子材料表现出丰富有趣的输运现象, 包括纵向负磁阻、量子干涉效应和螺旋磁效应等. 本文介绍狄拉克量子材料输运理论最新进展, 总结了基于狄拉克方程的相关量子输运理论和量子反常效应, 重点关注有质量的狄拉克费米子和量子反常半金属, 介绍了半磁拓扑绝缘体中宇称反常和半整数量子霍尔效应的实现.

关键词:

Abstract

Dirac quantum materials comprise a broad category of condensed matter systems characterized by low-energy excitations described by the Dirac equation. These excitations, which can manifest as either collective states or band structure effects, have been identified in a wide range of systems, from exotic quantum fluids to crystalline materials. Over the past several decades, they have sparked extensive experimental and theoretical investigations in various materials, such as topological insulators and topological semimetals. The study of Dirac quantum materials has also opened up new possibilities for topological quantum computing, giving rise to a burgeoning field of physics and offering a novel platform for realizing rich topological phases, including various quantum Hall effects and topological superconducting phases. Furthermore, the topologically non-trivial band structures of Dirac quantum materials give rise to plentiful intriguing transport phenomena, including longitudinal negative magnetoresistance, quantum interference effects, helical magnetic effects, and others. Currently, numerous transport phenomena in Dirac quantum materials remain poorly understood from a theoretical standpoint, such as linear magnetoresistance in weak fields, anomalous Hall effects in nonmagnetic materials, and three-dimensional quantum Hall effects. Studying these transport properties will not only deepen our understanding of Dirac quantum materials, but also provide important insights for their potential applications in spintronics and quantum computing. In this paper, quantum transport theory and quantum anomaly effects related to the Dirac equation are summarized, with emphasis on massive Dirac fermions and quantum anomalous semimetals. Additionally, the realization of parity anomaly and half-quantized quantum Hall effects in semi-magnetic topological insulators are also put forward. Finally, the key scientific issues of interest in the field of quantum transport theory are reviewed and discussed.

Keywords:

作者及机构信息

1.
电子科技大学物理学院, 成都　611731

2.
香港大学物理学系, 香港　999077

通信作者: 王焕文, wanghw@uestc.edu.cn ; 付博, fubo@gbu.edu.cn ; 沈顺清, sshen@hku.hk

基金项目: 国家重点研发计划(批准号: 2019YFA0308603)、香港特别行政区研究拨款委员会(批准号: C7012-21G, 17301220)、电子科技大学科研启动基金(批准号: Y030232059002011)和博士后国际交流计划(批准号: YJ20220059)资助的课题.

Authors and contacts

1.
School of Physics, University of Electronic Science and Technology of China, Chengdu 611731, China

2.
Department of Physics, The University of Hong Kong, Hong Kong 999077, China

Corresponding author: Wang Huan-Wen, wanghw@uestc.edu.cn ; Fu Bo, fubo@gbu.edu.cn ; Shen Shun-Qing, sshen@hku.hk

Funds: Project supported by the National Key R&D Program of China (Grant No. 2019YFA0308603), the Research Grants Council of the Hong Kong Government, University Grants Committee, China (Grant Nos. C7012-21G, 17301220), the Scientific Research Starting Foundation of University of Electronic Science and Technology of China (Grant No. Y030232059002011), and the International Postdoctoral Exchange Fellowship Program, China (Grant No. YJ20220059)

