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Journal of Experimental and Theoretical Physics

, Volume 119, Issue 6, pp 1042–1057 | Cite as

Andreev-Majorana bound states in superfluids

  • M. A. SilaevEmail author
  • G. E. Volovik
Special issue in honor of A.F. Andreev’s 75th birthday Issue Editor: I.A. Fomin

Abstract

We consider Andreev-Majorana (AM) bound states with zero energy on surfaces, interfaces, and vortices in different phases of the p-wave superfluids. We discuss the chiral superfluid 3He-A and time reversal invariant phases: superfluid 3He-B, planar and polar phases. The AM zero modes are determined by topology in the bulk and disappear at the quantum phase transition from the topological to nontopological state of the superfluid. The topology demonstrates the interplay of dimensions. In particular, the zero-dimensional Weyl points in chiral superfluids (the Berry phase monopoles in momentum space) give rise to the one-dimensional Fermi arc of AM bound states on the surface and to the one-dimensional flat band of AM modes in the vortex core. The one-dimensional nodal line in the polar phase produces a two-dimensional flat band of AM modes on the surface. The interplay of dimensions also connects the AM states in superfluids with different dimensions. For example, the topological properties of the spectrum of bound states in three-dimensional 3He-B are connected to the properties of the spectrum in the two-dimensional planar phase (thin film).

