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

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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.

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References

  1. C. K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Rev. Mod. Phys. 88, 035005 (2016), arXiv: 1505.03535.

    Article  ADS  Google Scholar 

  2. B. A. Bernevig, and T. L. Hughes, Topological Insulators and Topological Superconductors (Princeton University Press, Princeton, 2013).

    Book  Google Scholar 

  3. S.-Q. Shen, Topological Insulators, vol. 174 (Springer, Cham, 2012).

  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.

    Article  ADS  Google Scholar 

  5. C. Liu, T. L. Hughes, X. L. Qi, K. Wang, and S. C. Zhang, Phys. Rev. Lett. 100, 236601 (2008), arXiv: 0801.2831.

    Article  ADS  Google Scholar 

  6. H. Zhang, C. X. Liu, X. L. Qi, X. Dai, Z. Fang, and S. C. Zhang, Nat. Phys. 5, 438 (2009).

    Article  Google Scholar 

  7. Y. Ando, J. Phys. Soc. Jpn. 82, 102001 (2013), arXiv: 1304.5693.

    Article  ADS  Google Scholar 

  8. C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008), arXiv: 0707.1889.

    Article  ADS  Google Scholar 

  9. Q. F. Liang, Z. Wang, and X. Hu, Phys. Rev. B 89, 224514 (2014), arXiv: 1302.4337.

    Article  ADS  Google Scholar 

  10. L. Fu, and C. L. Kane, Phys. Rev. Lett. 100, 096407 (2008), arXiv: 0707.1692.

    Article  ADS  Google Scholar 

  11. R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev. Lett. 105, 077001 (2010), arXiv: 1002.4033.

    Article  ADS  Google Scholar 

  12. L. H. Wu, Q. F. Liang, Z. Wang, and X. Hu, J. Phys.-Conf. Ser. 393, 012018 (2012).

    Article  Google Scholar 

  13. J. Alicea, Rep. Prog. Phys. 75, 076501 (2012), arXiv: 1202.1293.

    Article  ADS  Google Scholar 

  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.

    Article  ADS  Google Scholar 

  15. Z. Z. Li, F. C. Zhang, and Q. H. Wang, Sci. Rep. 4, 6363 (2015).

    Article  Google Scholar 

  16. S. D. Sarma, M. Freedman, and C. Nayak, npj Quantum Inf. 1, 15001 (2015).

    Article  ADS  Google Scholar 

  17. M. Sato, and S. Fujimoto, J. Phys. Soc. Jpn. 85, 072001 (2016), arXiv: 1601.02726.

    Article  ADS  Google Scholar 

  18. R. Aguado, arXiv: 1711.00011.

  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.

    Article  ADS  Google Scholar 

  20. L. H. Hu, C. Li, D. H. Xu, Y. Zhou, and F. C. Zhang, Phys. Rev. B 94, 224501 (2016).

    Article  ADS  Google Scholar 

  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.

    Article  ADS  Google Scholar 

  22. P. Hosur, P. Ghaemi, R. S. K. Mong, and A. Vishwanath, Phys. Rev. Lett. 107, 097001 (2011), arXiv: 1012.0330.

    Article  ADS  Google Scholar 

  23. C. K. Chiu, P. Ghaemi, and T. L. Hughes, Phys. Rev. Lett. 109, 237009 (2012), arXiv: 1203.2958.

    Article  ADS  Google Scholar 

  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.

    Article  ADS  Google Scholar 

  25. G. Xu, B. Lian, P. Tang, X. L. Qi, and S. C. Zhang, Phys. Rev. Lett. 117, 047001 (2016), arXiv: 1511.06942.

    Article  ADS  Google Scholar 

  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.

    Article  ADS  Google Scholar 

  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.

    Article  ADS  Google Scholar 

  28. L. Fu, Phys. Rev. Lett. 103, 266801 (2009), arXiv: 0908.1418.

    Article  ADS  Google Scholar 

  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.

    Article  ADS  Google Scholar 

  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.

    Article  ADS  Google Scholar 

  31. L. Fu, and C. L. Kane, Phys. Rev. B 76, 045302 (2007).

    Article  ADS  Google Scholar 

  32. A. Y. Kitaev, Phys.-Usp. 44, 131 (2001).

    Article  ADS  Google Scholar 

  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.

  34. G. Xu, H. Weng, Z. Wang, X. Dai, and Z. Fang, Phys. Rev. Lett. 107, 186806 (2011), arXiv: 1106.3125.

    Article  ADS  Google Scholar 

  35. C. K. Chiu, M. J. Gilbert, and T. L. Hughes, Phys. Rev. B 84, 144507 (2011), arXiv: 1108.4711.

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Department of Physics, Zhejiang University, Hangzhou, 310027, China

    Chuang Li

  2. Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China

    Chuang Li, Lun-Hui Hu & Fu-Chun Zhang

  3. Department of Physics, University of California, San Diego, CA, 92093, USA

    Lun-Hui Hu

  4. CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing, 100190, China

    Fu-Chun Zhang

  5. Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing, 210093, China

    Fu-Chun Zhang

Authors
  1. Chuang Li
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  2. Lun-Hui Hu
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  3. Fu-Chun Zhang
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Correspondence to Lun-Hui Hu or Fu-Chun Zhang.

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Li, C., Hu, LH. & Zhang, FC. Hexagonal warping effect on Majorana zero modes at the ends of superconducting vortex lines in doped strong 3D topological insulators. Sci. China Phys. Mech. Astron. 62, 117411 (2019). https://doi.org/10.1007/s11433-019-9391-7

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  • DOI: https://doi.org/10.1007/s11433-019-9391-7

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