文章全文 : translate this paragraph

参考文献

[1]	Shen S Q 2017 Topological Insulators (Vol. 187) (2nd Ed.) (Singapore: Springer) pp17–32
[2]	Wehling T O, Black S, Annica M, Balatsky A V 2014 Adv. Phys. 63 1 Google Scholar
[3]	Volovik G E 2003 The Universe in a Helium Droplet (Oxford: Clarendon Press) pp454–456
[4]	Geim A K, Novoselov K S 2007 Nat. Mater. 6 183 Google Scholar
[5]	Castro Neto A H, Guinea F, Peres N M R, Novoselov K S, Geim A K 2009 Rev. Mod. Phys. 81 109 Google Scholar
[6]	Sarma S D, Adam S, Hwang E H, Rossi E 2011 Rev. Mod. Phys. 83 407 Google Scholar
[7]	Moore J E 2010 Nature 464 194 Google Scholar
[8]	Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057 Google Scholar
[9]	Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045 Google Scholar
[10]	Ando Y 2013 J. Phys. Soc. Jpn. 82 102001 Google Scholar
[11]	Manzeli S, Ovchinnikov D, Pasquier D, Yazyev O V, Kis A 2017 Nat. Rev. Mater. 2 17033 Google Scholar
[12]	Wang G, Chernikov A, Glazov, M M, Heinz T F, Marie X, Amand T, Urbaszek B 2018 Rev. Mod. Phys. 90 021001 Google Scholar
[13]	Fu L 2011 Phys. Rev. Lett. 106 106802 Google Scholar
[14]	Ando Y, Fu L 2015 Annu. Rev. Condens. Matter Phys. 6 361 Google Scholar
[15]	Armitage N P, Mele E J, Vishwanath A 2018 Rev. Mod. Phys. 90 015001 Google Scholar
[16]	Lv B Q, Qian T, Ding H 2021 Rev. Mod. Phys. 93 025002 Google Scholar
[17]	Fu L, Kane C L, Mele E J 2007 Phys. Rev. Lett. 98 106803 Google Scholar
[18]	Haldane F D M 1988 Phys. Rev. Lett. 61 2015 Google Scholar
[19]	Kane C L, Me le, E J 2005 Phys. Rev. Lett. 95 226801 Google Scholar
[20]	Fu L, Kane C L 2008 Phys. Rev. Lett. 100 096407 Google Scholar
[21]	Bansil A, Lin H, Das T 2016 Rev. Mod. Phys. 88 021004 Google Scholar
[22]	Hosur P, Qi X L 2013 CR Phys. 14 857 Google Scholar
[23]	Jia S, Xu S Y, Hasan M Z 2016 Nat. Mater. 15 1140 Google Scholar
[24]	Lu H Z, Shen S Q 2017 Front. Phys. 12 127201 Google Scholar
[25]	Wang S, Lin B C, Wang A Q, Yu D P, Liao Z M 2017 Adv. Phys. X 2 518 Google Scholar
[26]	Hu J, Xu S Y, Ni N, Mao Z Q 2019 Annu. Rev. Mater. Res. 49 207 Google Scholar
[27]	Culcer D, Keser A C, Li Y, Tkachov G 2020 2D Mater. 7 022007 Google Scholar
[28]	Kim H J, Kim K S, Wang J F, Sasaki M, Satoh N, Ohnishi A, Kitaura M, Yang M, Li L 2013 Phys. Rev. Lett. 111 246603 Google Scholar
[29]	Xiong J, Kushwaha S K, Liang T, Krizan J W, Hirschberger M, Wang W, Cava R J, Ong N P 2015 Science 350 413 Google Scholar
[30]	Li H, He H T, Lu H Z, Zhang H C, Liu H C, Ma R, Fan Z Y, Shen S Q, Wang J N 2016 Nat. Commun. 7 10301 Google Scholar
[31]	Li C Z, Wang L X, Liu H W, Wang J, Liao Z M, Yu D P 2015 Nat. Commun. 6 10137 Google Scholar
[32]	Zhang C L, Xu S Y, Belopolski I, Yuan Z, Lin Z, Tong B, Bian G, Alidoust N, Lee C C, Huang S M 2016 Nat. Commun. 7 10735 Google Scholar
[33]	Huang X, Zhao L, Long Y, Wang P, Chen D, Yang Z, Liang H, Xue M, Weng H, Fang Z 2015 Phys. Rev. X 5 031023 Google Scholar
[34]	Li Q, Kharzeev D E, Zhang C, Huang Y, Pletikosić I, Fedorov A, Zhong R, Schneeloch J, Gu G, Valla T 2016 Nat. Phys. 12 550 Google Scholar
[35]	Liang S, Lin J, Kushwaha S, Xing J, Ni N, Cava R J, Ong N P 2018 Phys. Rev. X 8 031002 Google Scholar
[36]	Wang J, Li H, Chang C, He K, Lee J S, Lu H, Sun Y, Ma X, Samarth N, Shen S Q 2012 Nano Res. 5 739 Google Scholar
[37]	He H T, Liu H C, Li B K, Guo X, Xu Z J, Xie M H, Wang J N 2013 Appl. Phys. Lett. 103 031606 Google Scholar
[38]	Assaf B, Phuphachong T, Kampert E, Volobuev V, Mandal P, Sánchez-Barriga J, Rader O, Bauer G, Springholz G, De Vaulchier L 2017 Phys. Rev. Lett. 119 106602 Google Scholar
[39]	Wiedmann S, Jost A, Fauqué B, Van Dijk J, Meijer M, Khouri T, Pezzini S, Grauer S, Schreyeck S, Brüne C 2016 Phys. Rev. B 94 081302 Google Scholar
[40]	Mutch J, Chen W C, Went P, Qian T, Wilson I Z, Andreev A, Chen C C, Chu J H 2019 Sci. Adv. 5 eaav9771 Google Scholar
[41]	Nielsen H, Ninomiya M 1983 Phys. Lett. B 130 389 Google Scholar
[42]	Son D T, Spivak B Z 2013 Phys. Rev. B 88 104412 Google Scholar
[43]	Burkov A A 2014 Phys. Rev. Lett. 113 247203 Google Scholar
[44]	Goswami P, Pixley J H, Sarma S D 2015 Phys. Rev. B 92 075205 Google Scholar
[45]	Gao Y, Yang S A, Niu Q 2017 Phys. Rev. B 95 165135 Google Scholar
[46]	Dai X, Du Z, Lu H Z 2017 Phys. Rev. Lett. 119 166601 Google Scholar
[47]	Andreev A V, Spivak B Z 2018 Phys. Rev. Lett. 120 026601 Google Scholar
[48]	Wang H W, Fu B, Shen S Q 2018 Phys. Rev. B 98 081202(R Google Scholar
[49]	Fu B, Wang H W, Shen S Q 2020 Phys. Rev. B 101 125203 Google Scholar
[50]	Wang H W, Fu B, Shen S Q 2021 Phys. Rev. B 104 L241111 Google Scholar
[51]	Gorkov L P, Larkin A I, Khmelnitskii D E 1979 JETP Lett. 30 228
[52]	Hikami S, Larkin A I, Nagaoka Y 1980 Prog. Theor. Phys. 63 707 Google Scholar
[53]	Chakravarty S, Schmid A 1986 Phys. Rep. 140 193 Google Scholar
[54]	Wu X S, Li X B, Song Z M, Berger C, de Heer W A 2007 Phys. Rev. Lett. 98 136801 Google Scholar
[55]	Tikhonenko F V, Kozikov A A, Savchenko A K, Gorbachev R V 2009 Phys. Rev. Lett. 103 226801 Google Scholar
[56]	Checkelsky J G, Hor Y S, Liu M H, Qu D X, Cava R J, Ong N P 2009 Phys. Rev. Lett. 103 246601 Google Scholar
[57]	Chen J, Qin H J, Yang F, Liu J, Guan T, Qu F M, Zhang G H, Shi J R, Xie X C, Yang C L 2010 Phys. Rev. Lett. 105 176602 Google Scholar
[58]	He H T, Wang G, Zhang T, Sou I K, Wong G K, Wang J N, Lu H Z, Shen S Q, Zhang F C 2011 Phys. Rev. Lett. 106 166805 Google Scholar
[59]	Wang J, DaSilva A M, Chang C Z, He K, Jain J K, Samarth N, Ma X C, Xue Q K, Chan M H W 2011 Phys. Rev. B 83 245438 Google Scholar
[60]	Liu M H, Zhang J S, Chang C Z, Zhang Z C, Feng X, Li K, H e, K, Wa ng, L L, Chen X, Dai Xi, Fang Z, Xue Q K, Ma X C, Wang Y Y 2012 Phys. Rev. Lett. 108 036805 Google Scholar
[61]	Liu H C, Lu H Z, He H T, Li B K, Liu S G, He Q L, Wang G, S ou, I K, Shen S Q, Wang J N 2014 ACS Nano 8 9616 Google Scholar
[62]	Li H, Wang H W, Li Y, Zhang H C, Zhang S, Pan X C, Jia B, Song F Q, Wang J N 2019 Nano Lett. 19 2450 Google Scholar
[63]	Tkac V, Vyborny K, Komanicky V, Warmuth J, Michiardi M, Ngankeu A S 2019 Phys. Rev. Lett. 123 036406 Google Scholar
[64]	Zhao B, Cheng P H, Pan H Y, Zhang S, Wang B G, Wang G H, Xiu F X, Song F Q 2016 Sci. Rep. 6 22377 Google Scholar
[65]	Nakamura H, Huang D, Merz J, Khalaf E, Ostrovsky P, Yaresko A, Samal D, Takagi H 2020 Nat. Commun. 11 1161 Google Scholar
[66]	Suzuura H, Ando T 2002 Phys. Rev. Lett. 89 266603 Google Scholar
[67]	McCann E, Kechedzhi K, Falko V I, Suzuura H, Ando T, Altshuler B L 2006 Phys. Rev. Lett. 97 146805 Google Scholar
[68]	Garate I, Glazman L 2012 Phys. Rev. B 86 035422 Google Scholar
[69]	Lu H Z, Shi J R, Shen S Q 2011 Phys. Rev. Lett. 107 076801 Google Scholar
[70]	Lu H Z, Shen S Q 2014 Phys. Rev. Lett. 112 146601 Google Scholar
[71]	Gornyi I V, Kachorovskii V Y, Ostrovsky P M 2014 Phys. Rev. B 90 085401 Google Scholar
[72]	Wang H W, Fu B, Shen S Q 2020 Phys. Rev. Lett. 124 206603 Google Scholar
[73]	Lu H Z, Shen S Q 2015 Phys. Rev. B 92 035203 Google Scholar
[74]	Dai X, Lu H Z, Shen S Q, Yao H 2016 Phys. Rev. B 93 161110 Google Scholar
[75]	Fu B, Wang H W, Shen S Q 2019 Phys. Rev. Lett. 122 246601 Google Scholar
[76]	Chen W, Lu H Z, Zilberberg O 2019 Phys. Rev. Lett. 122 196603 Google Scholar
[77]	Novoselov K, Geim A, Morozov S, Jiang D, Katsnelson M, Grigorieva I, Dubonos S, Firsov A 2005 Nature 438 197 Google Scholar
[78]	Zhang Y, Tan Y W, Stormer H L, Kim P 2005 Nature 438 201 Google Scholar
[79]	Gusynin V P, Sharapov S G 2005 Phys. Rev. Lett. 95 146801 Google Scholar
[80]	Chang C Z, Liu C X, MacDonald A H 2023 Rev. Mod. Phys. 95 011002 Google Scholar
[81]	Yu R, Zhang W, Zhang H J, Zhang S C, Dai X, Fang Z 2010 Science 329 61 Google Scholar
[82]	Chang C Z, Zhang J, Feng X, Shen J, Zhang Z, Guo M, Li K, Ou Y, Wei P, Wang L L 2013 Science 340 167 Google Scholar
[83]	Xu Y, Miotkowski I, Liu C, Tian J F, Nam H, Alidoust N, Hu J N, Shih C K, Hasan M Z, Chen Y P 2014 Nat. Phys. 10 956 Google Scholar
[84]	Zhang C, Zhang Y, Yuan X, Lu S, Zhang J, Narayan A, Liu Y, Zhang H, Ni Z, Liu R 2019 Nature 565 331 Google Scholar
[85]	Tang F D, Ren Y F, Wang P P, Zhong R D, Schneeloch J, Yang S Y A, Yang K, Lee P A, Gu G D, Qiao X H, Zhang L Y 2019 Nature 569 537 Google Scholar
[86]	Galeski S, Ehmcke T, Wawrzynczak R, Lozano P M, Cho K, Sharma A, Das S 2021 Nat. Commun. 12 3197 Google Scholar
[87]	Wang C M, Sun H P, Lu H Z, Xie X C 2017 Phys. Rev. Lett. 119 136806 Google Scholar
[88]	Qin F, Li S, Du Z Z, Wang C M, Zhang W Q, Yu D P, Lu H Z, Xie X C 2020 Phys. Rev. Lett. 125 206601 Google Scholar
[89]	Yoshimi R, Yasuda K, Tsukazaki A, Takahashi K S, Nagaosa N, Kawasaki M, Tokura Y 2015 Nat. Commun. 6 8530 Google Scholar
[90]	Zhang S, Pi L, Wang R, Yu G, Pan X C, Wei Z, Zhang J, Xi C, Bai Z 2017 Nat. Commun. 8 977 Google Scholar
[91]	Mogi M, Okamura Y, Kawamura M, Yoshimi R, Yoshimi K, Tsukazaki A, Takahashi K S 2022 Nat. Phys. 18 390 Google Scholar
[92]	Liang T, Lin J J, Gibson Q, Kushwaha S, Liu M H, Wang W D, Xiong H Y, Sobota J A, Hashimoto M, Kirchmann P S, Shen Z X, Cava R J, Ong N P 2018 Nat. Phys. 14 451 Google Scholar
[93]	Sun Z, Cao Z, Cui J, Zhu C, Ma D, Wang H, Zhuo W, Cheng Z, Wang Z, Wan X, Chen X H 2020 Npj Quantum Mater. 5 36 Google Scholar
[94]	Liu Y Z, Wang H C, Fu H X, et al. 2021 Phys. Rev. B 103 L201110 Google Scholar
[95]	Mutch J, Ma X, Wang C, Malinowski P, AyresSims J, Jiang Q, Liu Z, Xiao D, Yankowitz M, Chu J H 2021 arXiv: 2101.02681 [cond-mat]
[96]	Gourgout A, Leroux M, Smirr J L, et al. 2022 npj Quantum Mater. 7 71 Google Scholar
[97]	Lozano P M, Cardoso G, Aryal N, Nevola D, Gu G, Tsvelik A, Yin W, Li Q 2022 Phys. Rev. B 106 L081124 Google Scholar
[98]	Burkov A A 2017 Phys. Rev. B 96 041110 Google Scholar
[99]	Nandy S, Sharma G, Taraphder A, Tewari S 2017 Phys. Rev. Lett. 119 176804 Google Scholar
[100]	Taskin A A, Legg H F, Yang F, Sasaki S, Kanai Y, Matsumoto K, Rosch A, Ando Y 2017 Nat. Commun. 8 1340 Google Scholar
[101]	Li H, Wang H W, He H T, Wang J N, Shen S Q 2018 Phys. Rev. B 97 201110 Google Scholar
[102]	Wu M, Zheng G L, Chu W W, Liu Y Q, Gao W S, Zhang H W, Lu J W, Han Y Y, Zhou J H, Ning W, Tian M L 2018 Phys. Rev. B 98 161110 Google Scholar
[103]	Kumar N, Guin S N, Felser C, Shekhar C 2018 Phys. Rev. B 98 041103 Google Scholar
[104]	Li P, Zhang C H, Zhang J W, Wen Y, Zhang X X 2018 Phys. Rev. B 98 121108(R Google Scholar
[105]	Huang D, Nakamura H, Takagi H 2021 Phys. Rev. Research 3 013268 Google Scholar
[106]	Wu M, Tu D, Nie Y, Miao S, Gao W, Han Y, Zhu X D, Zhou J H, Ning W, Tian M L 2022 Nano Lett. 22 73 Google Scholar
[107]	Zhong J Y, Zhuang J C, Du Y 2023 Chin. Phys. B 32 047203 Google Scholar
