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手征马约拉纳费米子

王靖

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手征马约拉纳费米子

王靖

Chiral Majorana fermion

Wang Jing
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  • 手征马约拉纳费米子是具有手性的无质量费米子, 是其本身的反粒子, 只能存在于1+1维(即1维空间+1维时间)或者9+1维. 在凝聚态物理中, 1维手征马约拉纳费米子可看成1/2分数化的狄拉克费米子, 并作为二维拓扑态的边缘元激发. 奇数个手征马约拉纳费米子边缘态的存在也预示着体系中存在满足非阿贝尔量子统计的伊辛任意子. 手征马约拉纳费米子也可进行非阿贝尔编织, 理论上可用来实现容错量子计算, 因此近年来在凝聚态物理研究中引起了广泛的兴趣. 本文从二维拓扑态出发, 介绍手征拓扑超导态和量子反常霍尔态之间的深刻联系, 并由此得出量子反常霍尔平台转变与超导近邻实现手征马约拉纳费米子的方案, 最后以单通道手征马约拉纳费米子为例, 探讨其实现电子态的非阿贝尓量子门.
    The chiral Majorana fermion, is a massless fermionic particle being its own antiparticle, which was predicted to live in (1+1)D (i.e. one-dimensional space plus one-dimensional time) or (9+1)D. In condensed matter physics, one-dimensional (1D) chiral Majorana fermion can be viewed as the 1/2 of the chiral Dirac fermion, which could arise as the quasiparticle edge state of a two-dimensional (2D) topological state of matter. The appearance of an odd number of 1D chiral Majorana fermions on the edge implies that there exist the non-Abelian defects in the bulk. The chiral Majorana fermion edge state can be used to realize the non-Abelian quantum gate operations on electron states. Starting with the topological states in 2D, we illustrate the general and intimate relation between chiral topological superconductor and quantum anomalous Hall insulator, which leads to the theoretical prediction of the chiral Majorana fermion from the quantum anomalous Hall plateau transition in proximity to a conventional s-wave superconductor. We show that the propagation of chiral Majorana fermions leads to the same unitary transformation as that in the braiding of Majorana zero modes, and may be used for the topological quantum computation.
      通信作者: 王靖, wjingphys@fudan.edu.cn
    • 基金项目: 国家级-国家重点基础研究发展计划(2016YFA0300703,2019YFA0308404)
      Corresponding author: Wang Jing, wjingphys@fudan.edu.cn
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    Schnyder A P, Ryu S, Furusaki A, Ludwig A W W 2008 Phys. Rev. B 78 195125Google Scholar

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    Wang J, Lian B, Qi X L, Zhang S C 2015 Phys. Rev. B 92 081107Google Scholar

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    Nayak C, Simon S H, Stern A, Freedman M, Das Sarma S 2008 Rev. Mod. Phys. 80 1083Google Scholar

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    Lian B, Sun X Q, Vaezi A, Qi X L, Zhang S C 2018 Proc. Natl. Acad. Sci. U.S.A. 115 10938Google Scholar

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    Kitaev A 2006 Ann. Phys. 321 2Google Scholar

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    Wang J, Lian B 2018 Phys. Rev. Lett. 121 256801Google Scholar

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    Lian B, Wang J 2019 Phys. Rev.B 99 041404Google Scholar

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    Kayyalha M, Xiao D, Zhang R, et al. 2020 Science 367 64Google Scholar

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    Ji W, Wen X G 2018 Phys. Rev. Lett. 120 107002Google Scholar

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    Banerjee M, Heiblum M, Umansky V, et al. 2018 Nature (London) 559 205Google Scholar

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    Kasahara Y, Ohnishi T, Mizukami Y, et al. 2018 Nature (London) 559 227Google Scholar

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    Simon S H 2018 Phys. Rev. B 97 121406Google Scholar

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    Hu Y, Kane C L 2018 Phys. Rev. Lett. 120 066801Google Scholar

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    Akhmerov A R, Nilsson J, Beenakker C W 2009 Phys. Rev. Lett. 102 216404Google Scholar

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    Stern A, Halperin B I 2006 Phys. Rev. Lett. 96 016802Google Scholar

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    Bonderson P, Kitaev A, Shtengel K 2006 Phys. Rev. Lett. 96 016803Google Scholar

    [57]

    Lian B, Wang J, Zhang S C 2016 Phys. Rev. B 93 161401Google Scholar

    [58]

