Integer and fractional quantum Hall effect in a strip of stripes

Regular Article

Abstract

We study anisotropic stripe models of interacting electrons in the presence of magnetic fields in the quantum Hall regime with integer and fractional filling factors. The model consists of an infinite strip of finite width that contains periodically arranged stripes (forming supercells) to which the electrons are confined and between which they can hop with associated magnetic phases. The interacting electron system within the one-dimensional stripes are described by Luttinger liquids and shown to give rise to charge and spin density waves that lead to periodic structures within the stripe with a reciprocal wavevector 8kF in a mean field approximation. This wavevector gives rise to Umklapp scattering and resonant scattering that results in gaps and chiral edge states at all known integer and fractional filling factors ν. The integer and odd denominator filling factors arise for a uniform distribution of stripes, whereas the even denominator filling factors arise for a non-uniform stripe distribution. We focus on the ground state of the system, and identify the quantum Hall regime via the quantized Hall conductance. For this we calculate the Hall conductance via the Streda formula and show that it is given by σH = νe2/h for all filling factors. In addition, we show that the composite fermion picture follows directly from the condition of the resonant Umklapp scattering.

Keywords

Solid State and Materials 

References

  1. 1.
    F. Wilczek, Nat. Phys. 5, 614 (2009)CrossRefGoogle Scholar
  2. 2.
    M.Z. Hasan, C.L. Kane, Rev. Mod. Phys. 82, 3045 (2010)ADSCrossRefGoogle Scholar
  3. 3.
    X. Qi, S. Zhang, Rev. Mod. Phys. 83, 1057 (2011)ADSCrossRefGoogle Scholar
  4. 4.
    J. Alicea, Rep. Prog. Phys. 75, 076501 (2012)ADSCrossRefGoogle Scholar
  5. 5.
    K.V. Klitzing, G. Dorda, M. Pepper, Phys. Rev. Lett. 45, 494 (1980)ADSCrossRefGoogle Scholar
  6. 6.
    D.C. Tsui, H.L. Stormer, A.C. Gossard, Phys. Rev. Lett. 48, 1559 (1982)ADSCrossRefGoogle Scholar
  7. 7.
    R.E. Prange, S.M. Girvin, The Quantum Hall Effect (Springer, New York, 1990)Google Scholar
  8. 8.
    J.K. Jain, Composite Fermions (Cambridge University Press, Cambridge, 2007)Google Scholar
  9. 9.
    N. Read, D. Green, Phys. Rev. B 61, 10267 (2000)ADSCrossRefGoogle Scholar
  10. 10.
    C. Nayak, S.H. Simon, A. Stern, M. Freedman, S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008)ADSMATHCrossRefGoogle Scholar
  11. 11.
    L. Fu, C.L. Kane, Phys. Rev. Lett. 100, 096407 (2008)ADSCrossRefGoogle Scholar
  12. 12.
    Y. Tanaka, T. Yokoyama, N. Nagaosa, Phys. Rev. Lett. 103, 107002 (2009)ADSCrossRefGoogle Scholar
  13. 13.
    M. Sato, S. Fujimoto, Phys. Rev. B 79, 094504 (2009)ADSCrossRefGoogle Scholar
  14. 14.
    R.M. Lutchyn, J.D. Sau, S. Das Sarma, Phys. Rev. Lett. 105, 077001 (2010)ADSCrossRefGoogle Scholar
  15. 15.
    Y. Oreg, G. Refael, F. von Oppen, Phys. Rev. Lett. 105, 177002 (2010)ADSCrossRefGoogle Scholar
  16. 16.
    J. Klinovaja, S. Gangadharaiah, D. Loss, Phys. Rev. Lett. 108, 196804 (2012)ADSCrossRefGoogle Scholar
  17. 17.
    D. Sticlet, C. Bena, P. Simon, Phys. Rev. Lett. 108, 096802 (2012)ADSCrossRefGoogle Scholar
  18. 18.
    J. Klinovaja, G.J. Ferreira, D. Loss, Phys. Rev. B 86, 235416 (2012)ADSCrossRefGoogle Scholar
