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Majorana bound states are considered useful for realizing topological quantum computation since they obey the non-Abelian quantum statistics. Recent experiments have provided evidences for their existence in some superconducting systems, triggering significant interests from scientists in the field of condensed matter physics and related materials science. In this article, we briefly review the basic concepts and recent developments in the study of Majorana bound states. We first discuss about the origin of the nontrivial topology in superconducting systems within the Bogoliubov-de Gennes mean-field scheme. Then we show the construction of Majorana quasiparticle excitations from an electronic state, and the realization of non-Abelian statistics based on position exchanges of the Majorana bound states hosted in superconductivity vortices. Afterwards we talk about specific one-dimensional and two-dimensional topological superconductors, and propose possible experimental methods for detecting Majorana bound states and operating the Majorana qubits. In particular, a quantum device for Majorana braiding without moving vortices is introduced. Finally, perspectives of the study on Majorana bound states are provided.
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Keywords:
- Majorana bound state /
- topological superconductivity /
- non-Abelian statitistics /
- quantum computation
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图 1 (a) 量子霍尔效应及量子反常霍尔效应; (b)量子自旋霍尔效应; (c) 拓扑超导的体能带结构(红线和蓝线)和边缘态(绿色)的色散关系; (d)实空间边缘态的示意图
Figure 1. Schematic energy band structures for (a) quantum Hall effect and quantum anomalous Hall effect, (b) quantum spin Hall effect, (c) a topological superconductor and (d) sche-matic diagram of topological edge/surface states in real space.
图 2 拓扑超导约化能隙
$ {{g}}\left({{k}}\right)/\left|{{g}}\left({{k}}\right)\right| $ 的动量空间分布Figure 2. Distribution of normalized topological superconduc-tivity gap
$ {{g}}\left({{k}}\right)/\left|{{g}}\left({{k}}\right)\right| $ in momentum space.
图 3 利用拓扑超导量子涡旋里的Majorana束缚态实现非阿贝尔统计的示意图, 其中黑色箭号代表量子涡旋位置交换的轨迹, 当量子涡旋跨越红线时超导相位发生2π的不连续跳跃
Figure 3. Schematics of realization of non-Abelian statistics using Majorana bound states in vortex cores of a topological superconductor. Black arrows denote the exchanging paths of two quantum vortices. Superconducting phase takes a 2π jump when a vortex crosses the red cuts.
图 4 (a)具有自旋轨道耦合的半导体纳米线和s波超导的混合系统的示意图; (b)半导体纳米线在有限磁场(实线)和零磁场(虚线)下的色散关系
Figure 4. (a) Schematics of a heterostructure consisting of a spin-orbital coupling semiconductor nanowire and an s wave superconductor; (b) the band dispersion of the nanowire with finite magnetic field (solid lines) and zero magnetic field (dashed lines).
图 5 (a) 通过电压差控制Majorana量子比特的设计; (b) Majorana量子比特的两能级系统; (c)-(e) 量子比特在电流脉冲下的LZS震荡: (c)短脉冲, (d)长脉冲, (e)序列脉冲[25]
Figure 5. (a) Schematic design of a universal quantum gate for Majorana qubit, where the qubit is manipulated by voltage across the Josephson-Majorana junction; (b) the two energy levels of the Majorana qubit depending on the phase difference across the junction; (c)-(e) the LZS oscillation of Majorana qubit under current pulse: (c) a short pulse, (d) a long pulse, (e) a sequence of pulses[25].
图 8 (a)拓扑超导量子涡旋里的低能准粒子激发的自旋分辨波函数; (b)准粒子激发的自旋向上态密度和自旋向下态密度之比的能量-空间分布[29]
Figure 8. (a) Spin-resolved wavefunctions of the low energy quasiparticle states in the vortex core of a topological superconductor; (b) spectrum of the ratio between densities of states for the spin-up and spin-down components[29].
图 9 (a), (c), (e), (g)为利用栅极电压移动边界Majorana束缚态的示意图; (b), (d), (f)给出了与(a), (c), (e) 相对应的Majorana束缚态的波函数分布[32]
Figure 9. (a), (c), (e), (g) Schematic of the device which transports edge Majorana states using gate voltages; (b), (d), (f) corresponding wavefunctions of the edge Majorana states in (a), (c), (e)[32].