Keywords

Vortex Vortex Core Edge State Topological Insulator Point Vortex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    C. W. J. Beenakker, Phys. Rev. Lett. 112, 070604 (2014).ADSCrossRefGoogle Scholar
  2. 2.
    C. Chamon, R. Jackiw, Y. Nishida, S.-Y. Pi, and L. Santos, Phys. Rev. B: Condens. Matter 81, 224515 (2010).ADSCrossRefGoogle Scholar
  3. 3.
    T. Senthil and M. P. A. Fisher, Phys. Rev. B: Condens. Matter 61, 9690 (2000).ADSCrossRefGoogle Scholar
  4. 4.
    G. E. Volovik and M. A. Zubkov, Nucl. Phys. B 881, 514 (2014); arXiv:1402.5700.ADSCrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Phys. Rev. B: Condens. Matter 78, 195125 (2008).ADSCrossRefGoogle Scholar
  6. 6.
    A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, AIP Conf. Proc. 1134, 10 (2009); arXiv:0905.2029.ADSCrossRefGoogle Scholar
  7. 7.
    A. P. Schnyder, S. Ryu, and A. W. W. Ludwig, Phys. Rev. Lett. 102, 196804 (2009); arXiv:0901.1343.ADSCrossRefGoogle Scholar
  8. 8.
    A. Kitaev, AIP Conf. Proc. 1134, 22 (2009); arXiv:0901.2686.ADSCrossRefMathSciNetGoogle Scholar
  9. 9.
    G. E. Volovik, The Universe in a Helium Droplet (Clarendon, Oxford, 2003).zbMATHGoogle Scholar
  10. 10.
    G. E. Volovik, Lect. Notes Phys. 718, 31 (2007); arXiv:cond-mat/0601372.ADSCrossRefMathSciNetGoogle Scholar
  11. 11.
    P. Hořava, Phys. Rev. Lett. 95, 016405 (2005).ADSCrossRefGoogle Scholar
  12. 12.
    M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).ADSCrossRefGoogle Scholar
  13. 13.
    Xiao-Liang Qi and Shou-Cheng Zhang, Rev. Mod. Phys. 83, 1057 (2011).ADSCrossRefGoogle Scholar
  14. 14.
    M. A. Silaev and G. E. Volovik, J. Low Temp. Phys. 161, 460 (2010); arXiv:1005.4672.ADSCrossRefGoogle Scholar
  15. 15.
    G. E. Volovik, JETP Lett. 66(7), 522 (1997); arXiv:condmat/9709084.ADSCrossRefGoogle Scholar
  16. 16.
    Suk Bum Chung and Shou-Cheng Zhang, Phys. Rev. Lett. 103, 235301 (2009); arXiv:0907.4394.ADSCrossRefGoogle Scholar
  17. 17.
    G. E. Volovik, JETP Lett. 90(5), 398 (2009); arXiv:0907.5389.ADSCrossRefGoogle Scholar
  18. 18.
    Y. Nagato, S. Higashitani, and K. Nagai, J. Phys. Soc. Jpn. 78, 123603 (2009).ADSCrossRefGoogle Scholar
  19. 19.
    M. M. Salomaa and G. E. Volovik, Phys. Rev. B: Condens. Matter 37, 9298 (1988).ADSCrossRefMathSciNetGoogle Scholar
  20. 20.
    G. E. Volovik, JETP Lett. 90(8), 587 (2009); arXiv:0909.3084.ADSCrossRefMathSciNetGoogle Scholar
  21. 21.
    F. Wilczek, New J. Phys. 16, 082003 (2014).ADSCrossRefGoogle Scholar
  22. 22.
    D. A. Ivanov, Phys. Rev. Lett. 86, 268 (2001).ADSCrossRefGoogle Scholar
  23. 23.
    G. E. Volovik, JETP Lett. 70(9), 609 (1999); arXiv:condmat/9909426.ADSCrossRefGoogle Scholar
  24. 24.
    Y. Tsutsumi, M. Ichioka, and K. Machida, Phys. Rev. B: Condens. Matter 83, 094510 (2011).ADSCrossRefGoogle Scholar
  25. 25.
    Xiangang Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Phys. Rev. B: Condens. Matter 83, 205101 (2011).ADSCrossRefGoogle Scholar
  26. 26.
    A. A. Burkov and L. Balents, Phys. Rev. Lett. 107, 127205 (2011).ADSCrossRefGoogle Scholar
  27. 27.
    M. A. Silaev and G. E. Volovik, Phys. Rev. B: Condens. Matter 86, 214511 (2012); arXiv:1209.3368.ADSCrossRefGoogle Scholar
  28. 28.
    N. B. Kopnin and M. M. Salomaa, Phys. Rev. B: Condens. Matter 44, 9667 (1991).ADSCrossRefGoogle Scholar
  29. 29.
    Y. Tanaka and S. Kashiwaya, Phys. Rev. Lett. 74, 3451 (1995).ADSCrossRefGoogle Scholar