[108]	Gao A Y, Liu Y F, Hu C W, Qiu J X, Tzschaschel C, Ghosh B, Ho S C 2021 Nature 595 521 Google Scholar
[109]	Chen R, Sun H P, Gu M Q, Hua C B, Liu Q H, Lu H Z, Xie X C 2022 Natl. Sci. Rev. nwac140 Google Scholar
[110]	Zhai D W, Chen C, Xiao C, Yao W 2023 Nat. Commun. 14 1961 Google Scholar
[111]	Qu D X, Hor Y S, Xiong J, Cava R J, Ong N P 2010 Science 329 821 Google Scholar
[112]	Wang X, Du Y, Dou S, Zhang C 2012 Phys. Rev. Lett. 108 266806 Google Scholar
[113]	Zhang G, Qin H, Chen J, He X, Lu L, Li Y, Wu K 2011 Adv. Funct. Mater. 21 2351
[114]	Tang H, Liang D, Qiu R L J, Gao X P A 2011 ACS Nano 5 7510 Google Scholar
[115]	He H, Li B, Liu H, Guo X, Wang Z, Xie M, Wang J 2012 Appl. Phys. Lett. 100 032105 Google Scholar
[116]	He L P, Hong X C, Dong J K, Pan J, Zhang Z, Zhang J, Li S Y 2014 Phys. Rev. Lett. 113 246402 Google Scholar
[117]	Liang T, Gibson Q, Ali M N, Liu M, Cava R J, Ong N P 2015 Nat. Mater. 14 280 Google Scholar
[118]	Narayanan A, Watson M D, Blake S F, Bruyant N, Drigo L, Chen Y L, Prabhakaran D, Yan B, Felser C, Kong T, Canfield P C, Coldea A I 2015 Phys. Rev. Lett. 114 117201 Google Scholar
[119]	Feng J, Pang Y, Wu D, Wang Z J, Weng H M, Li J, Dai X, Fang Z, Shi Y, Lu L 2015 Phys. Rev. B 92 081306(R Google Scholar
[120]	Tian Y, Ghassemi N, Jr J H R 2021 Phys. Rev. Lett. 126 236401 Google Scholar
[121]	Fu B, Wang H W, Shen S Q 2020 Phys. Rev. Lett. 125 256601 Google Scholar
[122]	Fu B, Zou J Y, Hu Z A, Wang H W, Shen S Q 2022 npj Quantum Mater. 7 94 Google Scholar
[123]	Zou J Y, Fu B, Wang H W, Hu Z A, Shen S Q 2022 Phys. Rev. B 105 L201106 Google Scholar
[124]	Zou J Y, Chen R, Fu B, Wang H W, Hu Z A, Shen S Q 2023 Phys. Rev. B 107 125153 Google Scholar
[125]	Wang H W, Fu B, Zou J Y, Hu Z A, Shen S Q 2022 Phys. Rev. B 106 045111 Google Scholar
[126]	Shen S Q, Bao Y J, Ma M, Xie X C, Zhang F C 2005 Phys. Rev. B 71 155316 Google Scholar
[127]	Bjorken J D, Drell S D 1964 Relativistic Quantum Mechanics (New York: McGraw-Hill Inc.) pp45–60
[128]	Shen S Q, Shan W Y, Lu H Z 2011 SPIN 01 33 Google Scholar
[129]	Klitzing K V, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494 Google Scholar
[130]	Klitzing K v, Chakraborty T, Kim P, Madhavan V, Dai X, McIver J, Tokura Y, Savary L, Smirnova D, Rey A M, Felser C, Gooth J, Qi X L 2020 Nat. Rev. Phys. 2 397 Google Scholar
[131]	Schnyder A P, Ryu S, Furusaki A, Ludwig A W W 2008 Phys. Rev. B 78 195125 Google Scholar
[132]	Yang B J, Nagaosa N 2014 Nat. Commun. 5 4898 Google Scholar
[133]	Wilson K G 1975 New Phenomena in Subnuclear Physics (New York: Plenum) pp69–142
[134]	Rothe H J 2005 Lattice Gauge Theories: An Introduction (3rd Ed.) (Singapore: World Scientific) pp56–57
[135]	Zhang Y, Wang C, Yu L, Liu G, Liang A, Huang J 2017 Nat. Commun. 8 15512 Google Scholar
[136]	Zhang H J, Liu C X, Qi X L, Dai X, Fang Z, Zhang S C 2009 Nat. Phys. 5 438 Google Scholar
[137]	Xia Y Q, Qian D, Hsieh D, Wray L, Pal A, Lin H, Bansil A, Grauer D, Hor Y S, Cava R J, Hasan M Z 2009 Nat. Phys. 5 398 Google Scholar
[138]	Zee A 2010 Quantum Field Theory in a Nutshell (Vol. 7) (Princeton: Princeton University Press) pp99–100
[139]	Liu M H, Chang C Z, Zhang Z C, Zhang Y, Ruan W, He K, Wang L L, Chen X, Jia J F, Zhang S C, Xue Q K, Ma X C, Wang Y Y 2011 Phys. Rev. B 83 165440 Google Scholar
[140]	Takagaki Y, Jenichen B, Jahn U, Ramsteiner M, Friedl K J 2012 Phys. Rev. B 85 115314 Google Scholar
[141]	Jing Y M, Huang S Y, Zhang K, Wu J X, Guo Y F, Peng H L, Liu Z F, Xu H Q 2016 Nanoscale 8 1879 Google Scholar
[142]	Altshuler B L, Aronov A G, Lee P A 1980 Phys. Rev. Lett. 44 1288 Google Scholar
[143]	Lee P A, Ramakrishnan T V 1985 Rev. Mod. Phys. 57 287 Google Scholar
[144]	Liu Q, Liu C X, Xu C K, Qi X L, Zhang S C 2009 Phys. Rev. Lett. 102 156603 Google Scholar
[145]	Chen Y L, C hu, J H, Analytis J G, Liu Z K, Igarashi K, Kuo H H, Qi X L, Mo S K, Moore R G, Lu D H 2010 Science 329 659 Google Scholar
[146]	Tokura Y, Yasuda K, Tsukazaki A 2019 Nat. Rev. Phys. 1 126 Google Scholar
[147]	Okada S, Sambongi T, Ido M 1980 J. Phys. Soc. Jpn. 49 839 Google Scholar
[148]	Izumi M, Uchinokura K, Matsuura E 1981 Solid State Commun. 37 641 Google Scholar
[149]	DiSalvo F, Fleming R, Waszczak J 1981 Phys. Rev. B 24 2935 Google Scholar
[150]	Okada S, Sambongi T, Ido M, Tazuke Y, Aoki R, Fujita O 1982 J. Phys. Soc. Jpn. 51 460 Google Scholar
[151]	Bullett D 1982 Solid State Commun. 42 691 Google Scholar
[152]	Fjellvag H, Kjekshus A 1986 Solid State Commun. 60 91 Google Scholar
[153]	Chen Z G, Chen R, Zhong R, Schneeloch J, Zhang C, Huang Y, Qu F, Yu R, Li Q, Gu G, Wang N 2017 Proc. Natl. Acad. Sci. U.S.A. 114 816 Google Scholar
[154]	Manzoni G, Gragnaniello L, AutASs G, Kuhn T, Sterzi A, Cilento F, Zacchigna M, Enenkel V 2016 Phys. Rev. Lett. 117 237601 Google Scholar
[155]	Zhang J L, Wang C M, Guo C Y, Zhu X D, Zhang Y, Yang J Y, Wang Y Q, Qu Z, Pi L, Lu H Z, Tian M L 2019 Phys. Rev. Lett. 123 196602 Google Scholar
[156]	Jiang Y, Wang J, Zhao T, Dun Z L, Huang Q, Wu X S, Mourigal M, Zhou H D, Pan W, Ozerov M, Smirnov D, Jiang Z 2020 Phys. Rev. Lett. 125 046403 Google Scholar
[157]	Wang P, Ren Y, Tang F, Wang P, Hou T, Zeng H, Zhang L, Qiao Z H 2020 Phys. Rev. B 101 161201 Google Scholar
[158]	Zhao L X, Huang X C, Long Y J, Chen D, Liang H, Yang Z H, Xue M Q, Ren Z A, Weng H M, Fang Z, Dai X, Chen G F 2017 Chin. Phys. Lett. 34 037102 Google Scholar
[159]	Shahi P, Singh D J, Sun J P, Zhao L X, Chen G F, Lv Y Y, Li J, Yan J Q, Mandrus D G, Cheng J G 2018 Phys. Rev. X 8 021055 Google Scholar
[160]	Xu B, Zhao L, Marsik P, Sheveleva E, Lyzwa F, Dai Y, Chen G, Qiu X, Bernhard C 2018 Phys. Rev. Lett. 121 187401 Google Scholar
[161]	Wang C J 2021 Phys. Rev. Lett. 126 126601 Google Scholar
[162]	Wang Y G, Legg H F, Bömerich T, Park J, Biesenkamp S, Taskin A A, Braden M, Rosch A, Ando Y 2022 Phys. Rev. Lett. 128 176602 Google Scholar
[163]	Zhang Y, Wang C, Liu G, Liang A, Zhao L, Huang J, Gao Q, Shen B 2017 Sci. Bull. 62 950 Google Scholar
[164]	Adler S L 1969 Phys. Rev. 177 2426 Google Scholar
[165]	Bell J S, Jackiw R 1969 Nuovo Cimento A 60 47 Google Scholar
[166]	Fujikawa K 1979 Phys. Rev. Lett. 42 1195 Google Scholar
[167]	Peskin M E, Schroeder D V 1995 An Introduction to Quantum Field Theory (Cambridge: Perseus Books Publishing LLC) pp651–667
[168]	Weinberg S 1995 The Quantum Theory of Fields (Vol. 2) (Cambridge: Cambridge University Press) pp473–485
[169]	Dirac P A M 1958 The Principles of Quantum Mechanics (New York: Oxford University Press Inc.) pp253–267
[170]	Jackiw R, Johnson K 1969 Phys. Rev. 182 1459 Google Scholar
[171]	Kharzeev D E, Kikuchi Y, Meyer R, Tanizaki Y 2018 Phys. Rev. B 98 014305 Google Scholar
[172]	Yamamoto N, Yang D L 2021 Phys. Rev. D 103 125003 Google Scholar
[173]	Vilenkin A 1980 Phys. Rev. D 22 3080 Google Scholar
[174]	Fukushima K, Fukushima D E, Warringa H J 2008 Phys. Rev. D 78 074033 Google Scholar
[175]	Huang Z M, Zhou J H, Shen S Q 2017 Phys. Rev. B 96 085201 Google Scholar
[176]	Lu H Z, Zhang S B, Shen S Q 2015 Phys. Rev. B 92 045203 Google Scholar
[177]	Zhang S B, Lu H Z, Shen S Q 2016 New J. Phys. 18 053039 Google Scholar
[178]	Klier J, Gornyi I V, Mirlin A D 2017 Phys. Rev. B 96 214209 Google Scholar
[179]	Niemi A J, Semenoff G W 1983 Phys. Rev. Lett. 51 2077 Google Scholar
[180]	Redlich A N 1984 Phys. Rev. Lett. 52 18 Google Scholar
[181]	Jackiw R 1984 Phys. Rev. D 29 2375 Google Scholar
[182]	Boyanovsky D, Blankenbecler R, Yahalom R 1986 Nucl. Phys. B 270 483 Google Scholar
[183]	Schakel A M J 1991 Phys. Rev. D 43 1428 Google Scholar
[184]	Chu R L, Shi J R, Shen S Q 2011 Phys. Rev. B 84 085312 Google Scholar
[185]	Lapa M F 2019 Phys. Rev. B 99 235144 Google Scholar
[186]	Lu R, Sun H, Kumar S, Wang Y, Gu M, Zeng M, Hao Y J, Li J 2021 Phys. Rev. X 11 011039 Google Scholar
[187]	Essin A M, Moore J E, Vanderbilt D 2009 Phys. Rev. Lett. 102 146805 Google Scholar
[188]	Sitte M, Rosch A, Altman E, Fritz L 2012 Phys. Rev. Lett. 108 126807 Google Scholar
[189]	Wang J, Lian B, Qi X L, Zhang S C 2015 Phys. Rev. B 92 081107 Google Scholar
[190]	Gu M, Li J, Sun H, Zhao Y, Liu C, Liu J, Lu H, Liu Q 2021 Nat. Commun. 12 3524 Google Scholar
[191]	Mogi M, Kawamura M, Yoshimi R, Tsukazaki A, Kozuka Y, Shirakawa N, Takahashi K S, Kawasaki M, Tokura Y 2017 Nat. Mater. 16 516 Google Scholar
[192]	Mogi M, Kawamura M, Tsukazaki A, Yoshimi R, Takahashi K S, Kawasaki M, Tokura Y 2017 Sci. Adv. 3 eaao1669 Google Scholar
[193]	Xiao D, Jiang J, Shin J H, Wang W, Wang F, Zhao Y F, Liu C, Wu W, Chan M H W, Samarth N, Chang C Z 2018 Phys. Rev. Lett. 120 056801 Google Scholar
[194]	Zhang D, Shi M, Zhu T, Xing D, Zhang H, Wang J 2019 Phys. Rev. Lett. 122 206401 Google Scholar
[195]	Liu C, Wang Y, Li H, Wu Y, Li Y, Li J, He K, Xu Y, Zhang J, Wang Y 2020 Nat. Mat. 19 522 Google Scholar
[196]	Wilczek F 1987 Phys. Rev. Lett. 58 1799 Google Scholar
[197]	Qi X L, Hughes T L, Zhang S C 2008 Phys. Rev. B 78 195424 Google Scholar
[198]	Spaldin N A, Fiebig M 2005 Science 309 391 Google Scholar
[199]	Fiebig M 2005 J. Phys. D 38 R123 Google Scholar
[200]	Maciejko J, Qi X L, Drew H D, Zhang S C 2010 Phys. Rev. Lett. 105 166803 Google Scholar
[201]	Tse W K, MacDonald A H 2011 Phys. Rev. B 84 205327 Google Scholar
[202]	Li R, Wang J, Qi X L, Zhang S C 2010 Nat. Phys. 6 284 Google Scholar
[203]	Sekine A, Nomura K 2014 J Phys. Soc. Jpn. 83 104709 Google Scholar
[204]	Sekine A, Nomura K 2021 J. Appl. Phys. 129 141101 Google Scholar
[205]	Shoron O F, Kealhofer D A, Goyal M, Schumann T, Burkov A A, Stemmer S 2021 Appl. Phys. Lett. 119 171907 Google Scholar
[206]	Abrikosov A A 1998 Phys. Rev. B 58 2788 Google Scholar
[207]	Parish M M, Littlewood P B 2003 Nature 426 162 Google Scholar
[208]	Parish M M, Littlewood P B 2005 Phys. Rev. B 72 094417 Google Scholar
[209]	Cao H, Tian J, Miotkowski I, Shen T, Hu J, Qiao S, Chen Y P 2012 Phys. Rev. Lett. 108 216803 Google Scholar
[210]	Wang C M, Lei X L 2012 Phys. Rev. B 86 035442 Google Scholar
[211]	Avron J E, Seiler R, Simon B 1983 Phys. Rev. Lett. 51 51 Google Scholar
[212]	Halperin B I 1987 Jpn. J. Appl. Phys. 26 1913 Google Scholar
[213]	Zhao P L, Lu H Z, Xie X C 2021 Phys. Rev. Lett. 127 046602 Google Scholar
[214]	Nagaosa N, Sinova J, Onoda S, MacDonald A H, Ong N P 2010 Rev. Mod. Phys. 82 1539 Google Scholar
[215]	Chen R, Shen S Q 2023 arXiv: 2304.04229 [cond-mat]
[216]	Zhou H M, Li H L, Xu D H, Chen C Z, Sun Q F, Xie X C 2022 Phys. Rev. Lett. 129 096601 Google Scholar
[217]	Gong M, Liu H W, Jiang H, Chen C Z, Xie X C 2023 Natl. Sci. Rev 10 nwad025 Google Scholar
[218]	Goos F, Hanchen H 1947 Ann. Phys. 436 333 Google Scholar

