Advertisement

Hexagonal warping effect on Majorana zero modes at the ends of superconducting vortex lines in doped strong 3D topological insulators

  • Chuang Li
  • Lun-Hui HuEmail author
  • Fu-Chun ZhangEmail author
Article
  • 59 Downloads

Abstract

In a superconducting topological insulator, a superconducting vortex line can trap a one-dimensional topological band with localized Majorana zero modes at the ends. Here, we study the effect of hexagonal warping and its corresponding symmetry-breaking effect on vortex phase transition. We perform both analytical calculations based on a semiclassical formula and numerical calculations based on full quantum mechanics using the Bogoliubov-de Gennes equation. We find that the hexagonal warping term extends the topological region of the vortex line as the chemical potential changes and leads to MZMs, even in the absence of topological surface states.

Key words

topological insulator topological superconductor vortex phase transition Majorana zero modes hexagonal warping 

References

  1. [1]
    C. K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Rev. Mod. Phys. 88, 035005 (2016), arXiv: 1505.03535.ADSCrossRefGoogle Scholar
  2. [2]
    B. A. Bernevig, and T. L. Hughes, Topological Insulators and Topological Superconductors (Princeton University Press, Princeton, 2013).CrossRefGoogle Scholar
  3. [3]
    S.-Q. Shen, Topological Insulators, vol. 174 (Springer, Cham, 2012).Google Scholar
  4. [4]
    M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenkamp, X. L. Qi, and S. C. Zhang, Science 318, 766 (2007), arXiv: 0710.0582.ADSCrossRefGoogle Scholar
  5. [5]
    C. Liu, T. L. Hughes, X. L. Qi, K. Wang, and S. C. Zhang, Phys. Rev. Lett. 100, 236601 (2008), arXiv: 0801.2831.ADSCrossRefGoogle Scholar
  6. [6]
    H. Zhang, C. X. Liu, X. L. Qi, X. Dai, Z. Fang, and S. C. Zhang, Nat. Phys. 5, 438 (2009).CrossRefGoogle Scholar
  7. [7]
    Y. Ando, J. Phys. Soc. Jpn. 82, 102001 (2013), arXiv: 1304.5693.ADSCrossRefGoogle Scholar
  8. [8]
    C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008), arXiv: 0707.1889.ADSCrossRefGoogle Scholar
  9. [9]
    Q. F. Liang, Z. Wang, and X. Hu, Phys. Rev. B 89, 224514 (2014), arXiv: 1302.4337.ADSCrossRefGoogle Scholar
  10. [10]
    L. Fu, and C. L. Kane, Phys. Rev. Lett. 100, 096407 (2008), arXiv: 0707.1692.ADSCrossRefGoogle Scholar
  11. [11]
    R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev. Lett. 105, 077001 (2010), arXiv: 1002.4033.ADSCrossRefGoogle Scholar
  12. [12]
    L. H. Wu, Q. F. Liang, Z. Wang, and X. Hu, J. Phys.-Conf. Ser. 393, 012018 (2012).CrossRefGoogle Scholar
  13. [13]
    J. Alicea, Rep. Prog. Phys. 75, 076501 (2012), arXiv: 1202.1293.ADSCrossRefGoogle Scholar
  14. [14]
    S. Nadj-Perge, I. K. Drozdov, J. Li, H. Chen, S. Jeon, J. Seo, A. H. MacDonald, B. A. Bernevig, and A. Yazdani, Science 346, 602 (2014), arXiv: 1410.0682.ADSCrossRefGoogle Scholar
  15. [15]
    Z. Z. Li, F. C. Zhang, and Q. H. Wang, Sci. Rep. 4, 6363 (2015).CrossRefGoogle Scholar
  16. [16]
    S. D. Sarma, M. Freedman, and C. Nayak, npj Quantum Inf. 1, 15001 (2015).ADSCrossRefGoogle Scholar
  17. [17]
    M. Sato, and S. Fujimoto, J. Phys. Soc. Jpn. 85, 072001 (2016), arXiv: 1601.02726.ADSCrossRefGoogle Scholar