  30. 30.
    S. Ryu and Y. Hatsugai, Phys. Rev. Lett. 89, 077002 (2002).ADSCrossRefGoogle Scholar
  31. 31.
    A. P. Schnyder and S. Ryu, Phys. Rev. B: Condens. Matter 84, 060504(R) (2011); arXiv:1011.1438.ADSCrossRefGoogle Scholar
  32. 32.
    T. T. Heikkiland G. E. Volovik, JETP Lett. 93(2), 59 (2011); arXiv:1011.4185.ADSCrossRefGoogle Scholar
  33. 33.
    T. T. Heikkilä, N. B. Kopnin, and G. E. Volovik, JETP Lett. 94(3), 233 (2011); arXiv:1012.0905.ADSCrossRefGoogle Scholar
  34. 34.
    G. E. Volovik, JETP Lett. 93(2), 66 (2011); arXiv:1011.4665.ADSCrossRefGoogle Scholar
  35. 35.
    M. Sato, Y. Tanaka, K. Yada, and T. Yokoyama, Phys. Rev. B: Condens. Matter 83, 224511 (2011).ADSCrossRefGoogle Scholar
  36. 36.
    L. V. Levitin, R. G. Bennett, A. Casey, B. Cowan, J. Saunders, D. Drung, T. Schurig, J. M. Parpia, B. Ilic, and N. Zhelev, J. Low Temp. Phys. 175, 667 (2014).ADSCrossRefGoogle Scholar
  37. 37.
    H. So, Prog. Theor. Phys. 74, 585 (1985).ADSCrossRefGoogle Scholar
  38. 38.
    K. Ishikawa and T. Matsuyama, Z. Phys. C: Part. Fields 33, 41 (1986).CrossRefGoogle Scholar
  39. 39.
    K. Ishikawa and T. Matsuyama, Nucl. Phys. B 280, 523 (1987).ADSCrossRefGoogle Scholar
  40. 40.
    G. E. Volovik, Sov. Phys. JETP 94(9), 1804 (1988).Google Scholar
  41. 41.
    G. E. Volovik and V. M. Yakovenko, J. Phys.: Condens. Matter 1, 5263 (1989).ADSGoogle Scholar
  42. 42.
    D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982).ADSCrossRefGoogle Scholar
  43. 43.
    Q. Niu, D. J. Thouless, and Y.-Sh. Wu, Phys. Rev. B: Condens. Matter 31, 3372 (1985).ADSCrossRefMathSciNetGoogle Scholar
  44. 44.
    J. S. Korhonen and G. E. Volovik, JETP Lett. 55(6), 368 (1992).ADSGoogle Scholar
  45. 45.
    D. Vollhardt and P. Woelfe, The Superfluid Phases of Helium 3 (Taylor and Francis, New York, 1990).Google Scholar
  46. 46.
    R. G. Bennett, L. V. Levitin, A. Casey, B. Cowan, J. Parpia, and J. Saunders, J. Low Temp. Phys. 158, 163 (2010).ADSCrossRefGoogle Scholar
  47. 47.
    L. V. Levitin, R. G. Bennett, A. Casey, B. Cowan, J. Parpia, and J. Saunders, J. Low Temp. Phys. 158, 159 (2010).ADSCrossRefGoogle Scholar
  48. 48.
    L. Levitin, R. Bennett, A. Casey, B. Cowan, J. Saunders, D. Drung, Th. Schurig, and J. M. Parpia, Science (Washington) 340, 841 (2013).ADSCrossRefGoogle Scholar
  49. 49.
    L. V. Levitin, R. G. Bennett, E. V. Surovtsev, J. M. Parpia, B. Cowan, A. J. Casey, and J. Saunders, Phys. Rev. Lett. 111, 235304 (2013).ADSCrossRefGoogle Scholar
  50. 50.
    G. E. Volovik, Exotic Properties of Superfluid 3 He (World Scientific, Singapore, 1992).CrossRefGoogle Scholar
  51. 51.
    G. E. Volovik, AIP Conf. Proc. 194, 136 (1989).ADSCrossRefGoogle Scholar
  52. 52.
    J. A. Sauls, Phys. Rev. B: Condens. Matter 84, 214509 (2011).ADSCrossRefGoogle Scholar
  53. 53.
    H. Wu and J. A. Sauls, Phys. Rev. B: Condens. Matter 88, 184506 (2013).ADSCrossRefGoogle Scholar
  54. 54.
    B. A. Volkov, A. A. Gorbatsevich, Yu. V. Kopaev, and V. V. Tugushev, Sov. Phys. JETP 54(2), 391 (1981); B. A. Volkov and O. A. Pankratov, JETP Lett. 42 (4), 178 (1985).Google Scholar
  55. 55.
    K. Nagai, Y. Nagato, M. Yamamoto, and S. Higashitani, J. Phys. Soc. Jpn. 77, 111003 (2008).ADSCrossRefGoogle Scholar
  56. 56.