施引文献

图 1 有质量狄拉克费米子的内禀磁阻　(a)有质量狄拉克费米子的横向磁阻和纵向磁阻, 其中纵向磁阻为负, 横向磁阻为正; (b)无量纲系数随能带展宽的变化关系, $c_\alpha$在弱散射下趋于一个常数. 转载自文献[48]

Fig. 1. Intrinsic Magnetoresistivity in massive Dirac fermion: (a) Transversal and longitudinal magnetoresistivity, where the longitudinal one is negative and transversal one is positive; (b) dimensionless parameter $c_\alpha$ as functions of band broadening, here $c_\alpha$ tends to a constant in weak scattering. Reproduced with permission from Ref. [48]

下载: 全尺寸图片幻灯片

图 2 Mn掺杂的拓扑绝缘体薄膜中磁导在不同磁场强度下的温度依赖行为, 其中(a) $x_{\mathrm{Mn}} = 0{\text{%}}$和(b) $x_{\mathrm{Mn}} = 8{\text{%}}$. 图中的空心方块是从文献[63]中获得的实验数据, 实线是(25)式在不同磁场下的拟合结果. 转载自文献[72]

Fig. 2. Magnetoconductivity as a function of temperature at different magnetic field strength for two Mn-doped topological insulator thin films of (a) $x_{{\rm{Mn}}} = 0{\text{%}}$ and (b) $x_{{\rm{Mn}}} = 8{\text{%}}$. The open squares are the experimental data extract from Ref. [63]. The solid red lines are the fitting results at different magnetic filed B by using the formula in Eq. (25). Reproduced with permission from Ref. [72]

下载: 全尺寸图片幻灯片

图 3 库珀子能隙和权重因子关于能量的函数关系, 其中(a) 拓扑绝缘体$(mb>0)$, (b) 平庸绝缘体$(mb<0)$, (c)半金属($mb=0$). $F_\mathrm{tot}=\displaystyle\sum\nolimits_i F_i$是总的权重因子. 转载自文献[75]

Fig. 3. Dimensionless Cooperon gap $\ell_e^2/\ell_i^2$ and weighting factor $F_i$ as a function of the Fermi energy $\mu$ for (a) topological insulator ($mb>0$), (b) trivial insulator ($mb<0$), and (c) Dirac semimetal ($mb=0$). $F_\mathrm{tot}$ is the total weighting factors defined as $F_\mathrm{tot}=\displaystyle\sum\nolimits_i F_i$. Reproduced with permission from Ref. [75]

下载: 全尺寸图片幻灯片

图 4 (a) (27)式对实验中Cd₂As₃样品^[64]相对纵向磁阻的理论拟合; (b) 拟合得到的相干长度关于温度的函数, 可以被$\ell_{\phi}\propto T^{-0.75}$很好地拟合; (c)不同磁场强度下的相对磁阻关于温度的函数. 转载自文献[75]

Fig. 4. Theoretical fitting to the relative longitudinal magnetoresistance in a $\mathrm{Cd}_{2}\mathrm{As}_{3}$ sample^[64]; (b) fitted phase coherence length $\ell_\phi$ as a function of temperatures (open squares), which can be well-fitted by $\ell_{\phi}\propto T^{-0.75}$; (c) measured relative magnetoresistance as a function of temperatures at $B=1, 2, 3\;\mathrm{T}$. Reproduced with permission from Ref. [75]

下载: 全尺寸图片幻灯片

图 5 ZrTe₅和HfTe₅电阻反常效应　(a)关于ZrTe₅温度依赖的能谱在实验 (根据文献中ARPES测量得到) 和理论 (实线) 上的比较, 随着温度升高, 化学势由导带变化至价带; (b)—(d) 分别为不同载流子浓度下计算得到的电阻反常行为、霍尔系数和塞贝克系数. 转载自文献[121]

Fig. 5. Resistivity anomaly in ZrTe₅ and HfTe₅: (a) Comparison of experimental (according to the ARPES measurements in literature) and theoretical (solid lines) temperature-dependent energy spectrum. The chemical potential varies from valence band to conduction band with the increasing of temperature. (b)–(d) The resistivity anomaly, Hall coefficients, and Seebeck coefficient for several different carrier concentrations. Reproduced with permission from Ref. [121]

下载: 全尺寸图片幻灯片

图 6 不同温度下的(a)横向电阻, (b)霍尔电阻, (c)塞贝克系数和(d)能斯特系数的磁场依赖. 转载自文献[121]

Fig. 6. Magnetic field dependence of (a) the transverse magnetoresistance $\rho_xx$, (b) the Hall resistivity $\rho_{xy}$, (c) the Seebeck coefficient and (d) the Nernst coefficient for different temperatures. Reproduced with permission from Ref. [121]

下载: 全尺寸图片幻灯片

图 7 连续性方程(32)和(34)中系数$C_{\rm{h}}{\rm{}}$和$C_5$的比较. 转载自文献[50]

Fig. 7. Comparison of the coefficients $C_{{{{\rm{h}}}}/5}$ in the equations for the divergence of the helical current and axial vector currents in Eqs. (32) and (34). Reproduced with permission from Ref. [50]

下载: 全尺寸图片幻灯片

图 8 宇称反常半金属示意图　(a) Haldane模型: 无质量和有质量的狄拉克锥在动量空间分开; (b)三维半磁性拓扑绝缘体: 无质量和有质量的狄拉克锥在实空间分开; (c)宇称反常半金属中低能电子态的分布以及幂律衰减的边界流. 转载自文献[123]

Fig. 8. Illustration of parity anomaly semimetals: (a) Haldane model where massive and massless Dirac cone separated in momentum space; (b) semi-magnetic 3D topological insulator in which a massive and a massless Dirac cone separated in position space; (c) distribution of a set of low energy states and the power law decay edge current in the parity anomalous semimetal for open boundary condition. Reproduced with permission from Ref. [123]

下载: 全尺寸图片幻灯片

图 9 (a)关于磁化强度和体拓扑磁电效应关系的示意图, 外加电场产生了表面电流和磁化强度; (b)沿着x方向的局域电流密度关于位置z的函数; (c)轴子角$\theta$关于位置z的函数关系. 这里电场沿着y方向. 转载自文献[125]

Fig. 9. (a) Schematic diagram of the relation between magnetization and bulk topological magnetoelectric effect. A surface current is produced by an electric field due to the magnetization. (b) Local current density along the x–direction as a function of slab position z. (c) Spatial dependent $\theta$ along the z direction. The electric field is applied along the y-direction. Reproduced with permission from Ref. [125]

