Recent progress of transport theory in Dirac quantum materials

Wang Huan-Wen; Fu Bo; Shen Shun-Qing

doi:10.7498/aps.72.20230672

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:

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)

FullText HTML : translate this paragraph

References

[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

Cited By

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]

Figure 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]

DownLoad: Full-Size Img PowerPoint

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]

Figure 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]

DownLoad: Full-Size Img PowerPoint

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]

Figure 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]

DownLoad: Full-Size Img PowerPoint

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]

Figure 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]

DownLoad: Full-Size Img PowerPoint

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]

Figure 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]

DownLoad: Full-Size Img PowerPoint

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]

Figure 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]

DownLoad: Full-Size Img PowerPoint

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]

Figure 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]

DownLoad: Full-Size Img PowerPoint

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]

Figure 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]

DownLoad: Full-Size Img PowerPoint

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]

Figure 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]

DownLoad: Full-Size Img PowerPoint

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}$

Figure 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 $x\to0$

DownLoad: Full-Size Img PowerPoint

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 $({m}_{i})$	$\times$	$\checkmark$	$\times$	$\varDelta_{{\rm{M}}}$

DownLoad: 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	$\infty$
$t_{-}$	$\|1, -1\rangle$	$\dfrac{4\eta^{2}(1+\eta^{2})}{(1+3\eta^{2})^{2}}$	$\dfrac{4(1+\eta)^{2}\eta^{2}}{(1+3\eta^{2})(1-\eta)^{2}}$

