Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Bosonization for fermions and parafermions

  • 4 Accesses

Abstract

Parafermions are fractional excitations which can be regarded as generalizations of Majorana bound states, but in contrast to the latter they require electron-electron interactions. Compared to Majorana bound states, they offer richer non-Abelian braiding statistics, and have thus been proposed as building blocks for topologically protected universal quantum computation. In this review, we provide a pedagogical introduction to the field of parafermion bound states in one-dimensional systems. We present the necessary theoretical tools for their study, in particular bosonization and the renormalization-group technique, and show how those can be applied to study parafermions.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    J. Alicea, Rep. Prog. Phys. 75, 076501 (2012)

  2. 2.

    C.W.J. Beenakker, Ann. Rev. Cond. Mat. Phys. 4, 113 (2013)

  3. 3.

    M. Leijnse, K. Flensberg, Semicond. Sci. Technol. 27, 124003 (2012)

  4. 4.

    R.S. Deacon et al., Phys. Rev. X 7, 021011 (2017)

  5. 5.

    E. Bocquillon, R.S. Deacon, J. Wiedenmann, P. Leubner, T.M. Klapwijk, C. Brüne, K. Ishibashi, H. Buhmann, L.W. Molenkamp, Nat. Nanotechnol. 12, 137 (2016)

  6. 6.

    V. Mourik, K. Zuo, S.M. Frolov, S.R. Plissard, E.P.A.M. Bakkers, L.P. Kouwenhoven, Science 336, 1003 (2012)

  7. 7.

    S.M. Albrecht, A.P. Higginbotham, M. Madsen, F. Kuemmeth, T.S. Jespersen, J. Nygård, P. Krogstrup, C.M. Marcus, Nature 531, 206 (2016)

  8. 8.

    Ö. Gül et al., Nat. Nanotechnol. 13, 192 (2018)

  9. 9.

    S.D. Sarma, M. Freedman, C. Nayak, Phys. Today 59, 32 (2006)

  10. 10.

    C. Nayak, S.H. Simon, A. Stern, M. Freedman, S.D. Sarma, Rev. Mod. Phys. 80, 1083 (2008)

  11. 11.

    V. Lahtinen, J.K. Pachos, SciPost Phys. 3, 021 (2017)

  12. 12.

    J. Alicea, P. Fendley, Ann. Rev. Condens. Mat. Phys. 7, 119 (2016)

  13. 13.

    N.H. Lindner, E. Berg, G. Refael, A. Stern, Phys. Rev. X 2, 041002 (2012)

  14. 14.

    D.J. Clarke, J. Alicea, K. Shtengel, Nat. Comm. 4, 1348 (2013)

  15. 15.

    M. Barkeshli, C.-M. Jian, X.-L. Qi, Phys. Rev. B 87, 045130 (2013)

  16. 16.

    R.S.K. Mong et al., Phys. Rev. X 4, 011036 (2014)

  17. 17.

    M. Barkeshli, X.L. Qi, Phys. Rev. X 4, 041035 (2014)

  18. 18.

    C. Chen, F.J. Burnell, Phys. Rev. Lett. 116, 106405 (2016)

  19. 19.

    S. Groenendijk, A. Calzona, H. Tschirhart, E.G. Idrisov, T.L. Schmidt, Phys. Rev. B 100, 205424 (2019)

  20. 20.

    H. Ebisu, E. Sagi, Y. Tanaka, Y. Oreg, Phys. Rev. B 95, 075111 (2017)

  21. 21.

    Y. Alavirad, D. Clarke, A. Nag, J.D. Sau, Phys. Rev. Lett. 119, 217701 (2017)

  22. 22.

    J. Klinovaja, D. Loss, Phys. Rev. Lett. 112, 246403 (2013)

  23. 23.

    J. Klinovaja, D. Loss, Phys. Rev. B 90, 045118 (2014)

  24. 24.

    E. Sagi, Y. Oreg, Phys. Rev. B 90, 201102 (2014)

  25. 25.