下载: 全尺寸图片幻灯片

图 10 ${\cal{F}}(x)$关于x的函数关系, 其中蓝色虚线表示$ \mathcal{F}(x)= 1 $的位置. 插图是$x\leqslant3$的函数曲线, 绿色虚线是$x\to0$下的线性拟合${\cal{F}}(x)={x}/{2}$

Fig. 10. Function relation between ${\cal{F}}(x)$ and x, the dashed blue line indicates the position of $ \mathcal{F}(x)=1 $. Insert is the function curve for $x\leqslant3$, the dashed green line is the linear fitting with ${\cal{F}}(x)={x}/{2}$ for $x\to0$

下载: 全尺寸图片幻灯片

表 1 狄拉克哈密顿量中利用狄拉克伽马矩阵表示的16个物理量及无序根据时间反演(${\cal{T}}$)、宇称(${\cal{I}}$)以及手性对称性(${\cal{C}}$)的分类. 转载自文献[49]

Table 1. Various types of physical quantities and disorder represented by fermionic bilinears ($i = 1, $$ 2, 3$), their symmetries under time-reversal (${\cal{T}}$), parity (${\cal{I}}$), and continuous chiral rotation (${\cal{C}}$). Reproduced with permission from Ref. [49]

Bilinear ($\hat{\cal{S} }_{\mathtt{A} }\propto\bar{\varPsi}{\boldsymbol{\gamma}}^{\mathtt{A} }\varPsi$)	Physical quantity	${\cal{T}}$	${\cal{I}}$	${\cal{C}}$	Disorder
$\bar{\varPsi}{\boldsymbol{\gamma}}^{0}\varPsi$	Total charge $(J^{0})$	$\checkmark$	$\checkmark$	$\checkmark$	$\varDelta$
$\bar{\varPsi}{\boldsymbol{\gamma}}^{0}{\boldsymbol{\gamma}}^{5}\varPsi$	Axial charge $(J^{a0})$	$\checkmark$	$\times$	$\checkmark$	$\varDelta_{{\rm{a}}}$
$\bar{\varPsi}\varPsi$	Scalar mass $({n}_{\beta})$	$\checkmark$	$\checkmark$	$\times$	$\varDelta_{{\rm{m}}}$
$\bar{\varPsi}{\rm{i}}{\boldsymbol{\gamma}}^{5}\varPsi$	Pseudo-scalar density $({n}_{{\rm{P}}})$	$\times$	$\times$	$\times$	$\varDelta_{{\rm{P}}}$
$\bar{\varPsi}{\boldsymbol{\gamma}}^{i}\varPsi$	Current $(J^{i})$	$\times$	$\times$	$\checkmark$	$\varDelta_{{\rm{c}}}$
$\bar{\varPsi}\gamma^{i}\gamma^{5}\varPsi$	Axial current $(J^{ai})$	$\times$	$\checkmark$	$\checkmark$	$\varDelta_{{\rm{ac}}}$
$\bar{\varPsi}{\rm{i}}{\boldsymbol{\gamma}}^{0}{\boldsymbol{\gamma}}^{i}\varPsi$	Electric polarization $({p}_{i})$	$\checkmark$	$\times$	$\times$	$\varDelta_{{\rm{p}}}$
$\bar{\varPsi }{\boldsymbol{\gamma}}^{5}{\boldsymbol{\gamma} }^{0}{\boldsymbol{\gamma} }^{i}\varPsi$	Magnetization $({m}_{i})$	$\times$	$\checkmark$	$\times$	$\varDelta_{{\rm{M}}}$

下载: 导出CSV

表 2 4个库珀子通道$i = s, t_{0, \pm}$的库珀子能隙(以$ \ell_{e}^{-2} $为单位)和权重因子, 其中$\eta = mv^2/\mu$是狄拉克费米子的自旋极化. 转载自文献[72]

Table 2. Components of four Cooperon channels $i = s, t_{0, \pm}$ in the basis of spin-triplet and singlet $|s, s_{z}\rangle$, the Cooperon gap $\ell_{i}^{-2}$ in unit of the mean free path $\ell_{{\rm{e}}}^{-2}$ and the weighting factors $w_{i}$, where $\eta = mv^2/\mu$ is the orbital polarization of Dirac fermion. Reproduced with permission from Ref. [72]

i	Cooperon in $\|s, s_z\rangle$	$w_i$	$\ell_{\rm{e}}^2/\ell_i^2$
s	$\|0, 0\rangle$	$-\dfrac{(1-\eta^{2})^{2}}{2(1+3\eta^{2})^{2}}$	$\dfrac{(1-\eta^{2})\eta^{2}}{(1+\eta^{2})^{2}}$
$t_{+}$	$\|1, 1\rangle$	$\dfrac{4\eta^{2}(1+\eta^{2})}{(1+3\eta^{2})^{2}}$	$\dfrac{4(1-\eta)^{2}\eta^{2}}{(1+3\eta^{2})(1+\eta)^{2}}$
$t_{0}$	$\|1, 0\rangle$	0	$\infty$
$t_{-}$	$\|1, -1\rangle$	$\dfrac{4\eta^{2}(1+\eta^{2})}{(1+3\eta^{2})^{2}}$	$\dfrac{4(1+\eta)^{2}\eta^{2}}{(1+3\eta^{2})(1-\eta)^{2}}$

