Quantum Hall effect and the topological number in graphene

Recently, an unusual integer quantum Hall effect was observed in graphene in which the Hall conductivity is quantized as ${\sigma }_{xy}=(\pm 2,\pm 6,\pm 10,\dots )\times e^2∕h$, where $e$ is the electron charge and $h$ is the Planck constant. To explain this we consider the energy structure as a function of magnetic field (the Hofstadter butterfly diagram) on the honeycomb lattice and the Streda formula for Hall conductivity. The quantized Hall conductivities are identified as the topological TKNN integers [D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982); M. Kohmoto, Ann. Phys. (N.Y.) 160, 343 (1985)]. They are odd integers $\pm 1,\pm 3,\pm 5,\dots \times 2$ (spin degrees of freedom) when a uniform magnetic field is as high as $30\mathrm{T}$ for example. The gaps corresponding to even integers, $\pm 2,\pm 4,\pm 6,\dots$ are too small to be observed, but when the system is anisotropic, which is described by the generalized honeycomb lattice, and/or in an extremely strong magnetic field, quantization in even integers takes place as well. We also compare the results with those for the square lattice in an extremely strong magnetic field.