    N. Kainaris, S.T. Carr, Phys. Rev. B 92, 035139 (2015)

  26. 26.

    T.L. Schmidt, C.J. Pedder, Phys. Rev. B 94, 125420 (2016)

  27. 27.

    C.J. Pedder, T. Meng, R.P. Tiwari, T.L. Schmidt, Phys. Rev. B 94, 245414 (2016)

  28. 28.

    C.J. Pedder, T. Meng, R. Tiwari, T.L. Schmidt, Phys. Rev. B 96, 165429 (2017)

  29. 29.

    V. Kornich, C.J. Pedder, T.L. Schmidt, Phys. Rev. B 95, 045413 (2017)

  30. 30.

    E. Sela, A. Altland, A. Rosch, Phys. Rev. B 84, 085114 (2011)

  31. 31.

    J. Klinovaja, A. Yacoby, D. Loss, Phys. Rev. B 90, 155447 (2014)

  32. 32.

    F. Zhang, C.L. Kane, Phys. Rev. Lett. 113, 036401 (2014)

  33. 33.

    C.P. Orth, R.P. Tiwari, T. Meng, T.L. Schmidt, Phys. Rev. B 91, 081406(R) (2015)

  34. 34.

    R. Hevroni, V. Shelukhin, M. Karpovski, M. Goldstein, E. Sela, H. Shtrikman, A. Palevski, Phys. Rev. B 93, 035305 (2016)

  35. 35.

    N.T. Ziani, C. Fleckenstein, G. Dolcetto, B. Trauzettel, Phys. Rev. B 95, 205418 (2017)

  36. 36.

    Y. Vinkler-Aviv, P.W. Brouwer, F. von Oppen, Phys. Rev. B 96, 195421 (2017)

  37. 37.

    J. Klinovaja, D. Loss, Phys. Rev. B 92, 121410(R) (2015)

  38. 38.

    C. Fleckenstein, N.T. Ziani, B. Trauzettel, Phys. Rev. Lett. 122, 066801 (2019)

  39. 39.

    J. Strunz et al. (2019), https://arXiv:1905.08175 [cond-mat.mes-hall]

  40. 40.

    R. Blumenhagen, E. Plauschin, Introduction to Conformal Field Theory, Lecture Notes in Physics (Springer, Berlin Heidelberg, 2009)

  41. 41.

    P. Francesco, P. Mathieu, D. Senechal,Conformal Field Theory (Springer, 1997)

  42. 42.

    T. Giamarchi,Quantum Physics in One Dimension (Clarendon Press, Oxford, 2003)

  43. 43.

    D. Sénéchal (1999), https://arXiv:cond-mat/9908262 [cond-mat.str-el]

  44. 44.

    J. von Delft, H. Schoeller, Ann. Phys. 7, 225 (1998)

  45. 45.

    J.M. Luttinger, J. Math. Phys. 4, 1154 (1963)

  46. 46.

    S. Tomonaga, Prog. Theor. Phys. 5, 544 (1950)

  47. 47.

    F.D.M. Haldane, J. Phys. C: Solid State Phys. 14, 2585 (1981)

  48. 48.

    A. Imambekov, T.L. Schmidt, L.I. Glazman, Rev. Mod. Phys. 84, 1253 (2012)

  49. 49.

    A.Y. Kitaev, Physics-Uspekhi 44, 131 (2001)

  50. 50.

    A.B. Zamolodchikov, V.A. Fateev, Sov. Phys. JETP 62, 215 (1985)

  51. 51.

    R.S.K. Mong, D.J. Clarke, J. Alicea, N.H. Lindner, P. Fendley, J. Phys. A: Math. Theor. 47, 452001 (2014)

  52. 52.

    N. Read, E. Rezayi, Phys. Rev. B 59, 8084 (1999)

  53. 53.

    G. Moore, N. Read, Nucl. Phys. B 360, 362 (1991)

  54. 54.

    P. Rickhaus, M. Weiss, L. Marot, C. Schönenberger, Nano Letters 12, 1942 2012

  55. 55.