下载: 导出CSV

[1]	Shen S Q 2017 Topological Insulators (Vol. 187) (2nd Ed.) (Singapore: Springer) pp17–32
[2]	Wehling T O, Black S, Annica M, Balatsky A V 2014 Adv. Phys. 63 1 Google Scholar
[3]	Volovik G E 2003 The Universe in a Helium Droplet (Oxford: Clarendon Press) pp454–456
[4]	Geim A K, Novoselov K S 2007 Nat. Mater. 6 183 Google Scholar
[5]	Castro Neto A H, Guinea F, Peres N M R, Novoselov K S, Geim A K 2009 Rev. Mod. Phys. 81 109 Google Scholar
[6]	Sarma S D, Adam S, Hwang E H, Rossi E 2011 Rev. Mod. Phys. 83 407 Google Scholar
[7]	Moore J E 2010 Nature 464 194 Google Scholar
[8]	Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057 Google Scholar
[9]	Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045 Google Scholar
[10]	Ando Y 2013 J. Phys. Soc. Jpn. 82 102001 Google Scholar
[11]	Manzeli S, Ovchinnikov D, Pasquier D, Yazyev O V, Kis A 2017 Nat. Rev. Mater. 2 17033 Google Scholar
[12]	Wang G, Chernikov A, Glazov, M M, Heinz T F, Marie X, Amand T, Urbaszek B 2018 Rev. Mod. Phys. 90 021001 Google Scholar
[13]	Fu L 2011 Phys. Rev. Lett. 106 106802 Google Scholar
[14]	Ando Y, Fu L 2015 Annu. Rev. Condens. Matter Phys. 6 361 Google Scholar
[15]	Armitage N P, Mele E J, Vishwanath A 2018 Rev. Mod. Phys. 90 015001 Google Scholar
[16]	Lv B Q, Qian T, Ding H 2021 Rev. Mod. Phys. 93 025002 Google Scholar
[17]	Fu L, Kane C L, Mele E J 2007 Phys. Rev. Lett. 98 106803 Google Scholar
[18]	Haldane F D M 1988 Phys. Rev. Lett. 61 2015 Google Scholar
[19]	Kane C L, Me le, E J 2005 Phys. Rev. Lett. 95 226801 Google Scholar
[20]	Fu L, Kane C L 2008 Phys. Rev. Lett. 100 096407 Google Scholar
[21]	Bansil A, Lin H, Das T 2016 Rev. Mod. Phys. 88 021004 Google Scholar
[22]	Hosur P, Qi X L 2013 CR Phys. 14 857 Google Scholar
[23]	Jia S, Xu S Y, Hasan M Z 2016 Nat. Mater. 15 1140 Google Scholar
[24]	Lu H Z, Shen S Q 2017 Front. Phys. 12 127201 Google Scholar
[25]	Wang S, Lin B C, Wang A Q, Yu D P, Liao Z M 2017 Adv. Phys. X 2 518 Google Scholar
[26]	Hu J, Xu S Y, Ni N, Mao Z Q 2019 Annu. Rev. Mater. Res. 49 207 Google Scholar
[27]	Culcer D, Keser A C, Li Y, Tkachov G 2020 2D Mater. 7 022007 Google Scholar
[28]	Kim H J, Kim K S, Wang J F, Sasaki M, Satoh N, Ohnishi A, Kitaura M, Yang M, Li L 2013 Phys. Rev. Lett. 111 246603 Google Scholar
[29]	Xiong J, Kushwaha S K, Liang T, Krizan J W, Hirschberger M, Wang W, Cava R J, Ong N P 2015 Science 350 413 Google Scholar
[30]	Li H, He H T, Lu H Z, Zhang H C, Liu H C, Ma R, Fan Z Y, Shen S Q, Wang J N 2016 Nat. Commun. 7 10301 Google Scholar
[31]	Li C Z, Wang L X, Liu H W, Wang J, Liao Z M, Yu D P 2015 Nat. Commun. 6 10137 Google Scholar
[32]	Zhang C L, Xu S Y, Belopolski I, Yuan Z, Lin Z, Tong B, Bian G, Alidoust N, Lee C C, Huang S M 2016 Nat. Commun. 7 10735 Google Scholar
[33]	Huang X, Zhao L, Long Y, Wang P, Chen D, Yang Z, Liang H, Xue M, Weng H, Fang Z 2015 Phys. Rev. X 5 031023 Google Scholar
[34]	Li Q, Kharzeev D E, Zhang C, Huang Y, Pletikosić I, Fedorov A, Zhong R, Schneeloch J, Gu G, Valla T 2016 Nat. Phys. 12 550 Google Scholar
[35]	Liang S, Lin J, Kushwaha S, Xing J, Ni N, Cava R J, Ong N P 2018 Phys. Rev. X 8 031002 Google Scholar
[36]	Wang J, Li H, Chang C, He K, Lee J S, Lu H, Sun Y, Ma X, Samarth N, Shen S Q 2012 Nano Res. 5 739 Google Scholar
[37]	He H T, Liu H C, Li B K, Guo X, Xu Z J, Xie M H, Wang J N 2013 Appl. Phys. Lett. 103 031606 Google Scholar
[38]	Assaf B, Phuphachong T, Kampert E, Volobuev V, Mandal P, Sánchez-Barriga J, Rader O, Bauer G, Springholz G, De Vaulchier L 2017 Phys. Rev. Lett. 119 106602 Google Scholar
[39]	Wiedmann S, Jost A, Fauqué B, Van Dijk J, Meijer M, Khouri T, Pezzini S, Grauer S, Schreyeck S, Brüne C 2016 Phys. Rev. B 94 081302 Google Scholar
[40]	Mutch J, Chen W C, Went P, Qian T, Wilson I Z, Andreev A, Chen C C, Chu J H 2019 Sci. Adv. 5 eaav9771 Google Scholar
[41]	Nielsen H, Ninomiya M 1983 Phys. Lett. B 130 389 Google Scholar
[42]	Son D T, Spivak B Z 2013 Phys. Rev. B 88 104412 Google Scholar
[43]	Burkov A A 2014 Phys. Rev. Lett. 113 247203 Google Scholar
[44]	Goswami P, Pixley J H, Sarma S D 2015 Phys. Rev. B 92 075205 Google Scholar
[45]	Gao Y, Yang S A, Niu Q 2017 Phys. Rev. B 95 165135 Google Scholar
[46]	Dai X, Du Z, Lu H Z 2017 Phys. Rev. Lett. 119 166601 Google Scholar
[47]	Andreev A V, Spivak B Z 2018 Phys. Rev. Lett. 120 026601 Google Scholar
[48]	Wang H W, Fu B, Shen S Q 2018 Phys. Rev. B 98 081202(R Google Scholar
[49]	Fu B, Wang H W, Shen S Q 2020 Phys. Rev. B 101 125203 Google Scholar
[50]	Wang H W, Fu B, Shen S Q 2021 Phys. Rev. B 104 L241111 Google Scholar
[51]	Gorkov L P, Larkin A I, Khmelnitskii D E 1979 JETP Lett. 30 228
[52]	Hikami S, Larkin A I, Nagaoka Y 1980 Prog. Theor. Phys. 63 707 Google Scholar
[53]	Chakravarty S, Schmid A 1986 Phys. Rep. 140 193 Google Scholar
[54]	Wu X S, Li X B, Song Z M, Berger C, de Heer W A 2007 Phys. Rev. Lett. 98 136801 Google Scholar
[55]	Tikhonenko F V, Kozikov A A, Savchenko A K, Gorbachev R V 2009 Phys. Rev. Lett. 103 226801 Google Scholar
[56]	Checkelsky J G, Hor Y S, Liu M H, Qu D X, Cava R J, Ong N P 2009 Phys. Rev. Lett. 103 246601 Google Scholar
[57]	Chen J, Qin H J, Yang F, Liu J, Guan T, Qu F M, Zhang G H, Shi J R, Xie X C, Yang C L 2010 Phys. Rev. Lett. 105 176602 Google Scholar
[58]	He H T, Wang G, Zhang T, Sou I K, Wong G K, Wang J N, Lu H Z, Shen S Q, Zhang F C 2011 Phys. Rev. Lett. 106 166805 Google Scholar
[59]	Wang J, DaSilva A M, Chang C Z, He K, Jain J K, Samarth N, Ma X C, Xue Q K, Chan M H W 2011 Phys. Rev. B 83 245438 Google Scholar
[60]	Liu M H, Zhang J S, Chang C Z, Zhang Z C, Feng X, Li K, H e, K, Wa ng, L L, Chen X, Dai Xi, Fang Z, Xue Q K, Ma X C, Wang Y Y 2012 Phys. Rev. Lett. 108 036805 Google Scholar
[61]	Liu H C, Lu H Z, He H T, Li B K, Liu S G, He Q L, Wang G, S ou, I K, Shen S Q, Wang J N 2014 ACS Nano 8 9616 Google Scholar
[62]	Li H, Wang H W, Li Y, Zhang H C, Zhang S, Pan X C, Jia B, Song F Q, Wang J N 2019 Nano Lett. 19 2450 Google Scholar
[63]	Tkac V, Vyborny K, Komanicky V, Warmuth J, Michiardi M, Ngankeu A S 2019 Phys. Rev. Lett. 123 036406 Google Scholar
[64]	Zhao B, Cheng P H, Pan H Y, Zhang S, Wang B G, Wang G H, Xiu F X, Song F Q 2016 Sci. Rep. 6 22377 Google Scholar
[65]	Nakamura H, Huang D, Merz J, Khalaf E, Ostrovsky P, Yaresko A, Samal D, Takagi H 2020 Nat. Commun. 11 1161 Google Scholar
[66]	Suzuura H, Ando T 2002 Phys. Rev. Lett. 89 266603 Google Scholar
[67]	McCann E, Kechedzhi K, Falko V I, Suzuura H, Ando T, Altshuler B L 2006 Phys. Rev. Lett. 97 146805 Google Scholar
[68]	Garate I, Glazman L 2012 Phys. Rev. B 86 035422 Google Scholar
[69]	Lu H Z, Shi J R, Shen S Q 2011 Phys. Rev. Lett. 107 076801 Google Scholar
[70]	Lu H Z, Shen S Q 2014 Phys. Rev. Lett. 112 146601 Google Scholar
[71]	Gornyi I V, Kachorovskii V Y, Ostrovsky P M 2014 Phys. Rev. B 90 085401 Google Scholar
[72]	Wang H W, Fu B, Shen S Q 2020 Phys. Rev. Lett. 124 206603 Google Scholar
[73]	Lu H Z, Shen S Q 2015 Phys. Rev. B 92 035203 Google Scholar
[74]	Dai X, Lu H Z, Shen S Q, Yao H 2016 Phys. Rev. B 93 161110 Google Scholar
[75]	Fu B, Wang H W, Shen S Q 2019 Phys. Rev. Lett. 122 246601 Google Scholar
[76]	Chen W, Lu H Z, Zilberberg O 2019 Phys. Rev. Lett. 122 196603 Google Scholar
[77]	Novoselov K, Geim A, Morozov S, Jiang D, Katsnelson M, Grigorieva I, Dubonos S, Firsov A 2005 Nature 438 197 Google Scholar
[78]	Zhang Y, Tan Y W, Stormer H L, Kim P 2005 Nature 438 201 Google Scholar
[79]	Gusynin V P, Sharapov S G 2005 Phys. Rev. Lett. 95 146801 Google Scholar
[80]	Chang C Z, Liu C X, MacDonald A H 2023 Rev. Mod. Phys. 95 011002 Google Scholar
[81]	Yu R, Zhang W, Zhang H J, Zhang S C, Dai X, Fang Z 2010 Science 329 61 Google Scholar
[82]	Chang C Z, Zhang J, Feng X, Shen J, Zhang Z, Guo M, Li K, Ou Y, Wei P, Wang L L 2013 Science 340 167 Google Scholar
[83]	Xu Y, Miotkowski I, Liu C, Tian J F, Nam H, Alidoust N, Hu J N, Shih C K, Hasan M Z, Chen Y P 2014 Nat. Phys. 10 956 Google Scholar
[84]	Zhang C, Zhang Y, Yuan X, Lu S, Zhang J, Narayan A, Liu Y, Zhang H, Ni Z, Liu R 2019 Nature 565 331 Google Scholar
[85]	Tang F D, Ren Y F, Wang P P, Zhong R D, Schneeloch J, Yang S Y A, Yang K, Lee P A, Gu G D, Qiao X H, Zhang L Y 2019 Nature 569 537 Google Scholar
[86]	Galeski S, Ehmcke T, Wawrzynczak R, Lozano P M, Cho K, Sharma A, Das S 2021 Nat. Commun. 12 3197 Google Scholar
[87]	Wang C M, Sun H P, Lu H Z, Xie X C 2017 Phys. Rev. Lett. 119 136806 Google Scholar
[88]	Qin F, Li S, Du Z Z, Wang C M, Zhang W Q, Yu D P, Lu H Z, Xie X C 2020 Phys. Rev. Lett. 125 206601 Google Scholar
[89]	Yoshimi R, Yasuda K, Tsukazaki A, Takahashi K S, Nagaosa N, Kawasaki M, Tokura Y 2015 Nat. Commun. 6 8530 Google Scholar
[90]	Zhang S, Pi L, Wang R, Yu G, Pan X C, Wei Z, Zhang J, Xi C, Bai Z 2017 Nat. Commun. 8 977 Google Scholar
[91]	Mogi M, Okamura Y, Kawamura M, Yoshimi R, Yoshimi K, Tsukazaki A, Takahashi K S 2022 Nat. Phys. 18 390 Google Scholar
[92]	Liang T, Lin J J, Gibson Q, Kushwaha S, Liu M H, Wang W D, Xiong H Y, Sobota J A, Hashimoto M, Kirchmann P S, Shen Z X, Cava R J, Ong N P 2018 Nat. Phys. 14 451 Google Scholar
[93]	Sun Z, Cao Z, Cui J, Zhu C, Ma D, Wang H, Zhuo W, Cheng Z, Wang Z, Wan X, Chen X H 2020 Npj Quantum Mater. 5 36 Google Scholar
[94]	Liu Y Z, Wang H C, Fu H X, et al. 2021 Phys. Rev. B 103 L201110 Google Scholar
[95]	Mutch J, Ma X, Wang C, Malinowski P, AyresSims J, Jiang Q, Liu Z, Xiao D, Yankowitz M, Chu J H 2021 arXiv: 2101.02681 [cond-mat]
[96]	Gourgout A, Leroux M, Smirr J L, et al. 2022 npj Quantum Mater. 7 71 Google Scholar
[97]	Lozano P M, Cardoso G, Aryal N, Nevola D, Gu G, Tsvelik A, Yin W, Li Q 2022 Phys. Rev. B 106 L081124 Google Scholar
[98]	Burkov A A 2017 Phys. Rev. B 96 041110 Google Scholar
[99]	Nandy S, Sharma G, Taraphder A, Tewari S 2017 Phys. Rev. Lett. 119 176804 Google Scholar
[100]	Taskin A A, Legg H F, Yang F, Sasaki S, Kanai Y, Matsumoto K, Rosch A, Ando Y 2017 Nat. Commun. 8 1340 Google Scholar
[101]	Li H, Wang H W, He H T, Wang J N, Shen S Q 2018 Phys. Rev. B 97 201110 Google Scholar
[102]	Wu M, Zheng G L, Chu W W, Liu Y Q, Gao W S, Zhang H W, Lu J W, Han Y Y, Zhou J H, Ning W, Tian M L 2018 Phys. Rev. B 98 161110 Google Scholar
[103]	Kumar N, Guin S N, Felser C, Shekhar C 2018 Phys. Rev. B 98 041103 Google Scholar
[104]	Li P, Zhang C H, Zhang J W, Wen Y, Zhang X X 2018 Phys. Rev. B 98 121108(R Google Scholar
[105]	Huang D, Nakamura H, Takagi H 2021 Phys. Rev. Research 3 013268 Google Scholar
[106]	Wu M, Tu D, Nie Y, Miao S, Gao W, Han Y, Zhu X D, Zhou J H, Ning W, Tian M L 2022 Nano Lett. 22 73 Google Scholar
[107]	Zhong J Y, Zhuang J C, Du Y 2023 Chin. Phys. B 32 047203 Google Scholar
