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本文研究了一维公度势和非公度势调制下的p波超导量子线系统的拓扑相变. 在公度势调制下, 通过计算
$Z_2$ 拓扑不变量确定系统的相图, 指出系统的拓扑相变强烈地依赖于调制参数$\alpha$ 和相移$\delta$ . 在非公度势调制下, 以$\alpha=(\sqrt{5}-1)/2$ ,$\delta=0$ 为例, 计算系统的低能激发谱、$Z_2$ 拓扑不变量以及逆参与率等, 发现p波配对强度$\varDelta\in(0,0.33)$ 时, 系统存在拓扑非平庸超导相, 拓扑平庸超导相和拓扑平庸局域相的转变. 而当p波配对强度$\varDelta > 0.33$ 时, 系统存在拓扑非平庸超导相和拓扑平庸局域相的转变.-
关键词:
- p波超导体 /
- Majorana费米子 /
- Z2拓扑不变量
We consider a one-dimensional p-wave superconducting quantum wire with the modulated chemical potential, which is described by$\hat{H}= \displaystyle\sum\nolimits_{i}\left[ \left( -t\hat{c}_{i}^{\dagger }\hat{c}_{i+1}+\Delta \hat{c}_{i}\hat{c}_{i+1}+ h.c.\right) +V_{i}\hat{n}_{i}\right]$ ,$V_{i}=V\dfrac{\cos \left( 2{\text{π}} i\alpha + \delta \right) }{1-b\cos \left( 2{\text{π}} i\alpha+\delta \right) }$ and can be solved by the Bogoliubov-de Gennes method. When$b=0$ ,$\alpha$ is a rational number, the system undergoes a transition from topologically nontrivial phase to topologically trivial phase which is accompanied by the disappearance of the Majorana fermions and the changing of the$Z_2$ topological invariant of the bulk system. We find the phase transition strongly depends on the strength of potential V and the phase shift$\delta$ . For some certain special parameters$\alpha$ and$\delta$ , the critical strength of the phase transition is infinity. For the incommensurate case, i.e.$\alpha=(\sqrt{5}-1)/2$ , the phase diagram is identified by analyzing the low-energy spectrum, the amplitudes of the lowest excitation states, the$Z_2$ topological invariant and the inverse participation ratio (IPR) which characterizes the localization of the wave functions. Three phases emerge in such case for$\delta=0$ , topologically nontrivial superconductor, topologically trivial superconductor and topologically trivial Anderson insulator. For a topologically nontrivial superconductor, it displays zero-energy Majorana fermions with a$Z_2$ topological invariant. By calculating the IPR, we find the lowest excitation states of the topologically trivial superconductor and topologically trivial Anderson insulator show different scaling features. For a topologically trivial superconductor, the IPR of the lowest excitation state tends to zero with the increase of the size, while it keeps a finite value for different sizes in the trivial Anderson localization phase.-
Keywords:
- p-wave superconductor /
- Majorana fermions /
- Z2 topological invariant
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图 1 Hofstadter蝴蝶谱: 随
$\alpha$ 变化的能谱, 红色点是零能Majorana费米子b = 0.5, L = 120,$\varDelta=0.2$ , V = 1.5, δ = 0Fig. 1. Hofstadter butterfly: the energy spectrum varying with
$\alpha$ . The red dotted point denotes the Majorana Fermion.$b=0.5,\; L=120,\; \varDelta=0.2, \;V=1.5$ and$\delta=0$ 
图 2 在
$b=0.5$ 时, 参数$\varDelta-V$ 平面的拓扑相图 (a)$\alpha= $ $1/2, L=120$ ; (b)$\alpha=1/3, L=120$ ; (c)$\alpha= ( \sqrt{5}-1) /2$ , L = 2584Fig. 2. Topological phase diagram in
$\varDelta-V$ plane with$b=0.5$ . (a)$\alpha=1/2, L=120$ ; (b)$\alpha=1/3, L=120$ ; (c)$\alpha= ( \sqrt{5}-1) /2$ , L = 2584
图 3 在开边界条件下, 本征能量随相移
$\delta$ 的变化.$b=0.5$ ,$\varDelta=0.2$ ,$L=2584$ Fig. 3. Energy varying with phase shift
$\delta$ with$b=0.5$ ,$\varDelta=0.2$ and$L=2584$ under open boundary condition.