    F. Amet et al., Science 352, 966 (2016)

  56. 56.

    G.-H. Lee, K.F. Huang, D.K. Efetov, D.S. Wei, S. Hart, T. Taniguchi, K. Watanabe, A. Yacoby, P. Kim, Nat. Phys. 13, 693 (2017)

  57. 57.

    P. Fendley, J. Stat. Mech. 2012, P11020 (2012)

  58. 58.

    E. Cobanera, G. Ortiz, Phys. Rev. A 89, 012328 (2014)

  59. 59.

    A. Hutter, J.R. Wootton, D. Loss, Phys. Rev. X 5, 041040 (2015)

  60. 60.

    A. Hutter, D. Loss, Phys. Rev. B 93, 125105 (2016)

  61. 61.

    Y. Zhuang, H.J. Changlani, N.M. Tubman, T.L. Hughes, Phys. Rev. B 92, 035154 (2015)

  62. 62.

    F. Iemini, C. Mora, L. Mazza, Phys. Rev. Lett. 118, 170402 (2017)

  63. 63.

    A. Calzona, T. Meng, M. Sassetti, T.L. Schmidt, Phys. Rev. B 98, 201110(R) (2018)

  64. 64.

    L. Mazza, F. Iemini, M. Dalmonte, C. Mora, Phys. Rev. B 98, 201109 (2018)

  65. 65.

    A. Chew, D.F. Mross, J. Alicea, Phys. Rev. B 98, 085143 (2018)

  66. 66.

    M.C. Strinati, S. Sahoo, K. Shtengel, E. Sela, Phys. Rev. B 99, 245101 (2019)

  67. 67.

    J. Alicea, Y. Oreg, G. Refael, F. von Oppen, M.P.A. Fisher, Nat. Phys. 7, 412 (2011)

  68. 68.

    L. Fidkowski, A. Kitaev, Phys. Rev. B 83, 075103 (2011)

  69. 69.

    A.M. Turner, F. Pollmann, E. Berg, Phys. Rev. B 83, 075102 (2011)

  70. 70.

    X. Chen, Z.C. Gu, X.G. Wen, Phys. Rev. B 84, 235128 (2011)

  71. 71.

    D.C. Mattis, E.H. Lieb, J. Math. Phys. 6, 304 (1965)

  72. 72.

    L. Fu, C.L. Kane, Phys. Rev. B 79, 161408 (2009)

  73. 73.

    H.B. Nielsen, M. Ninomiya, Phys. Lett. B 105, 219 (1981)

  74. 74.

    J. Cardy,Scaling and Renormalization in Statistical Physics (Cambridge University Press, 1996)

  75. 75.

    R. Shankar, Rev. Mod. Phys. 66, 129 (1994)

  76. 76.

    G. Mahan,Many-particle physics (Plenum, New York, 1990)

  77. 77.

    F.A. Smirnov, Form Factors in Completely Integrable Models of Quantum Field Theory, Advanced Series in Mathematical Physics (World Scientific, 1992)

  78. 78.

    A.B. Zamolodchikov, A.B. Zamolodchikov, Ann. Phys. 120, 253 (1979)

  79. 79.

    A.B. Zamolodchikov, Int. J. Mod. Phys. A 10, 1125 (1995)

  80. 80.

    S. Lukyanov, A. Zamolodchikov, Nucl. Phys. B 493, 571 (1997)

  81. 81.

    S. Lukyanov, A. Zamolodchikov, Nucl. Phys. B 607, 437 (2001)

  82. 82.

    S. Coleman, Phys. Rev. D 11, 2088 (1975)

  83. 83.

    A. Luther, V.J. Emery, Phys. Rev. Lett. 33, 589 (1974)

  84. 84.

    T. Meng, Eur. Phys. J Special topics, in this issue

Download references

Author information

Correspondence to Thomas L. Schmidt.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Schmidt, T.L. Bosonization for fermions and parafermions. Eur. Phys. J. Spec. Top. 229, 621–636 (2020). https://doi.org/10.1140/epjst/e2019-900112-y

Download citation