DownLoad: 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]	Ji Yu-Xuan, Zhang Ming-Kai, Li Yan. Dual-band semi-Dirac cones in two-dimensional photonic crystal and zero-index material. Acta Physica Sinica, 2024, 73(18): 181101. doi: 10.7498/aps.73.20240800
[2]	Zhang Ming-Mei, Guo Ya-Tao, Fu Xu-Ri, Li Meng-Lei, Ren Bao-Cang, Zheng Jun, Yuan Rui-Yang. Spin-switching effect and giant magnetoresistance in quantum structure of monolayer MoS₂ nanoribbons with ferromagnetic electrode. Acta Physica Sinica, 2023, 72(15): 157202. doi: 10.7498/aps.72.20230483
[3]	Hu Ju-Gang, Jia Zhen-Yu, Li Shao-Chun. Electron transport property of epitaixial bilayer graphene on SiC substrate. Acta Physica Sinica, 2022, 71(12): 127204. doi: 10.7498/aps.71.20220062
[4]	Zhang Wei-Xi, Li Yong, Tian Chang-Hai, She Yan-Chao. Room-temperature quantum anomalous Hall effect in monolayer BaPb with large magnetocrystalline anisotropy energies. Acta Physica Sinica, 2021, 70(15): 157502. doi: 10.7498/aps.70.20210014
[5]	Zhu Yan-Qing, Zhang Dan-Wei, Zhu Shi-Liang. Simulating Dirac, Weyl and Maxwell equations with cold atoms in optical lattices. Acta Physica Sinica, 2019, 68(4): 046701. doi: 10.7498/aps.68.20181929
[6]	Liu Jie, Wang Lu, Sun Ling, Wang Wen-Qi, Wu Hai-Yan, Jiang Yang, Ma Zi-Guang, Wang Wen-Xin, Jia Hai-Qiang, Chen Hong. Anomalous light-to-electricity conversion of low dimensional semiconductor in p-n junction and interband transition quantum well infrared detector. Acta Physica Sinica, 2018, 67(12): 128101. doi: 10.7498/aps.67.20180588
[7]	Liu Song, Luo Chun-Rong, Zhai Shi-Long, Chen Huai-Jun, Zhao Xiao-Peng. Inverse Doppler effect of acoustic metamaterial with negative mass density. Acta Physica Sinica, 2017, 66(2): 024301. doi: 10.7498/aps.66.024301
[8]	Wang Xiao, Chen Li-Chao, Liu Yan-Hong, Shi Yun-Long, Sun Yong. Effect of longitudinal mode on the transmission properties near the Dirac-like point of the photonic crystals. Acta Physica Sinica, 2015, 64(17): 174206. doi: 10.7498/aps.64.174206
[9]	Cao Hui-Xian, Mei Jun. Semi-Dirac points in two-dimensional phononic crystals. Acta Physica Sinica, 2015, 64(19): 194301. doi: 10.7498/aps.64.194301
[10]	Huang Xue-Qin, Chan Che-Ting. Dirac-like cones at k=0. Acta Physica Sinica, 2015, 64(18): 184208. doi: 10.7498/aps.64.184208
[11]	Kang Yong-Qiang, Gao Peng, Liu Hong-Mei, Zhang Chun-Min, Shi Yun-Long. Resonant modes in photonic double quantum well structures with single-negative materials. Acta Physica Sinica, 2015, 64(6): 064207. doi: 10.7498/aps.64.064207
[12]	Shi Wang-Lin, Liu Xing-Ye, Liu Zhen-Xing. Dirac radiation of Vaidya-Bonner-de Sitter black hole. Acta Physica Sinica, 2004, 53(7): 2396-2400. doi: 10.7498/aps.53.2396
[13]	Xu Xiao-hua, Jiang Hao, Fu Rong-tang, Sun Xin. . Acta Physica Sinica, 2000, 49(3): 426-429. doi: 10.7498/aps.49.426
[14]	CAO JIANG-LING, PENG FANG-ZHI. DIRAC PARTICLE RADIATION FROM EINSTEIN-MAXWELL-DILATON BLACK PLANE. Acta Physica Sinica, 1998, 47(2): 177-182. doi: 10.7498/aps.47.177
[15]	DONG ZHENG-CHAO. QUANTUM ANALYTICAL THEORY FOR GIANT MAGNETORESISTANCE IN A MAGNETIC SANDWICH STRUCTURE. Acta Physica Sinica, 1997, 46(9): 1801-1807. doi: 10.7498/aps.46.1801
[16]	YI LIN, YAO KAI-LUN. QUANTUM TRANSPORT EQUATION FOR SPIN GLASS SYSTEMS. Acta Physica Sinica, 1994, 43(6): 1024-1028. doi: 10.7498/aps.43.1024
[17]	TAN WEI-HAN, LI YU-FANG, ZHANG WEI-PING. THE FORMAL SOLUTIONS FOR FOKKER-PLANCK EQUATION WITH ZERO OR NEGATIVE DIFFUSION COEFFICIENTS AND THEIR APPLICATIONS TO QUANTUM OPTICS. Acta Physica Sinica, 1988, 37(3): 396-407. doi: 10.7498/aps.37.396
[18]	JIANG QI-DU, LIU FU-SUI. COUPLING PAIR MODEL FOR NEGATIVE MAGNETORESISTANCE OF MAGNETIC ALLOY WITH SLIGHTLY HIGH CONCENTRATION OF IMPURITY. Acta Physica Sinica, 1986, 35(2): 177-187. doi: 10.7498/aps.35.177
[19]	WU HANG-SHENG, GU YI-MING. A POSSIBLE EXPLANATION FOR THE NEGATIVE MAGNETO-RESISTANCE EFFECT IN THE RAPIDLY COOLED Al-Si ALLOYS. Acta Physica Sinica, 1982, 31(8): 1126-1129. doi: 10.7498/aps.31.1126
[20]	GUAN WEI-YAN, CHEN XI-SEN, WANG ZU-LUN, YI SUN-SHENG. NEGATIVE MAGNETORESISTIVITY OF RAPIDLY QUENCHED SUPERCONDUCTING ALLOY AI-11.3 at% Si. Acta Physica Sinica, 1981, 30(9): 1284-1286. doi: 10.7498/aps.30.1284

Catalog

Vol. 72, Issue 17 - 2023-09-05

Metrics

Abstract views: 5456
PDF Downloads: 387
Cited By: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

Search

Article

留言板