    Zhao L, Arnault E G, Bondarev A, et al. 2020 Nat. Phys.Google Scholar

  • 图 1  二维体系中的拓扑态. (上) 手征拓扑超导态与量子霍尔态的对应, 在这两个体系中, 时间反演对称性破缺, 同时存在手征边界态; (下) 螺旋拓扑超导态与量子自旋霍尔态的对应, 这两个体系同时保持时间反演对称性, 且存在螺旋边界态. 从边界态的自由度来看, (QSH) = (QH)2 = (Helical SC)2 = (Chiral SC)4, 其中QSH = 量子自旋霍尔态, QH = 量子霍尔态, Helical SC = 螺旋拓扑超导态, Chiral SC = 手征拓扑超导态, 指数1, 2, 4指这几种拓扑物质中边界态自由度之间的关系. 取自文献[26]

    Fig. 1.  Topological states in 2D. Top row: Schematic comparison of a 2D chiral superconductor and the QH/QAH state. In both systems, TR symmetry is broken and the edge states carry a definite chirality. Bottom row: Schematic comparison of a 2D TR-invariant TSC and the QSH insulator. Both systems preserve TR symmetry and have a helical pair of edge states, where opposite spin states counterpropagate. The dashed lines show that the edge states of the superconductors are Majorana fermions so that the E < 0 part of the quasiparticle spectra is redundant. In terms of the edge-state degrees of freedom, we have (QSH) = (QH/QAH)2 = (Helical SC)2 = (Chiral SC)4. The QAH state can be obtained from the QSH state by magnetic doping, and the chiral TSC state can be obtained from the QAH state by proximity contact with a conventional superconductor. The superscripts 1, 2, 4 denote relation of the number of degree of freedom of edge states in these topological matter. Adapted from Ref. [26], APS.

    图 2  手征马约拉纳费米子 (a) 基本想法: 将量子反常霍尔的手征狄拉克费米子一分为二得到手征马约拉纳费米子; (b) 实现手征马约拉纳费米子的量子反常霍尔绝缘体-超导体的异质结器件. 取自文献[41]

    Fig. 2.  Chiral Majorana fermion: (a) Basic idea: the quantum anomalous Hall chiral edge state splits into two chiral Majorana fermions; (b) the hybrid quantum anomalous Hall-superconductor device for chiral Majorana fermion. Adapted from Ref. [41], APS.

    图 3  手征马约拉纳费米子实现电子态的非阿贝尔量子门操作 (a) 量子反常霍尔绝缘体-手征拓扑超导-量子反常霍尔绝缘体的异质结器件实现电子态的非阿贝尔量子门, 其等价于实现单比特ZH量子门. 其中Z是泡利-Z门, H是Hadamard门; (b) Corbino异质结器件测量手征马约拉纳费米子量子相干. 取自文献[38]

    Fig. 3.  Braiding of chiral Majorana fermion: (a) The QAH-TSC-QAH device realize the non-Abelian gate which is equivalent to a Hadamard gate H followed by a Pauli-Z gate Z; (b) quantum interference in the QAH-TSC-QAH-TSC Corbino junction. Adapted from Ref. [38], PNAS.

  • [1]

    Anderson P W 1997 Basic Notions of Condensed Matter Physics (Boulder: Westview Press) pp57–87

    [2]

    Landau L D, Lifshitz E M 1980 Statistical Physics (Oxford: Pergamon Press) pp1-10

    [3]

    von Klitzing K, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494Google Scholar

    [4]

    Thouless D J, Kohmoto M, Nightingale M P, den Nijs M 1982 Phys. Rev. Lett. 49 405Google Scholar

    [5]

    Wen X G 1995 Adv. Phys. 44 405Google Scholar

    [6]

    Laughlin R B 1981 Phys. Rev. B 23 5632Google Scholar

    [7]

    Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 226801Google Scholar

    [8]

    Bernevig B A, Hughes T L, Zhang S C 2006 Science 314 1757Google Scholar

    [9]

    König M, Wiedmann S, Brüne C, et al. 2007 Science 318 766Google Scholar

    [10]

    Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045Google Scholar

    [11]

    Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057Google Scholar

    [12]

    Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 146802Google Scholar

    [13]

    Qi X L, Hughes T L, Zhang S C 2008 Phys. Rev. B 78 195424Google Scholar

    [14]

    Moore J E, Balents L 2007 Phys. Rev. B 75 121306Google Scholar

    [15]

    Fu L, Kane C L, Mele E J 2007 Phys. Rev. Lett. 98 106803Google Scholar

    [16]

    Roy R 2009 Phys. Rev. B 79 195322Google Scholar

    [17]

    Haldane F D M 1988 Phys. Rev. Lett. 61 2015Google Scholar

    [18]

    Qi X L, Wu Y S, Zhang S C 2006 Phys. Rev. B 74 085308Google Scholar

    [19]

    Liu C X, Qi X L, Dai X, Fang Z, Zhang S C 2008 Phys. Rev. Lett. 101 146802Google Scholar

    [20]

    Yu R, Zhang W, Zhang H J, et al. 2010 Science 329 61Google Scholar

    [21]

