The European Physical Journal Special Topics

, Volume 148, Issue 1, pp 133–141 | Cite as

Topological aspects of graphene

Dirac fermions and the bulk-edge correspondence in magnetic fields
  • Y. Hatsugai
  • T. Fukui
  • H. Aoki


Topological aspects of the electronic properties of graphene, including edge effects, with the tight-binding model on a honeycomb lattice and its extensions to show the following: (i) Presence of the pair of massless Dirac dispersions, which is the origin of anomalous properties including a peculiar quantum Hall effect (QHE), is not accidental to honeycomb, but is generic for a class of two-dimensional lattices that interpolate between square and π-flux lattices. Topological stability guarantees persistence of the peculiar QHE. (ii) While we have the massless Dirac dispersion only around E=0, the anomalous QHE associated with the Dirac cone unexpectedly persists for a wide range of the chemical potential. The range is bounded by van Hove singularities, at which we predict a transition to the ordinary fermion behaviour accompanied by huge jumps in the QHE with a sign change. (iii) We establish a coincidence between the quantum Hall effect in the bulk and the quantum Hall effect for the edge states, which is another topological effect. We have also explicitly shown that the E=0 edge states in honeycomb in zero magnetic field persist in magnetic field. (iv) We have also identified a topological origin of the fermion doubling in terms of the chiral symmetry.


European Physical Journal Special Topic Chiral Symmetry Edge State Dirac Fermion Hall Conductivity 
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  1. G. Semenoff, Phys. Rev. Lett. 53, 2449 (1984) Google Scholar
  2. F.D.M. Haldane, Phys. Rev. Lett. 61, 2015 (1987) Google Scholar
  3. C.L. Kane, E.J. Mele, Phys. Rev. Lett. 95, 146802 (2005) Google Scholar
  4. A.W.W. Ludwig, M.P.A. Fisher, R. Shankar, G. Grinstein, Phys. Rev. B 50, 7526 (1993) Google Scholar
  5. Y. Hatsugai, M. Kohmoto, Phys. Rev. B 42, 8282 (1990) Google Scholar
  6. V.P. Gusynin, S. Sharapov, Phys. Rev. Lett. 95, 146801 (2005) Google Scholar
  7. K.S. Novoselov, Nature 438, 197 (2005); Nat. Phys. 2, 177 (2006) Google Scholar
  8. Y. Zhang, Y.-W. Tan, H.L. Stormer, P. Kim, Nature 438, 201 (2005) Google Scholar
  9. Y. Zheng, T. Ando, Phys. Rev. B 65, 245420 (2002) Google Scholar
  10. T. Ando, J. Phys. Soc. Jpn. 74, 777 (2005) Google Scholar
  11. M. Fujita, K. Wakabayashi, K. Nakada, K. Kusakabe, J. Phys. Soc. Jpn. 65, 1920 (1996) Google Scholar
  12. K. Wakabayashi, M. Fujita, H. Ajiki, M. Sigrist, Phys. Rev. B 59, 8271 (1999) Google Scholar
  13. X.G. Wen, Phys. Rev. B 40, 7387 (1989) Google Scholar
  14. Y. Hatsugai, J. Phys. Soc. Jpn. 73, 2604 (2004a) Google Scholar
  15. Y. Hatsugai, J. Phys. Soc. Jpn. 74, 1374 (2004b) Google Scholar
  16. A. Shapere, F. Wilczek (eds.), Geometric Phases in Physics (World Scientific, 1989) Google Scholar
  17. R.B. Laughlin, Phys. Rev. B 23, 5632 (1981) Google Scholar
  18. B.I. Halperin, Phys. Rev. B 25, 2185 (1982) Google Scholar
  19. Y. Hatsugai, Phys. Rev. B 48, 11851 (1993a) Google Scholar
  20. Y. Hatsugai, Phys. Rev. Lett. 71, 3697 (1993b) Google Scholar
  21. Y. Hatsugai, J. Phys. Condens. Matter 9, 2507 (1997) Google Scholar
  22. E. Witten, Phys. Lett. B 117, 324 (1982) Google Scholar
  23. A.J. Niemi, G. Semenoff, Phys. Rev. Lett. 51, 2077 (1983) Google Scholar
  24. S. Ryu, Y. Hatsugai, Phys. Rev. Lett. 89, 077002 (2002) Google Scholar
  25. S. Ryu, Y. Hatsugai, Physica C 388-389, 90 (2003) Google Scholar
  26. Y. Hatsugai, T. Fukui, H. Aoki, Phys. Rev. B 74, 205414 (2006) Google Scholar
  27. Y. Hatsugai, T. Fukui, H. Aoki (unpublished) Google Scholar
  28. H.B. Nielsen, M. Ninomiya, Nucl. Phys. 185, 20 (1981) Google Scholar
  29. H. Aoki, T. Ando, Solid State Commun. 38, 1079 (1981) Google Scholar
  30. D.J. Thouless, M. Kohmoto, P. Nightingale, M. den Nijs, Phys. Rev. Lett. 49, 405 (1982) Google Scholar
  31. T. Fukui, Y. Hatsugai, H. Suzuki, J. Phys. Soc. Jpn. 74, 1674 (2005) Google Scholar
  32. Y. Hatsugai, T. Fukui, H. Suzuki (2005), cond-mat/0507466 Google Scholar
  33. D.N. Sheng, L. Sheng, Z.Y. Weng, cond-mat/0602190 Google Scholar
  34. K. Kusakabe, Y. Takagi, Mol. Cryst. Liq. Cryst. 387, 7 (2002) Google Scholar
  35. H. Aoki, Surf. Sci. 263, 137 (1992) Google Scholar

Copyright information

© EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2007

Authors and Affiliations

  • Y. Hatsugai
    • 1
  • T. Fukui
    • 2
  • H. Aoki
    • 3
  1. 1.Department of Applied PhysicsUniversity of TokyoTokyoJapan
  2. 2.Department of Mathematical SciencesIbaraki UniversityMitoJapan
  3. 3.Department of PhysicsUniversity of TokyoTokyoJapan

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