Quantum Hall effect and the topological number in graphene

Recently unusual integer quantum Hall effect was observed in graphene in which the Hall conductivity is quantized as $\sigma_{xy}=(\pm 2, \pm 6, \pm 10, >...) \times \frac{e^2}{h}$, where $e$ is the electron charge and $h$ is the Planck constant. %\cite{Novoselov2005,Zheng2005}, %although it can be explained in the argument of massless Dirac fermions, 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 conductivity is obtained to be odd integer, $\pm1, \pm3, \pm5, ...$ times two (spin degrees of freedom) when a uniform magnetic field is as high as 30T for example. When the system is anisotropic and described by the generalized honeycomb lattice, Hall conductivity can be quantized to be any integer number. We also compare the results with those for the square lattice under extremely strong magnetic field.Comment: 4 pages, 10 figure

Correlations in one-dimensional disordered electronic systems with interaction

We investigate the effects of randomness in a strongly correlated electron model in one-dimension at half-filling. The ground state correlation functions are exactly written by products of 3$\times$3 transfer matrices and are evaluated numerically. The correlation lengths depend on randomness when the interaction is effectively weak. On the contrary, they are completely insensitive to randomness when the interaction is effectively strong.Comment: 7 pages, Revte

Zero modes and the edge states of the honeycomb lattice

The honeycomb lattice in the cylinder geometry with zigzag edges, bearded edges, zigzag and bearded edges (zigzag-bearded), and armchair edges are studied. The tight-binding model with nearest-neighbor hoppings is used. Edge states are obtained analytically for these edges except the armchair edges. It is shown, however, that edge states for the armchair edges exist when the the system is anisotropic. These states have not been known previously. We also find strictly localized states, uniformly extended states and states with macroscopic degeneracy.Comment: 6 pages 8 figure

Gauge fields, quantized fluxes and monopole confinement of the honeycomb lattice

Electron hopping models on the honeycomb lattice are studied. The lattice consists of two triangular sublattices, and it is non-Bravais. The dual space has non-trivial topology. The gauge fields of Bloch electrons have the U(1) symmetry and thus represent superconducting states in the dual space. Two quantized Abrikosov fluxes exist at the Dirac points and have fluxes $2pi$ and $-2pi$, respectively. We define the non-Abelian SO(3) gauge theory in the extended 3$d$ dual space and it is shown that a monopole and anti-monoplole solution is stable. The SO(3) gauge group is broken down to U(1) at the 2$d$ boundary.The Abrikosov fluxes are related to quantized Hall conductance by the topological expression. Based on this, monopole confinement and deconfinement are discussed in relation to time reversal symmetry and QHE. The Jahn-Teller effect is briefly discussed.Comment: 10 pages, 11 figure

On Berry Phase in Bloch States

We comment on the relation between Berry phase and quantized Hall conductivities for charge and spin currents in some Bloch states, such as Bloch electrons in the presence of electromagnetic fields and quasiparticles in the vortex states of superfluid $^3$He. One can find out that the arguments presented here are closely related to the spontaneous polarization in crystalline dielectrics and the adiabatic pumping.Comment: 2 pages, 1 figure, to be published in the proceedings of LT23 in Hiroshima, Aug. 200