Abstract.
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.
Similar content being viewed by others
References
G. Semenoff, Phys. Rev. Lett. 53, 2449 (1984)
F.D.M. Haldane, Phys. Rev. Lett. 61, 2015 (1987)
C.L. Kane, E.J. Mele, Phys. Rev. Lett. 95, 146802 (2005)
A.W.W. Ludwig, M.P.A. Fisher, R. Shankar, G. Grinstein, Phys. Rev. B 50, 7526 (1993)
Y. Hatsugai, M. Kohmoto, Phys. Rev. B 42, 8282 (1990)
V.P. Gusynin, S. Sharapov, Phys. Rev. Lett. 95, 146801 (2005)
K.S. Novoselov, Nature 438, 197 (2005); Nat. Phys. 2, 177 (2006)
Y. Zhang, Y.-W. Tan, H.L. Stormer, P. Kim, Nature 438, 201 (2005)
Y. Zheng, T. Ando, Phys. Rev. B 65, 245420 (2002)
T. Ando, J. Phys. Soc. Jpn. 74, 777 (2005)
M. Fujita, K. Wakabayashi, K. Nakada, K. Kusakabe, J. Phys. Soc. Jpn. 65, 1920 (1996)
K. Wakabayashi, M. Fujita, H. Ajiki, M. Sigrist, Phys. Rev. B 59, 8271 (1999)
X.G. Wen, Phys. Rev. B 40, 7387 (1989)
Y. Hatsugai, J. Phys. Soc. Jpn. 73, 2604 (2004a)
Y. Hatsugai, J. Phys. Soc. Jpn. 74, 1374 (2004b)
A. Shapere, F. Wilczek (eds.), Geometric Phases in Physics (World Scientific, 1989)
R.B. Laughlin, Phys. Rev. B 23, 5632 (1981)
B.I. Halperin, Phys. Rev. B 25, 2185 (1982)
Y. Hatsugai, Phys. Rev. B 48, 11851 (1993a)
Y. Hatsugai, Phys. Rev. Lett. 71, 3697 (1993b)
Y. Hatsugai, J. Phys. Condens. Matter 9, 2507 (1997)
E. Witten, Phys. Lett. B 117, 324 (1982)
A.J. Niemi, G. Semenoff, Phys. Rev. Lett. 51, 2077 (1983)
S. Ryu, Y. Hatsugai, Phys. Rev. Lett. 89, 077002 (2002)
S. Ryu, Y. Hatsugai, Physica C 388-389, 90 (2003)
Y. Hatsugai, T. Fukui, H. Aoki, Phys. Rev. B 74, 205414 (2006)
Y. Hatsugai, T. Fukui, H. Aoki (unpublished)
H.B. Nielsen, M. Ninomiya, Nucl. Phys. 185, 20 (1981)
H. Aoki, T. Ando, Solid State Commun. 38, 1079 (1981)
D.J. Thouless, M. Kohmoto, P. Nightingale, M. den Nijs, Phys. Rev. Lett. 49, 405 (1982)
T. Fukui, Y. Hatsugai, H. Suzuki, J. Phys. Soc. Jpn. 74, 1674 (2005)
Y. Hatsugai, T. Fukui, H. Suzuki (2005), cond-mat/0507466
D.N. Sheng, L. Sheng, Z.Y. Weng, cond-mat/0602190
K. Kusakabe, Y. Takagi, Mol. Cryst. Liq. Cryst. 387, 7 (2002)
H. Aoki, Surf. Sci. 263, 137 (1992)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hatsugai, Y., Fukui, T. & Aoki, H. Topological aspects of graphene. Eur. Phys. J. Spec. Top. 148, 133–141 (2007). https://doi.org/10.1140/epjst/e2007-00233-5
Issue Date:
DOI: https://doi.org/10.1140/epjst/e2007-00233-5