    S. Murakawa, Y. Tamura, Y. Wada, M. Wasai, M. Saitoh, Y. Aoki, R. Nomura, Y. Okuda, Y. Nagato, M. Yamamoto, S. Higashitani, and K. Nagai, Phys. Rev. Lett. 103, 155301 (2009).ADSCrossRefGoogle Scholar
  57. 57.
    J. P. Davis, J. Pollanen, H. Choi, J. A. Sauls, W. P. Halperin, and A. B. Vorontsov, Phys. Rev. Lett. 101, 085301 (2008).ADSCrossRefGoogle Scholar
  58. 58.
    Y. Aoki, Y. Wada, M. Saitoh, R. Nomura, Y. Okuda, Y. Nagato, M. Yamamoto, S. Higashitani, and K. Nagai, Phys. Rev. Lett. 95, 075301 (2005).ADSCrossRefGoogle Scholar
  59. 59.
    H. Choi, J. P. Davis, J. Pollanen, and W. P. Halperin, Phys. Rev. Lett. 96, 125301 (2006).ADSCrossRefGoogle Scholar
  60. 60.
    Y. M. Bunkov, J. Low Temp. Phys. 175, 385 (2014).ADSCrossRefGoogle Scholar
  61. 61.
    Yu. Makhlin, M. Silaev, and G. E. Volovik, Phys. Rev. B: Condens. Matter 89, 174502 (2014); arXiv:1312.2677.ADSCrossRefGoogle Scholar
  62. 62.
    M. A. Silaev and G. E. Volovik, JETP Lett. 95(1), 25 (2012).ADSCrossRefGoogle Scholar
  63. 63.
    V. Gurarie, Phys. Rev. B: Condens. Matter 83, 085426 (2011).ADSCrossRefGoogle Scholar
  64. 64.
    A. M. Essin and V. Gurarie, Phys. Rev. B: Condens. Matter 84, 125132 (2011).ADSCrossRefGoogle Scholar
  65. 65.
    L. Fidkowski and A. Kitaev, Phys. Rev. B: Condens. Matter 81, 134509 (2010); L. Fidkowski and A. Kitaev, Phys. Rev. B: Condens. Matter 83, 075103 (2011).ADSCrossRefGoogle Scholar
  66. 66.
    T. Mizushima, M. Sato, and K. Machida, Phys. Rev. Lett. 109, 165301 (2012).ADSCrossRefGoogle Scholar
  67. 67.
    T. Mizushima, Phys. Rev. B: Condens. Matter 86, 094518 (2012).ADSCrossRefGoogle Scholar
  68. 68.
    G. E. Volovik, JETP Lett. 91(4), 201 (2010).ADSCrossRefGoogle Scholar
  69. 69.
    T.-L. Ho, J. R. Fulco, J. R. Schrieffer, and F. Wilczek, Phys. Rev. Lett. 52, 1524 (1984).ADSCrossRefGoogle Scholar
  70. 70.
    M. M. Salomaa and G. E. Volovik, J. Low Temp. Phys. 74, 319 (1989).ADSCrossRefGoogle Scholar
  71. 71.
    M. Nakahara, J. Phys. C: Solid State Phys. 19, L195 (1986).ADSCrossRefGoogle Scholar
  72. 72.
    C. Caroli, P. G. de Gennes, and J. Matricon, Phys. Lett. 9, 307 (1964).ADSCrossRefzbMATHGoogle Scholar
  73. 73.
    N. Read and D. Green, Phys. Rev. B: Condens. Matter 61, 10267 (2000).ADSCrossRefGoogle Scholar
  74. 74.
    M. Sato, A. Yamakage, and T. Mizushima, Physica E (Amsterdam) 55, 20 (2014).ADSCrossRefGoogle Scholar
  75. 75.
    M. M. Salomaa and G. E. Volovik, Rev. Mod. Phys. 59, 533 (1987).ADSCrossRefGoogle Scholar
  76. 76.
    T. Sh. Misirpashaev and G. E. Volovik, Physica B (Amsterdam) 210, 338 (1995).ADSCrossRefGoogle Scholar
  77. 77.
    Y. Nishida, Phys. Rev. D: Part., Fields, Gravitation, Cosmol. 81, 074004 (2010).CrossRefGoogle Scholar
  78. 78.
    M. A. Silaev, JETP Lett. 90(5), 391 (2009).ADSCrossRefMathSciNetGoogle Scholar
  79. 79.
    G. E. Volovik, J. Phys.: Condens. Matter 3, 357 (1991).ADSGoogle Scholar
  80. 80.
    R. Jackiw and P. Rossi, Nucl. Phys. B 190, 681 (1981).ADSCrossRefGoogle Scholar
  81. 81.
    G. E. Volovik, JETP Lett. 91(2), 55 (2010); arXiv:0912.0502.ADSCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Inc. 2014

Authors and Affiliations

  1. 1.Low Temperature LaboratoryAalto UniversityAaltoFinland
  2. 2.Institute for Physics of MicrostructuresRussian Academy of SciencesNizhni NovgorodRussia
  3. 3.Landau Institute for Theoretical PhysicsRussian Academy of SciencesChernogolovka, Moscow oblastRussia

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