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

doi:10.7498/aps.69.20191868

Abstract

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:

Authors and contacts

1.
Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China
2.
State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China

Corresponding author: Xu Zhi-Hao, xuzhihao@sxu.edu.cn

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References

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Cited By

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

Figure 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 $\delta=0$

DownLoad: Full-Size Img PowerPoint

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 = 2584

Figure 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

DownLoad: Full-Size Img PowerPoint

图 3 在开边界条件下, 本征能量随相移$\delta$的变化. $b=0.5$, $\varDelta=0.2$, $L=2584$

Figure 3. Energy varying with phase shift $\delta$ with $b=0.5$, $\varDelta=0.2$ and $L=2584$ under open boundary condition.

DownLoad: Full-Size Img PowerPoint

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$, $b=0.5$, $\varDelta=0.2$, $L=2584$

Figure 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). $\alpha=(\sqrt{5}-1) /2$, $b=0.5$, $\varDelta=0.2$, $L=2584$.

DownLoad: Full-Size Img PowerPoint

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 $b=0.5$, $\alpha= (\sqrt{5}-1) /2$, $\varDelta=0.2$, $L=2584$

Figure 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, $\alpha= (\sqrt{5}-1)/2$, $b=0.5$, $\varDelta=0.2$, $L=2584$.

DownLoad: Full-Size Img PowerPoint

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) $\varDelta=0.8$

Figure 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) $\varDelta=0.8$

DownLoad: Full-Size Img PowerPoint

[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
[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 74 Google Scholar
[11]	Nadj-Perge S, Drozdov I K, Bernevig B A, Yazdani A 2013 Phys. Rev. B 88 020407 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 Google Scholar
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[19]	Fu L, Kane C L 2008 Phys. Rev. Lett. 100 096407 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
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[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
[36]	Chen C Z, Xie Y M, Liu J, Lee P A, Law K T 2018 Phys. Rev. B 97 104504 Google Scholar
[37]	Lang L J, Chen S 2012 Phys. Rev. B. 86 205135 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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[41]	Wakatsuki R, Ezawa M, Tanaka Y, Nagaosa N 2014 Phys. Rev. B 90 014505 Google Scholar
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[47]	Gennes P G d (translated by Pincus P A) 1999 Superconductivity of Metals and Alloys (Boulder: Westview Press) pp137–160
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[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
[56]	Choy T P, Edge J M, Akhmerov A R, Beenakker C W J 2011 Phys. Rev. B 84 195442 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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