图 4 (a)在开边界和周期性边界条件下最低激发态能量
$E_{1}$ 随准周期调制强度V的变化及其空间分布$\phi_1$ (b)和$\psi_1$ (c),$\alpha= $ $(\sqrt{5}-1)/2$ ,$b=0.5$ ,$\varDelta=0.2$ ,$L=2584$ Fig. 4. (a) The lowest excitation energies,
$E_{1}$ , varying with the quasi-periodic modulation amplitude, V, under OBC and PBC, respectively. The spatial distribution of the lowest excited state$\phi_1$ (b),$\psi_1$ (c).$\alpha=(\sqrt{5}-1) /2$ ,$b=0.5$ ,$\varDelta=0.2$ ,$L=2584$ .
图 5 (a)
$Z_{2}$ 拓扑不变量随非公度势强度的变化; (b) V = 2时$\mathrm{IPR}_{1}$ 的标度分析; (c)$V=3$ 时$\mathrm{IPR}_{1}$ 的标度分析$b=0.5$ ,$\alpha= (\sqrt{5}-1) /2$ ,$\varDelta=0.2$ ,$L=2584$ Fig. 5. (a)
$Z_{2}$ topological invariant varying with the strength of the potential V; (b) the scaling of$\mathrm{IPR}_{1}$ $V=2$ ; (c) the scaling of$\mathrm{IPR}_{1}$ $V=3$ . Here,$\alpha= (\sqrt{5}-1)/2$ ,$b=0.5$ ,$\varDelta=0.2$ ,$L=2584$ .
图 6
${\rm {IPR}}$ 随准周期调制强度V和本征能量$E_{n}$ 的变化$\alpha=(\sqrt{5}-1) /2$ ,$b=0.5, L=144, \delta=0$ (a)$\varDelta=0$ ; (b)$\varDelta=0.01$ ; (c)$\varDelta=0.5$ ; (d)$\varDelta=0.8$ Fig. 6.
${\rm {IPR}}$ varying with the amplitude of quasi-periodic modulation V and energy$E_{n}$ .$\alpha=(\sqrt{5}-1) /2$ ,$b=0.5, L=144$ ,$\delta=0$ : (a)$\varDelta=0$ ; (b)$\varDelta=0.01$ ; (c)$\varDelta=0.5$ ; (d)$\varDelta=0.8$ -
[1] Alicea J 2012 Rep. Prog. Phys. 75 076501
Google Scholar
[2] Beenakker C W J 2013 Ann. Rev. Con. Mat. Phys. 4 113
Google Scholar
[3] Wilczek F 2009 Nat. Phys. 5 614
Google Scholar
[4] Elliott S R, Franz M 2015 Rev. Mod. Phys. 87 137
Google Scholar
[5] Kitaev A Y 2001 Phys. Usp. 44 131
Google Scholar
[6] Mourik V, Zuo K, Frolov S M, Plissard S R, Bakkers E P A M, Kouwenhoven L P 2012 Science 336 1003
Google Scholar
[7] Chen J, Yu P, Stenger J, Hocevar M, Car D, Plissard S R, Bakkers E P A M, Stanescu T D, Frolov S M 2017 Sci. Adv. 3 e1701476
Google Scholar
[8] Albrecht S M, Higginbotham A P, Madsen M, Kuemmeth F, Jespersen T S, Nygard J, Krogstrup P, Marcus C M 2016 Nature 531 206
Google Scholar
[9] Deng M T, Vaitiekenas S, Hansen E B, Danon J, Leijnse M, Flensberg K, Nygard J, Krogstrup P, Marcus C M 2016 Science 354 1557
Google Scholar
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Google Scholar
[12] Nadj-Perge 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
[13] Jeon S, Xie Y L, Li Jian, Wang Z J, Bernevig B A, Yazdani A 2017 Science 358 772
[14] Hell M, Leijnse M, Flensberg K 2017 Phys. Rev. Lett. 118 107701
Google Scholar
[15] Pientka F, Keselman A, Berg E, Yacoby A, Stern A, Halperin B I 2017 Phys. Rev. X 7 021032
[16] Fornieri A, Whiticar A M, Setiawan F, Marín E P, Asbjórn C C D, Keselman A, Gronin S, Thomas C, Wang T, Kallaher R, Gardner G C, Berg E, Manfra M J, Stern A, Marcus C M, Nichele F 2019 Nature 569 89
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Google Scholar
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Google Scholar
[20] Ivanov D A 2001 Phys. Rev. Lett. 86 268
Google Scholar
[21] Zhu S L, Shao L B, Wang Z D, Duan L M 2011 Phys. Rev. Lett. 106 100404
Google Scholar