  18. [18]
    R. Aguado, arXiv: 1711.00011.Google Scholar
  19. [19]
    J. P. Xu, M. X. Wang, Z. L. Liu, J. F. Ge, X. Yang, C. Liu, Z. A. Xu, D. Guan, C. L. Gao, D. Qian, Y. Liu, Q. H. Wang, F. C. Zhang, Q. K. Xue, and J. F. Jia, Phys. Rev. Lett. 114, 017001 (2015), arXiv: 1312.7110.ADSCrossRefGoogle Scholar
  20. [20]
    L. H. Hu, C. Li, D. H. Xu, Y. Zhou, and F. C. Zhang, Phys. Rev. B 94, 224501 (2016).ADSCrossRefGoogle Scholar
  21. [21]
    H. H. Sun, K. W. Zhang, L. H. Hu, C. Li, G. Y. Wang, H. Y. Ma, Z. A. Xu, C. L. Gao, D. D. Guan, Y. Y. Li, C. Liu, D. Qian, Y. Zhou, L. Fu, S. C. Li, F. C. Zhang, and J. F. Jia, Phys. Rev. Lett. 116, 257003 (2016), arXiv: 1603.02549.ADSCrossRefGoogle Scholar
  22. [22]
    P. Hosur, P. Ghaemi, R. S. K. Mong, and A. Vishwanath, Phys. Rev. Lett. 107, 097001 (2011), arXiv: 1012.0330.ADSCrossRefGoogle Scholar
  23. [23]
    C. K. Chiu, P. Ghaemi, and T. L. Hughes, Phys. Rev. Lett. 109, 237009 (2012), arXiv: 1203.2958.ADSCrossRefGoogle Scholar
  24. [24]
    Y. S. Hor, J. G. Checkelsky, D. Qu, N. P. Ong, and R. J. Cava, J. Phys. Chem. Solids 72, 572 (2011), arXiv: 1006.0317.ADSCrossRefGoogle Scholar
  25. [25]
    G. Xu, B. Lian, P. Tang, X. L. Qi, and S. C. Zhang, Phys. Rev. Lett. 117, 047001 (2016), arXiv: 1511.06942.ADSCrossRefGoogle Scholar
  26. [26]
    P. Zhang, K. Yaji, T. Hashimoto, Y. Ota, T. Kondo, K. Okazaki, Z. Wang, J. Wen, G. D. Gu, H. Ding, and S. Shin, Science 360, 182 (2018), arXiv: 1706.05163.ADSCrossRefGoogle Scholar
  27. [27]
    D. Wang, L. Kong, P. Fan, H. Chen, S. Zhu, W. Liu, L. Cao, Y. Sun, S. Du, J. Schneeloch, R. Zhong, G. Gu, L. Fu, H. Ding, and H. J. Gao, Science 362, 333 (2018), arXiv: 1706.06074.ADSCrossRefGoogle Scholar
  28. [28]
    L. Fu, Phys. Rev. Lett. 103, 266801 (2009), arXiv: 0908.1418.ADSCrossRefGoogle Scholar
  29. [29]
    Z. Alpichshev, J. G. Analytis, J. H. Chu, I. R. Fisher, Y. L. Chen, Z. X. Shen, A. Fang, and A. Kapitulnik, Phys. Rev. Lett. 104, 016401 (2010), arXiv: 0908.0371.ADSCrossRefGoogle Scholar
  30. [30]
    C. X. Liu, X. L. Qi, H. J. Zhang, X. Dai, Z. Fang, and S. C. Zhang, Phys. Rev. B 82, 045122 (2010), arXiv: 1005.1682.ADSCrossRefGoogle Scholar
  31. [31]
    L. Fu, and C. L. Kane, Phys. Rev. B 76, 045302 (2007).ADSCrossRefGoogle Scholar
  32. [32]
    A. Y. Kitaev, Phys.-Usp. 44, 131 (2001).ADSCrossRefGoogle Scholar
  33. [33]
    S. S. Qin, L. H. Hu, C. C. Le, J. F. Zeng, F. C. Zhang, C. Fang, and J. P. Hu, arXiv: 1901.04932.Google Scholar
  34. [34]
    G. Xu, H. Weng, Z. Wang, X. Dai, and Z. Fang, Phys. Rev. Lett. 107, 186806 (2011), arXiv: 1106.3125.ADSCrossRefGoogle Scholar
  35. [35]
    C. K. Chiu, M. J. Gilbert, and T. L. Hughes, Phys. Rev. B 84, 144507 (2011), arXiv: 1108.4711.ADSCrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of PhysicsZhejiang UniversityHangzhouChina
  2. 2.Kavli Institute for Theoretical SciencesUniversity of Chinese Academy of SciencesBeijingChina
  3. 3.Department of PhysicsUniversity of CaliforniaSan DiegoUSA
  4. 4.CAS Center for Excellence in Topological Quantum ComputationUniversity of Chinese Academy of SciencesBeijingChina
  5. 5.Collaborative Innovation Center of Advanced MicrostructuresNanjing UniversityNanjingChina

Personalised recommendations