    Chang C Z, Zhang J, Feng X, et al. 2013 Science 340 167Google Scholar

    [22]

    Wang J, Lian B, Zhang S C 2015 Phys. Scr. 2015 014003

    [23]

    Mogi M, Yoshimi R, Tsukazaki A, et al. 2015 Appl. Phys. Lett. 107 182401Google Scholar

    [24]

    Deng Y, Yu Y, Shi M Z, et al. 2020 Science 367 895Google Scholar

    [25]

    Read N, Green D 2000 Phys. Rev. B 61 10267Google Scholar

    [26]

    Qi X L, Hughes T L, Raghu S, Zhang S C 2009 Phys. Rev. Lett. 102 187001Google Scholar

    [27]

    Schnyder A P, Ryu S, Furusaki A, Ludwig A W W 2008 Phys. Rev. B 78 195125Google Scholar

    [28]

    Kitaev A 2009 AIP Conf. Proc. 1134 22

    [29]

    Wang J, Lian B, Qi X L, Zhang S C 2015 Phys. Rev. B 92 081107Google Scholar

    [30]

    Liu Q, Liu C X, Xu C, Qi X L, Zhang S C 2009 Phys. Rev. Lett. 102 156603Google Scholar

    [31]

    Zhang D, Shi M, Zhu T, et al. 2019 Phys. Rev. Lett. 122 206401Google Scholar

    [32]

    Li J, Li Y, Du S, et al. 2019 Sci. Adv. 5 eaaw5685Google Scholar

    [33]

    Gong Y, Guo J, Li J, et al. 2019 Chin. Phys. Lett. 36 076801Google Scholar

    [34]

    Otrokov M M, Klimovskikh I I, Bentmann H, et al. 2019 Nature (London) 576 416Google Scholar

    [35]

    Moore G, Read N 1991 Nucl. Phys. B 360 362Google Scholar

    [36]

    Wen X G 1993 Phys. Rev. Lett. 70 355Google Scholar

    [37]

    Nayak C, Simon S H, Stern A, Freedman M, Das Sarma S 2008 Rev. Mod. Phys. 80 1083Google Scholar

    [38]

    Lian B, Sun X Q, Vaezi A, Qi X L, Zhang S C 2018 Proc. Natl. Acad. Sci. U.S.A. 115 10938Google Scholar

    [39]

    Kitaev A 2006 Ann. Phys. 321 2Google Scholar

    [40]

    Qi X L, Hughes T L, Zhang S C 2010 Phys. Rev. B 82 184516Google Scholar

    [41]

    Wang J, Zhou Q, Lian B, Zhang S C 2015 Phys. Rev. B 92 064520Google Scholar

    [42]

    Wang J 2016 Phys. Rev. B 94 214502Google Scholar

    [43]

    Chung S B, Qi X L, Maciejko J, Zhang S C 2011 Phys. Rev. B 83 100512Google Scholar

    [44]

    Wang J, Lian B 2018 Phys. Rev. Lett. 121 256801Google Scholar

    [45]

    Lian B, Wang J 2019 Phys. Rev.B 99 041404Google Scholar

    [46]

    Fu L, Kane C L 2009 Phys. Rev. Lett. 102 216403Google Scholar

    [47]

    He Q L, Pan L, Stern A L, et al. 2017 Science 357 294Google Scholar

    [48]

    Kayyalha M, Xiao D, Zhang R, et al. 2020 Science 367 64Google Scholar

    [49]

    Ji W, Wen X G 2018 Phys. Rev. Lett. 120 107002Google Scholar

    [50]

    Banerjee M, Heiblum M, Umansky V, et al. 2018 Nature (London) 559 205Google Scholar

    [51]

    Kasahara Y, Ohnishi T, Mizukami Y, et al. 2018 Nature (London) 559 227Google Scholar

    [52]

    Simon S H 2018 Phys. Rev. B 97 121406Google Scholar

    [53]

    Hu Y, Kane C L 2018 Phys. Rev. Lett. 120 066801Google Scholar

    [54]

    Akhmerov A R, Nilsson J, Beenakker C W 2009 Phys. Rev. Lett. 102 216404Google Scholar

    [55]

    Stern A, Halperin B I 2006 Phys. Rev. Lett. 96 016802Google Scholar

    [56]

    Bonderson P, Kitaev A, Shtengel K 2006 Phys. Rev. Lett. 96 016803Google Scholar

    [57]

    Lian B, Wang J, Zhang S C 2016 Phys. Rev. B 93 161401Google Scholar

    [58]

    Zhao L, Arnault E G, Bondarev A, et al. 2020 Nat. Phys.Google Scholar

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  • 收稿日期:  2020-04-11
  • 修回日期:  2020-05-13
  • 刊出日期:  2020-06-05

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