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[1] Klitzing K V, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494
Google Scholar
[2] Thouless D J, Kohmoto M, Nightingale M P, den Nijs M 1982 Phys. Rev. Lett. 49 405
Google Scholar
[3] Haldane F D M 1988 Phys. Rev. Lett. 61 2015
Google Scholar
[4] Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045
Google Scholar
[5] Qi XL, Zhang SC 2011 Rev. Mod. Phys. 83 1057
Google Scholar
[6] Nayak C, Simon S H, Stern A, Freedman M, Das Sarma S 2008 Rev. Mod. Phys. 80 1083
Google Scholar
[7] Read N, Green D 2000 Phys. Rev. B 61 10267
Google Scholar
[8] Sato M 2014 BUTSURI 69 297
[9] Alicea J 2012 Rep. Prog. Phys. 75 076501
Google Scholar
[10] Beenakker C W J 2013 Ann. Rev. Cond. Matt. Phys. 4 113
Google Scholar
[11] Berry M V 1984 Proc. Roy. Soc. London A: Math. Phys. Sci. 392 45
[12] Weng H, Yu R, Hu X, Dai X, Fang Z 2015 Adv.Phys. 64 227
Google Scholar
[13] Majorana E 2008 Il Nuovo Cimento (1924-1942) 14 171
[14] Ivanov D A 2001 Phys. Rev. Lett. 86 268
Google Scholar
[15] Kitaev A Y 2001 Physics-Uspekhi 44 131
Google Scholar
[16] Fu L, Kane C L 2008 Phys. Rev. Lett. 100 096407
Google Scholar
[17] Sato M, Takahashi Y, Fujimoto S 2009 Phys. Rev. Lett. 103 020401
Google Scholar
[18] Sau J D, Lutchyn R M, Tewari S, Das Sarma S 2010 Phys. Rev. Lett. 104 040502
Google Scholar
[19] Lutchyn R M, Sau J D, Das Sarma S 2010 Phys. Rev. Lett. 105 077001
Google Scholar
[20] Mourik V, Zuo K, Frolov S M, Plissard S R, Bakkers E P A M, Kouwenhoven L P 2012 Science 336 1003
Google Scholar
[21] NadjPerge S, Drozdov I K, Li J, Chen H, Jeon S, Seo J, MacDonald A H, Bernevig B A, Yazdani A 2014 Science 346 602
Google Scholar
[22] Xu J P, Wang M X, Liu Z L, Ge J F, Yang X, Liu C, Xu Z A, Guan D, Gao C L, Qian D, Liu Y, Wang Q H, Zhang F C, Xue Q K, Jia J F 2015 Phys. Rev. Lett. 114 017001
Google Scholar
[23] Wiedenmann J, Bocquillon E, Deacon R S, Hartinger S, Herrmann O, Klapwijk T M, Maier L, Ames C, Brüne C, Gould C, Oiwa A, Ishibashi K, Tarucha S, Buhmann H, Molenkamp L W 2016 Nat. Comm. 7 10303
Google Scholar
[24] Albrecht S M, Higginbotham A P, Madsen M, Kuemmeth F, Jespersen T S, Nygård J, Krogstrup P, Marcus C M 2016 Nature 531 206
Google Scholar
[25] Wang Z, Huang W C, Liang Q F, Hu X 2018 Sci. Rep. 8 7920
Google Scholar
[26] Hu X, Lin SZ 2010 Supercond. Sci. Tech. 23 053001
Google Scholar
[27] Fu L 2010 Phys. Rev. Lett. 104 056402
Google Scholar
[28] Wang Z, Hu XY, Liang Q F, Hu X 2013 Phys. Rev. B 87 214513
Google Scholar
[29] Kawakami T, Hu X 2015 Phys. Rev. Lett. 115 177001
Google Scholar
[30] Teo J C Y, Kane C L 2010 Phys. Rev. B 82 115120
Google Scholar
[31] Sun H H, Zhang K W, Hu L H, Li C, Wang G Y, Ma H Y, Xu Z A, Gao C L, Guan D D, Li Y Y, Liu C, Qian D, Zhou Y, Fu L, Li SC, Zhang F C, Jia J F 2016 Phys. Rev. Lett. 116 257003
Google Scholar
[32] Liang Q F, Wang Z, Hu X 2012 Europhys. Lett. 99 50004
Google Scholar
[33] Wu LH, Liang Q F, Hu X 2014 Sci.Tech. Adv. Mat. 15 064402
Google Scholar
[34] Liang Q F, Wang Z, Hu X 2014 Phys. Rev. B 89 224514
Google Scholar
[35] Maeno Y, Kittaka S, Nomura T, Yonezawa S, Ishida K 2011 J. Phys. Soc. Japan 81 011009
[36] Tsutsumi Y, Ishikawa M, Kawakami T, Mizushima T, Sato M, Ichioka M, Machida K 2013 J. Phys. Soc. Japan 82 113707
Google Scholar
[37] Sakano M, Okawa K, Kanou M, Sanjo H, Okuda T, Sasagawa T, Ishizaka K 2015 Nat. Comm. 6 8595
Google Scholar
[38] Yonezawa S, Tajiri K, Nakata S, Nagai Y, Wang Z, Segawa K, Ando Y, Maeno Y 2016 Nat. Phys. 13 123
[39] Matano K, Kriener M, Segawa K, Ando Y, Zheng G Q 2016 Nat. Phys. 12 852
Google Scholar
[40] Mizushima T, Tsutsumi Y, Kawakami T, Sato M, Ichioka M, Machida K 2016 J. Phys. Soc. Japan 85 022001
Google Scholar
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