  19. 19.
    J. Klinovaja, D. Loss, Phys. Rev. X 3, 011008 (2013)Google Scholar
  20. 20.
    R.B. Laughlin, Phys. Rev. B 23, 5632 (1981)ADSCrossRefGoogle Scholar
  21. 21.
    D.J. Thouless, M. Kohmoto, M.P. Nightingale, M. den Nijs, Phys. Rev. Lett. 49, 405 (1982)ADSCrossRefGoogle Scholar
  22. 22.
    B.I. Halperin, Phys. Rev. B 25, 2185 (1982)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    R.B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983)ADSCrossRefGoogle Scholar
  24. 24.
    A.S. Sorensen, E. Demler, M.D. Lukin, Phys. Rev. Lett. 94, 086803 (2005)ADSCrossRefGoogle Scholar
  25. 25.
    M. Hafezi, A.S. Sorensen, E. Demler, M.D. Lukin, Phys. Rev. A 76, 023613 (2007)ADSCrossRefGoogle Scholar
  26. 26.
    T. Kitagawa, E. Berg, M. Rudner, E. Demler, Phys. Rev. B 82, 235114 (2010)ADSCrossRefGoogle Scholar
  27. 27.
    J. Dalibard, F. Gerbier, G. Juzeliunas, P. Ohberg, Rev. Mod. Phys. 83, 1523 (2011)ADSCrossRefGoogle Scholar
  28. 28.
    T. Neupert, L. Santos, C. Chamon, C. Mudry, Phys. Rev. Lett. 106, 236804 (2011)ADSCrossRefGoogle Scholar
  29. 29.
    D. Xiao, W. Zhu, Y. Ran, N. Nagaosa, S. Okamoto, Nat. Commun. 2, 596 (2011)ADSCrossRefGoogle Scholar
  30. 30.
    J.W. Venderbos, S. Kourtis, J. van den Brink, M. Daghofer, Phys. Rev. Lett. 108, 126405 (2012)ADSCrossRefGoogle Scholar
  31. 31.
    N.R. Cooper, J. Dalibard, Phys. Rev. Lett. 110, 185301 (2013)ADSCrossRefGoogle Scholar
  32. 32.
    N.Y. Yao, A.V. Gorshkov, C.R. Laumann, A.M. Lauchli, J. Ye, M.D. Lukin, Phys. Rev. Lett. 110, 185302 (2013)ADSCrossRefGoogle Scholar
  33. 33.
    Y.E. Kraus, Z. Ringel, O. Zilberberg, Phys. Rev. Lett. 111, 226401 (2013)ADSCrossRefGoogle Scholar
  34. 34.
    J. Klinovaja, D. Loss, Phys. Rev. Lett. 111, 196401 (2013)ADSCrossRefGoogle Scholar
  35. 35.
    C.L. Kane, R. Mukhopadhyay, T.C. Lubensky, Phys. Rev. Lett. 88, 036401 (2001)ADSCrossRefGoogle Scholar
  36. 36.
    J.C.Y. Teo, C.L. Kane, Phys. Rev. B 89, 085101 (2014)ADSCrossRefGoogle Scholar
  37. 37.
    A.G. Lebed, J. Exp. Theor. Phys. Lett. 43, 174 (1986)Google Scholar
  38. 38.
    D. Poilblanc, G. Montambaux, M. Heritier, P. Lederer, Phys. Rev. Lett. 58, 270 (1987)ADSCrossRefGoogle Scholar
  39. 39.
    L.P. Gor’kov, A.G. Lebed, Phys. Rev. B 51, 3285 (1995)ADSCrossRefGoogle Scholar
  40. 40.
    V.M. Yakovenko, Phys. Rev. B 43, 11353 (1991)ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    D.-H. Lee, Phys. Rev. B 50, 10788 (1994)ADSCrossRefGoogle Scholar
  42. 42.
    V.M. Yakovenko, in The Physics of Organic Superconductors and Conductors, edited by A.G. Lebed (Springer, 2008), Vol. 110, p. 529Google Scholar
  43. 43.
    K. Kobayashi, H. Satsukawa, J. Yamada, T. Terashima, S. Uji, Phys. Rev. Lett. 112, 116805 (2014)ADSCrossRefGoogle Scholar
  44. 44.
    R. Jackiw, C. Rebbi, Phys. Rev. D 13, 3398 (1976)ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    W.P. Su, J.R. Schrieffer, A.J. Heeger, Phys. Rev. Lett. 42, 1698 (1979)ADSCrossRefGoogle Scholar
  46. 46.
    S. Kivelson, J.R. Schrieffer, Phys. Rev. B 25, 6447 (1982)ADSCrossRefGoogle Scholar
  47. 47.
    S. Gangadharaiah, L. Trifunovic, D. Loss, Phys. Rev. Lett. 108, 136803 (2012)ADSCrossRefGoogle Scholar
  48. 48.
    J. Klinovaja, P. Stano, D. Loss, Phys. Rev. Lett. 109, 236801 (2012)ADSCrossRefGoogle Scholar