[108]	Gao A Y, Liu Y F, Hu C W, Qiu J X, Tzschaschel C, Ghosh B, Ho S C 2021 Nature 595 521 Google Scholar
[109]	Chen R, Sun H P, Gu M Q, Hua C B, Liu Q H, Lu H Z, Xie X C 2022 Natl. Sci. Rev. nwac140 Google Scholar
[110]	Zhai D W, Chen C, Xiao C, Yao W 2023 Nat. Commun. 14 1961 Google Scholar
[111]	Qu D X, Hor Y S, Xiong J, Cava R J, Ong N P 2010 Science 329 821 Google Scholar
[112]	Wang X, Du Y, Dou S, Zhang C 2012 Phys. Rev. Lett. 108 266806 Google Scholar
[113]	Zhang G, Qin H, Chen J, He X, Lu L, Li Y, Wu K 2011 Adv. Funct. Mater. 21 2351
[114]	Tang H, Liang D, Qiu R L J, Gao X P A 2011 ACS Nano 5 7510 Google Scholar
[115]	He H, Li B, Liu H, Guo X, Wang Z, Xie M, Wang J 2012 Appl. Phys. Lett. 100 032105 Google Scholar
[116]	He L P, Hong X C, Dong J K, Pan J, Zhang Z, Zhang J, Li S Y 2014 Phys. Rev. Lett. 113 246402 Google Scholar
[117]	Liang T, Gibson Q, Ali M N, Liu M, Cava R J, Ong N P 2015 Nat. Mater. 14 280 Google Scholar
[118]	Narayanan A, Watson M D, Blake S F, Bruyant N, Drigo L, Chen Y L, Prabhakaran D, Yan B, Felser C, Kong T, Canfield P C, Coldea A I 2015 Phys. Rev. Lett. 114 117201 Google Scholar
[119]	Feng J, Pang Y, Wu D, Wang Z J, Weng H M, Li J, Dai X, Fang Z, Shi Y, Lu L 2015 Phys. Rev. B 92 081306(R Google Scholar
[120]	Tian Y, Ghassemi N, Jr J H R 2021 Phys. Rev. Lett. 126 236401 Google Scholar
[121]	Fu B, Wang H W, Shen S Q 2020 Phys. Rev. Lett. 125 256601 Google Scholar
[122]	Fu B, Zou J Y, Hu Z A, Wang H W, Shen S Q 2022 npj Quantum Mater. 7 94 Google Scholar
[123]	Zou J Y, Fu B, Wang H W, Hu Z A, Shen S Q 2022 Phys. Rev. B 105 L201106 Google Scholar
[124]	Zou J Y, Chen R, Fu B, Wang H W, Hu Z A, Shen S Q 2023 Phys. Rev. B 107 125153 Google Scholar
[125]	Wang H W, Fu B, Zou J Y, Hu Z A, Shen S Q 2022 Phys. Rev. B 106 045111 Google Scholar
[126]	Shen S Q, Bao Y J, Ma M, Xie X C, Zhang F C 2005 Phys. Rev. B 71 155316 Google Scholar
[127]	Bjorken J D, Drell S D 1964 Relativistic Quantum Mechanics (New York: McGraw-Hill Inc.) pp45–60
[128]	Shen S Q, Shan W Y, Lu H Z 2011 SPIN 01 33 Google Scholar
[129]	Klitzing K V, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494 Google Scholar
[130]	Klitzing K v, Chakraborty T, Kim P, Madhavan V, Dai X, McIver J, Tokura Y, Savary L, Smirnova D, Rey A M, Felser C, Gooth J, Qi X L 2020 Nat. Rev. Phys. 2 397 Google Scholar
[131]	Schnyder A P, Ryu S, Furusaki A, Ludwig A W W 2008 Phys. Rev. B 78 195125 Google Scholar
[132]	Yang B J, Nagaosa N 2014 Nat. Commun. 5 4898 Google Scholar
[133]	Wilson K G 1975 New Phenomena in Subnuclear Physics (New York: Plenum) pp69–142
[134]	Rothe H J 2005 Lattice Gauge Theories: An Introduction (3rd Ed.) (Singapore: World Scientific) pp56–57
[135]	Zhang Y, Wang C, Yu L, Liu G, Liang A, Huang J 2017 Nat. Commun. 8 15512 Google Scholar
[136]	Zhang H J, Liu C X, Qi X L, Dai X, Fang Z, Zhang S C 2009 Nat. Phys. 5 438 Google Scholar
[137]	Xia Y Q, Qian D, Hsieh D, Wray L, Pal A, Lin H, Bansil A, Grauer D, Hor Y S, Cava R J, Hasan M Z 2009 Nat. Phys. 5 398 Google Scholar
[138]	Zee A 2010 Quantum Field Theory in a Nutshell (Vol. 7) (Princeton: Princeton University Press) pp99–100
[139]	Liu M H, Chang C Z, Zhang Z C, Zhang Y, Ruan W, He K, Wang L L, Chen X, Jia J F, Zhang S C, Xue Q K, Ma X C, Wang Y Y 2011 Phys. Rev. B 83 165440 Google Scholar
[140]	Takagaki Y, Jenichen B, Jahn U, Ramsteiner M, Friedl K J 2012 Phys. Rev. B 85 115314 Google Scholar
[141]	Jing Y M, Huang S Y, Zhang K, Wu J X, Guo Y F, Peng H L, Liu Z F, Xu H Q 2016 Nanoscale 8 1879 Google Scholar
[142]	Altshuler B L, Aronov A G, Lee P A 1980 Phys. Rev. Lett. 44 1288 Google Scholar
[143]	Lee P A, Ramakrishnan T V 1985 Rev. Mod. Phys. 57 287 Google Scholar
[144]	Liu Q, Liu C X, Xu C K, Qi X L, Zhang S C 2009 Phys. Rev. Lett. 102 156603 Google Scholar
[145]	Chen Y L, C hu, J H, Analytis J G, Liu Z K, Igarashi K, Kuo H H, Qi X L, Mo S K, Moore R G, Lu D H 2010 Science 329 659 Google Scholar
[146]	Tokura Y, Yasuda K, Tsukazaki A 2019 Nat. Rev. Phys. 1 126 Google Scholar
[147]	Okada S, Sambongi T, Ido M 1980 J. Phys. Soc. Jpn. 49 839 Google Scholar
[148]	Izumi M, Uchinokura K, Matsuura E 1981 Solid State Commun. 37 641 Google Scholar
[149]	DiSalvo F, Fleming R, Waszczak J 1981 Phys. Rev. B 24 2935 Google Scholar
[150]	Okada S, Sambongi T, Ido M, Tazuke Y, Aoki R, Fujita O 1982 J. Phys. Soc. Jpn. 51 460 Google Scholar
[151]	Bullett D 1982 Solid State Commun. 42 691 Google Scholar
[152]	Fjellvag H, Kjekshus A 1986 Solid State Commun. 60 91 Google Scholar
[153]	Chen Z G, Chen R, Zhong R, Schneeloch J, Zhang C, Huang Y, Qu F, Yu R, Li Q, Gu G, Wang N 2017 Proc. Natl. Acad. Sci. U.S.A. 114 816 Google Scholar
[154]	Manzoni G, Gragnaniello L, AutASs G, Kuhn T, Sterzi A, Cilento F, Zacchigna M, Enenkel V 2016 Phys. Rev. Lett. 117 237601 Google Scholar
[155]	Zhang J L, Wang C M, Guo C Y, Zhu X D, Zhang Y, Yang J Y, Wang Y Q, Qu Z, Pi L, Lu H Z, Tian M L 2019 Phys. Rev. Lett. 123 196602 Google Scholar
[156]	Jiang Y, Wang J, Zhao T, Dun Z L, Huang Q, Wu X S, Mourigal M, Zhou H D, Pan W, Ozerov M, Smirnov D, Jiang Z 2020 Phys. Rev. Lett. 125 046403 Google Scholar
[157]	Wang P, Ren Y, Tang F, Wang P, Hou T, Zeng H, Zhang L, Qiao Z H 2020 Phys. Rev. B 101 161201 Google Scholar
[158]	Zhao L X, Huang X C, Long Y J, Chen D, Liang H, Yang Z H, Xue M Q, Ren Z A, Weng H M, Fang Z, Dai X, Chen G F 2017 Chin. Phys. Lett. 34 037102 Google Scholar
[159]	Shahi P, Singh D J, Sun J P, Zhao L X, Chen G F, Lv Y Y, Li J, Yan J Q, Mandrus D G, Cheng J G 2018 Phys. Rev. X 8 021055 Google Scholar
[160]	Xu B, Zhao L, Marsik P, Sheveleva E, Lyzwa F, Dai Y, Chen G, Qiu X, Bernhard C 2018 Phys. Rev. Lett. 121 187401 Google Scholar
[161]	Wang C J 2021 Phys. Rev. Lett. 126 126601 Google Scholar
[162]	Wang Y G, Legg H F, Bömerich T, Park J, Biesenkamp S, Taskin A A, Braden M, Rosch A, Ando Y 2022 Phys. Rev. Lett. 128 176602 Google Scholar
[163]	Zhang Y, Wang C, Liu G, Liang A, Zhao L, Huang J, Gao Q, Shen B 2017 Sci. Bull. 62 950 Google Scholar
[164]	Adler S L 1969 Phys. Rev. 177 2426 Google Scholar
[165]	Bell J S, Jackiw R 1969 Nuovo Cimento A 60 47 Google Scholar
[166]	Fujikawa K 1979 Phys. Rev. Lett. 42 1195 Google Scholar
[167]	Peskin M E, Schroeder D V 1995 An Introduction to Quantum Field Theory (Cambridge: Perseus Books Publishing LLC) pp651–667
[168]	Weinberg S 1995 The Quantum Theory of Fields (Vol. 2) (Cambridge: Cambridge University Press) pp473–485
[169]	Dirac P A M 1958 The Principles of Quantum Mechanics (New York: Oxford University Press Inc.) pp253–267
[170]	Jackiw R, Johnson K 1969 Phys. Rev. 182 1459 Google Scholar
[171]	Kharzeev D E, Kikuchi Y, Meyer R, Tanizaki Y 2018 Phys. Rev. B 98 014305 Google Scholar
[172]	Yamamoto N, Yang D L 2021 Phys. Rev. D 103 125003 Google Scholar
[173]	Vilenkin A 1980 Phys. Rev. D 22 3080 Google Scholar
[174]	Fukushima K, Fukushima D E, Warringa H J 2008 Phys. Rev. D 78 074033 Google Scholar
[175]	Huang Z M, Zhou J H, Shen S Q 2017 Phys. Rev. B 96 085201 Google Scholar
[176]	Lu H Z, Zhang S B, Shen S Q 2015 Phys. Rev. B 92 045203 Google Scholar
[177]	Zhang S B, Lu H Z, Shen S Q 2016 New J. Phys. 18 053039 Google Scholar
[178]	Klier J, Gornyi I V, Mirlin A D 2017 Phys. Rev. B 96 214209 Google Scholar
[179]	Niemi A J, Semenoff G W 1983 Phys. Rev. Lett. 51 2077 Google Scholar
[180]	Redlich A N 1984 Phys. Rev. Lett. 52 18 Google Scholar
[181]	Jackiw R 1984 Phys. Rev. D 29 2375 Google Scholar
[182]	Boyanovsky D, Blankenbecler R, Yahalom R 1986 Nucl. Phys. B 270 483 Google Scholar
[183]	Schakel A M J 1991 Phys. Rev. D 43 1428 Google Scholar
[184]	Chu R L, Shi J R, Shen S Q 2011 Phys. Rev. B 84 085312 Google Scholar
[185]	Lapa M F 2019 Phys. Rev. B 99 235144 Google Scholar
[186]	Lu R, Sun H, Kumar S, Wang Y, Gu M, Zeng M, Hao Y J, Li J 2021 Phys. Rev. X 11 011039 Google Scholar
[187]	Essin A M, Moore J E, Vanderbilt D 2009 Phys. Rev. Lett. 102 146805 Google Scholar
[188]	Sitte M, Rosch A, Altman E, Fritz L 2012 Phys. Rev. Lett. 108 126807 Google Scholar
[189]	Wang J, Lian B, Qi X L, Zhang S C 2015 Phys. Rev. B 92 081107 Google Scholar
[190]	Gu M, Li J, Sun H, Zhao Y, Liu C, Liu J, Lu H, Liu Q 2021 Nat. Commun. 12 3524 Google Scholar
[191]	Mogi M, Kawamura M, Yoshimi R, Tsukazaki A, Kozuka Y, Shirakawa N, Takahashi K S, Kawasaki M, Tokura Y 2017 Nat. Mater. 16 516 Google Scholar
[192]	Mogi M, Kawamura M, Tsukazaki A, Yoshimi R, Takahashi K S, Kawasaki M, Tokura Y 2017 Sci. Adv. 3 eaao1669 Google Scholar
[193]	Xiao D, Jiang J, Shin J H, Wang W, Wang F, Zhao Y F, Liu C, Wu W, Chan M H W, Samarth N, Chang C Z 2018 Phys. Rev. Lett. 120 056801 Google Scholar
[194]	Zhang D, Shi M, Zhu T, Xing D, Zhang H, Wang J 2019 Phys. Rev. Lett. 122 206401 Google Scholar
[195]	Liu C, Wang Y, Li H, Wu Y, Li Y, Li J, He K, Xu Y, Zhang J, Wang Y 2020 Nat. Mat. 19 522 Google Scholar
[196]	Wilczek F 1987 Phys. Rev. Lett. 58 1799 Google Scholar
[197]	Qi X L, Hughes T L, Zhang S C 2008 Phys. Rev. B 78 195424 Google Scholar
[198]	Spaldin N A, Fiebig M 2005 Science 309 391 Google Scholar
[199]	Fiebig M 2005 J. Phys. D 38 R123 Google Scholar
[200]	Maciejko J, Qi X L, Drew H D, Zhang S C 2010 Phys. Rev. Lett. 105 166803 Google Scholar
[201]	Tse W K, MacDonald A H 2011 Phys. Rev. B 84 205327 Google Scholar
[202]	Li R, Wang J, Qi X L, Zhang S C 2010 Nat. Phys. 6 284 Google Scholar
[203]	Sekine A, Nomura K 2014 J Phys. Soc. Jpn. 83 104709 Google Scholar
[204]	Sekine A, Nomura K 2021 J. Appl. Phys. 129 141101 Google Scholar
[205]	Shoron O F, Kealhofer D A, Goyal M, Schumann T, Burkov A A, Stemmer S 2021 Appl. Phys. Lett. 119 171907 Google Scholar
[206]	Abrikosov A A 1998 Phys. Rev. B 58 2788 Google Scholar
[207]	Parish M M, Littlewood P B 2003 Nature 426 162 Google Scholar
[208]	Parish M M, Littlewood P B 2005 Phys. Rev. B 72 094417 Google Scholar
[209]	Cao H, Tian J, Miotkowski I, Shen T, Hu J, Qiao S, Chen Y P 2012 Phys. Rev. Lett. 108 216803 Google Scholar
[210]	Wang C M, Lei X L 2012 Phys. Rev. B 86 035442 Google Scholar
[211]	Avron J E, Seiler R, Simon B 1983 Phys. Rev. Lett. 51 51 Google Scholar
[212]	Halperin B I 1987 Jpn. J. Appl. Phys. 26 1913 Google Scholar
[213]	Zhao P L, Lu H Z, Xie X C 2021 Phys. Rev. Lett. 127 046602 Google Scholar
[214]	Nagaosa N, Sinova J, Onoda S, MacDonald A H, Ong N P 2010 Rev. Mod. Phys. 82 1539 Google Scholar
[215]	Chen R, Shen S Q 2023 arXiv: 2304.04229 [cond-mat]
[216]	Zhou H M, Li H L, Xu D H, Chen C Z, Sun Q F, Xie X C 2022 Phys. Rev. Lett. 129 096601 Google Scholar
[217]	Gong M, Liu H W, Jiang H, Chen C Z, Xie X C 2023 Natl. Sci. Rev 10 nwad025 Google Scholar
[218]	Goos F, Hanchen H 1947 Ann. Phys. 436 333 Google Scholar

