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一维化学势调制的p波超导体中的拓扑量子相变

武璟楠 徐志浩 陆展鹏 张云波

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一维化学势调制的p波超导体中的拓扑量子相变

武璟楠, 徐志浩, 陆展鹏, 张云波

Topological quantum phase transitions in one-dimensional p-wave superconductors with modulated chemical potentials

Wu Jing-Nan, Xu Zhi-Hao, Lu Zhan-Peng, Zhang Yun-Bo
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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.
    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.
      通信作者: 徐志浩, xuzhihao@sxu.edu.cn
    • 基金项目: 国家级-国家自然科学基金(11604188)
      Corresponding author: Xu Zhi-Hao, xuzhihao@sxu.edu.cn
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    Zeng Q B, Yang Y B, Xu Y 2019 arXiv: 1901.08060 [cond-mat.mes-hall]

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    Okuma N, Sato M 2019 Phys. Rev. Lett. 123 097701Google Scholar

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    Hegde S S, Vishveshwara S 2016 Phys. Rev. Lett. 94 115166

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    Wakatsuki R, Ezawa M, Tanaka Y, Nagaosa N 2014 Phys. Rev. B 90 014505Google Scholar

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    Sen A, Damle K, Moessner R 2012 Phys. Rev. B 86 205134Google Scholar

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    Zhu J X 2016 Bogoliubov-de Gennes Method and Its Applications (New Mexico: Springer) p3

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    Gennes P G d (translated by Pincus P A) 1999 Superconductivity of Metals and Alloys (Boulder: Westview Press) pp137–160

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    Lieb E, Schultz T, Mattis D 1961 Ann. Phys. 16 407Google Scholar

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    Kraus Y E, Lahini Y, Ringel Z, Verbin M, Zliberberg O 2012 Phys. Rev. Lett. 109 106402Google Scholar

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    Liu T, Yan H Y, Guo H 2017 Phys. Rev. B 96 174207Google Scholar

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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 = 0

    Fig. 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  $b=0.5$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 = 2584

    Fig. 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$$\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 076501Google Scholar

    [2]

    Beenakker C W J 2013 Ann. Rev. Con. Mat. Phys. 4 113Google Scholar

    [3]

    Wilczek F 2009 Nat. Phys. 5 614Google Scholar

    [4]

    Elliott S R, Franz M 2015 Rev. Mod. Phys. 87 137Google Scholar

    [5]

    Kitaev A Y 2001 Phys. Usp. 44 131Google Scholar

    [6]

    Mourik V, Zuo K, Frolov S M, Plissard S R, Bakkers E P A M, Kouwenhoven L P 2012 Science 336 1003Google 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 e1701476Google 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 206Google 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 1557Google Scholar

    [10]

    Zhang H, Liu C X, Gazibegovic S, Xu D, Logan J A, Wang G Z, N Loo van, Bommer J D S, Moor M W A d, Car D, Veld R L M O H, Veldhoven P J, Koelling S, Verheijen M A, Pendharkar M, Pennachio D J, Shojaei B, Lee J S, Palmstrøm C J, Bakkers E P A M, Sarma S D, Kouwenhoven L P 2018 Nature 556 74Google Scholar

    [11]

    Nadj-Perge S, Drozdov I K, Bernevig B A, Yazdani A 2013 Phys. Rev. B 88 020407Google 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 602Google 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 107701Google 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 89Google Scholar

    [17]

    Cook A, Franz M 2011 Phys. Rev. B 84 201105Google Scholar

    [18]

    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 CH, Qian D, Zhou Y, Fu L, Li S C, Zhang F C, Jia J F 2016 Phys. Rev. Lett. 116 257003Google Scholar

    [19]

    Fu L, Kane C L 2008 Phys. Rev. Lett. 100 096407Google Scholar

    [20]

    Ivanov D A 2001 Phys. Rev. Lett. 86 268Google Scholar

    [21]

    Zhu S L, Shao L B, Wang Z D, Duan L M 2011 Phys. Rev. Lett. 106 100404Google 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 1083Google 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 220402Google Scholar

    [25]

    Hubener H, Sentef M A, Giovannini U D, Kemper A F, Rubio A 2017 Nat. Commun. 8 13940Google 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 173901Google Scholar

    [27]

    Cadez T, Mondaini R, Sacramento P D 2019 Phys. Rev. B 99 014301Google 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 213902Google 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 097701Google Scholar

    [33]

    Ezawa M 2019 Phys. Rev. B 100 045407Google 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 174305Google Scholar

    [36]

    Chen C Z, Xie Y M, Liu J, Lee P A, Law K T 2018 Phys. Rev. B 97 104504Google Scholar

    [37]

    Lang L J, Chen S 2012 Phys. Rev. B. 86 205135Google Scholar

    [38]

    Cai X M, Lang L J, Chen S, Wang Y P 2013 Phys. Rev. Lett. 110 176403Google Scholar

    [39]

    Hegde S S, Vishveshwara S 2016 Phys. Rev. Lett. 94 115166

    [40]

    DeGottardi W, Thakurathi M, Vishveshwara S, Sen D 2013 Phys. Rev. B 88 165111Google Scholar

    [41]

    Wakatsuki R, Ezawa M, Tanaka Y, Nagaosa N 2014 Phys. Rev. B 90 014505Google Scholar

    [42]

    Lang L J, Cai X M, Chen S 2012 Phys. Rev. Lett. 108 220401Google Scholar

    [43]

    Aubry S, André G 1980 Ann. Isr. Phys. Soc. 3 133

    [44]

    Ganeshan S, Pixley J H, Sarma S D 2015 Phys. Rev. Lett 114 146601Google Scholar

    [45]

    Sen A, Damle K, Moessner R 2012 Phys. Rev. B 86 205134Google 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 407Google Scholar

    [49]

    Kraus Y E, Lahini Y, Ringel Z, Verbin M, Zliberberg O 2012 Phys. Rev. Lett. 109 106402Google Scholar

    [50]

    Hofstadter D R 1976 Phys. Rev. B 14 2239Google Scholar

    [51]

    Zhou B, Shen S Q 2011 Phys. Rev. B 84 054532Google Scholar

    [52]

    Liu T, Yan H Y, Guo H 2017 Phys. Rev. B 96 174207Google Scholar

    [53]

    Akhmerov A R, Dahlhaus J P, Hassler F, Wimmer M, Beenakker C W J 2011 Phys. Rev. Lett. 106 057001Google Scholar

    [54]

    Fulga I C, Hassler F, Akhmerov A R, Beenakker C W J 2011 Phys. Rev. B 83 155429Google Scholar

    [55]

    Snyman I, Tworzydlo J, Beenakker C W J 2008 Phys. Rev. B 78 045118Google Scholar

    [56]

    Choy T P, Edge J M, Akhmerov A R, Beenakker C W J 2011 Phys. Rev. B 84 195442Google Scholar

    [57]

    Thouless D J 1974 Phys. Rep. 13 93Google Scholar

    [58]

    Kohmoto M 1983 Phys. Rev. Lett. 51 1198Google Scholar

    [59]

    Schreiber M 1985 J. Phys. C 18 2493Google Scholar

    [60]

    Hashimoto Y, Niizeki K, Okabe Y 1992 J. Phys. A 25 5211Google Scholar

    [61]

    Ingolda G L, Wobst A, Aulbach Ch, Hanggi P 2002 Eur. Phys. J. B 30 175Google Scholar

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出版历程
  • 收稿日期:  2019-12-10
  • 修回日期:  2020-01-15
  • 刊出日期:  2020-04-05

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