  49. 49.
    J. Klinovaja, D. Loss, Phys. Rev. Lett. 110, 126402 (2013)ADSCrossRefGoogle Scholar
  50. 50.
    D.R. Hofstadter, Phys. Rev. B 14, 2239 (1976)ADSCrossRefGoogle Scholar
  51. 51.
    P.K. Lam, S.M. Girvin, Phys. Rev. B 30, 473 (1984)ADSCrossRefGoogle Scholar
  52. 52.
    S. Kivelson, C. Kallin, D.P. Arovas, J.R. Schrieffer, Phys. Rev. Lett. 56, 873 (1986)ADSCrossRefGoogle Scholar
  53. 53.
    Z. Tesanovic, F. Axel, B.I. Halperin, Phys. Rev. B 39, 8525 (1989)ADSMathSciNetCrossRefGoogle Scholar
  54. 54.
    A.A. Koulakov, M.M. Fogler, B.I. Shklovskii, Phys. Rev. Lett. 76, 499 (1996)ADSCrossRefGoogle Scholar
  55. 55.
    C. Chang, C. Toeke, G. Jeon, J.K. Jain, Phys. Rev. B 73, 155323 (2006)ADSCrossRefGoogle Scholar
  56. 56.
    M.P. Lilly, K.B. Cooper, J.P. Eisenstein, L.N. Pfeiffer, K.W. West, Phys. Rev. Lett. 83, 824 (1999)ADSCrossRefGoogle Scholar
  57. 57.
    I.V. Kukushkin, V. Umansky, K.V. Klitzing, J.H. Smet, Phys. Rev. Lett. 106, 206804 (2011)ADSCrossRefGoogle Scholar
  58. 58.
    C. Wexler, O. Ciftja, Int. J. Mod. Phys. B 20, 747 (2006)ADSCrossRefGoogle Scholar
  59. 59.
    B.E. Feldman, A.J. Levin, B. Krauss, D. Abanin, B.I. Halperin, J.H. Smet, A. Yacoby, Phys. Rev. Lett. 111, 076802 (2013)ADSCrossRefGoogle Scholar
  60. 60.
    J. Klinovaja, D. Loss, Phys. Rev. B 86, 085408 (2012)ADSCrossRefGoogle Scholar
  61. 61.
    R. Peierls, More Surprises in Theoretical Physics (Princeton, 1991)Google Scholar
  62. 62.
    B. Braunecker, G.I. Japaridze, J. Klinovaja, D. Loss, Phys. Rev. B 82, 045127 (2010)ADSCrossRefGoogle Scholar
  63. 63.
    S. Bravyi, D.P. DiVincenzo, D. Loss, Ann. Phys. 326, 2793 (2011)ADSMATHMathSciNetCrossRefGoogle Scholar
  64. 64.
    G.F. Giuliani, G. Vignale, Quantum Theory of the Electron Liquid (Cambridge University Press, Cambridge, 2005)Google Scholar
  65. 65.
    N.W. Ashcroft, D.N. Mermin, Solid State Physics (Saunders College, Philadelphia, 1976)Google Scholar
  66. 66.
    P. Streda, J. Phys. C 15, L717 (1982)ADSCrossRefGoogle Scholar
  67. 67.
    D.J. Thouless, M. Kohmoto, M.P. Nightingale, M. den Nijs, Phys. Rev. Lett. 49, 405 (1982)ADSCrossRefGoogle Scholar
  68. 68.
    J.E. Avron, R. Seiler, Phys. Rev. Lett. 54, 259 (1985)ADSMathSciNetCrossRefGoogle Scholar
  69. 69.
    P.M. Morse, H. Feshbach, Methods of Theoretical Physics (MacGraw-Hill, New York, 1958)Google Scholar
  70. 70.
    R. Willett, J.P. Eisenstein, H.L. Störmer, D.C. Tsui, A.C. Gossard, J.H. English, Phys. Rev. Lett. 59, 1776 (1987)ADSCrossRefGoogle Scholar
  71. 71.
    W. Pan, J.-S. Xia, V. Shvarts, D.E. Adams, H.L. Stormer, D.C. Tsui, L.N. Pfeiffer, K.W. Baldwin, K.W. West, Phys. Rev. Lett. 83, 3530 (1999)ADSCrossRefGoogle Scholar
  72. 72.
    M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, U. Sen, Adv. Phys. 56, 243 (2007)ADSCrossRefGoogle Scholar
  73. 73.
    T. Giamarchi, Quantum Physics in One Dimension (Clarendon Press, Oxford, 2004)Google Scholar
  74. 74.
    Strongly Correlated Electronic Materials: The Los Alamos Symposium 1993, edited by K.S. Bedell et al. (Addison-Wesley, Reading, 1994)Google Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of PhysicsHarvard UniversityCambridgeUSA
  2. 2.Department of PhysicsUniversity of BaselBaselSwitzerland

Personalised recommendations