[1]	纪雨萱, 张明楷, 李妍. 二维光子晶体中的双波段半狄拉克锥与零折射率材料. 物理学报, 2024, 73(18): 181101. doi: 10.7498/aps.73.20240800
[2]	张明媚, 郭亚涛, 付旭日, 李梦蕾, 任宝藏, 郑军, 袁瑞玚. 铁磁电极单层二硫化钼纳米带量子结构中的自旋开关效应和巨磁阻. 物理学报, 2023, 72(15): 157202. doi: 10.7498/aps.72.20230483
[3]	胡聚罡, 贾振宇, 李绍春. 碳化硅衬底上外延双层石墨烯的电输运性质. 物理学报, 2022, 71(12): 127204. doi: 10.7498/aps.71.20220062
[4]	张蔚曦, 李勇, 田昌海, 佘彦超. 具有大磁晶各向异性能的单层BaPb的室温量子反常霍尔效应. 物理学报, 2021, 70(15): 157502. doi: 10.7498/aps.70.20210014
[5]	朱燕清, 张丹伟, 朱诗亮. 用光晶格模拟狄拉克、外尔和麦克斯韦方程. 物理学报, 2019, 68(4): 046701. doi: 10.7498/aps.68.20181929
[6]	刘洁, 王禄, 孙令, 王文奇, 吴海燕, 江洋, 马紫光, 王文新, 贾海强, 陈弘. 基于p-n结中反常光电转换现象的新型带间跃迁量子阱红外探测器. 物理学报, 2018, 67(12): 128101. doi: 10.7498/aps.67.20180588
[7]	刘松, 罗春荣, 翟世龙, 陈怀军, 赵晓鹏. 负质量密度声学超材料的反常多普勒效应. 物理学报, 2017, 66(2): 024301. doi: 10.7498/aps.66.024301
[8]	王晓, 陈立潮, 刘艳红, 石云龙, 孙勇. 纵模对光子晶体中类狄拉克点传输特性的影响. 物理学报, 2015, 64(17): 174206. doi: 10.7498/aps.64.174206
[9]	黄学勤, 陈子亭. k=0处的类狄拉克锥. 物理学报, 2015, 64(18): 184208. doi: 10.7498/aps.64.184208
[10]	曹惠娴, 梅军. 声子晶体中的半狄拉克点研究. 物理学报, 2015, 64(19): 194301. doi: 10.7498/aps.64.194301
[11]	康永强, 高鹏, 刘红梅, 张淳民, 石云龙. 单负材料组成一维光子晶体双量子阱结构的共振模. 物理学报, 2015, 64(6): 064207. doi: 10.7498/aps.64.064207
[12]	史旺林, 刘兴业, 刘振兴. Vaidya-Bonner-de Sitter黑洞对狄拉克粒子的热辐射. 物理学报, 2004, 53(7): 2396-2400. doi: 10.7498/aps.53.2396
[13]	徐晓华, 姜浩, 傅荣堂, 孙鑫. 耦合量子点的反常极化. 物理学报, 2000, 49(3): 426-429. doi: 10.7498/aps.49.426
[14]	曹江陵, 彭方志. Einstein-Maxwell-Dilaton黑膜的狄拉克粒子辐射. 物理学报, 1998, 47(2): 177-182. doi: 10.7498/aps.47.177
[15]	董正超. 磁三明治结构巨磁阻的量子解析理论. 物理学报, 1997, 46(9): 1801-1807. doi: 10.7498/aps.46.1801
[16]	易林, 姚凯伦. 自旋玻璃系统的量子输运方程. 物理学报, 1994, 43(6): 1024-1028. doi: 10.7498/aps.43.1024
[17]	谭维翰, 李宇舫, 张卫平. 具有零或负扩散系数的Fokker-Planck方程的形式解及其在量子光学中的应用. 物理学报, 1988, 37(3): 396-407. doi: 10.7498/aps.37.396
[18]	江启杜, 刘福绥. 较高杂质浓度磁性合金负磁阻的关联对模型. 物理学报, 1986, 35(2): 177-187. doi: 10.7498/aps.35.177
[19]	吴杭生, 顾一鸣. 急冷Al-Si合金负磁阻效应的一个可能解释. 物理学报, 1982, 31(8): 1126-1129. doi: 10.7498/aps.31.1126
[20]	管惟炎, 陈熙琛, 王祖仑, 易孙圣. 急冷Al-Si超导合金的负磁阻效应. 物理学报, 1981, 30(9): 1284-1286. doi: 10.7498/aps.30.1284

计量

文章访问数: 5447
PDF下载量: 387
被引次数: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

搜索

留言板