[22] Lindner N H, Berg E, Refael G, Stern A 2012 Phys. Rev. X 2 041002
[23] Nayak C, Simon S H, Stern A, Freedman M, Sarma S D 2008 Rev. Mod. Phys. 80 1083
Google Scholar
[24] Jiang L, Kitagawa T, Alicea J, Akhmerov A R, Pekker D, Refael G, Cirac J I, Demler E, Lukin M D, Zoller P 2011 Phys. Rev. Lett. 106 220402
Google Scholar
[25] Hubener H, Sentef M A, Giovannini U D, Kemper A F, Rubio A 2017 Nat. Commun. 8 13940
Google Scholar
[26] Cheng Q, Pan Y, Wang H, Zhang C, Yu D, Gover A, Zhang H, Li T, Zhou L, Zhu S 2019 Phys. Rev. Lett. 122 173901
Google Scholar
[27] Cadez T, Mondaini R, Sacramento P D 2019 Phys. Rev. B 99 014301
Google Scholar
[28] Wang H Y, Zhuang L, Liu W M 2019 arXiv: 1910.10911 [cond-mat.mes-hall]
[29] Takata K, Notomi M 2018 Phys. Rev. Lett. 121 213902
Google Scholar
[30] Zhou L 2019 arXiv: 1911.11978 [cond-mat.mes-hall]
[31] Zeng Q B, Yang Y B, Xu Y 2019 arXiv: 1901.08060 [cond-mat.mes-hall]
[32] Okuma N, Sato M 2019 Phys. Rev. Lett. 123 097701
Google Scholar
[33] Ezawa M 2019 Phys. Rev. B 100 045407
Google Scholar
[34] Wu Y J, Liu H W, Liu J, Jiang H, Xie X C https://doi.org/10.1093/nsr/nwz189 [2020-1-8]
[35] Amorim C S, Ebihara K, Yamakage A, Tanaka Y, Sato M 2015 Phys. Rev. B 91 174305
Google Scholar
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Google Scholar
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Google Scholar
[38] Cai X M, Lang L J, Chen S, Wang Y P 2013 Phys. Rev. Lett. 110 176403
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[43] Aubry S, André G 1980 Ann. Isr. Phys. Soc. 3 133
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Google Scholar
[45] Sen A, Damle K, Moessner R 2012 Phys. Rev. B 86 205134
Google Scholar
[46] Zhu J X 2016 Bogoliubov-de Gennes Method and Its Applications (New Mexico: Springer) p3
[47] Gennes P G d (translated by Pincus P A) 1999 Superconductivity of Metals and Alloys (Boulder: Westview Press) pp137–160
[48] Lieb E, Schultz T, Mattis D 1961 Ann. Phys. 16 407
Google Scholar
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Google Scholar
[50] Hofstadter D R 1976 Phys. Rev. B 14 2239
Google Scholar
[51] Zhou B, Shen S Q 2011 Phys. Rev. B 84 054532
Google Scholar
[52] Liu T, Yan H Y, Guo H 2017 Phys. Rev. B 96 174207
Google Scholar
[53] Akhmerov A R, Dahlhaus J P, Hassler F, Wimmer M, Beenakker C W J 2011 Phys. Rev. Lett. 106 057001
Google Scholar
[54] Fulga I C, Hassler F, Akhmerov A R, Beenakker C W J 2011 Phys. Rev. B 83 155429
Google Scholar
[55] Snyman I, Tworzydlo J, Beenakker C W J 2008 Phys. Rev. B 78 045118
Google Scholar
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Google Scholar
[57] Thouless D J 1974 Phys. Rep. 13 93
Google Scholar
[58] Kohmoto M 1983 Phys. Rev. Lett. 51 1198
Google Scholar
[59] Schreiber M 1985 J. Phys. C 18 2493
Google Scholar
[60] Hashimoto Y, Niizeki K, Okabe Y 1992 J. Phys. A 25 5211
Google Scholar
[61] Ingolda G L, Wobst A, Aulbach Ch, Hanggi P 2002 Eur. Phys. J. B 30 175
Google Scholar
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