## Surface-Quantized Anomalous Hall Current and the Magnetoelectric Effect
in Magnetically Disordered Topological Insulators

We study theoretically the role of quenched magnetic disorder at the surface of a topological insulator by numerical simulation and scaling analysis. It is found that all the surface states are localized while the transverse conductivity is quantized to be $\pm {{e^2} \over {2 h}}$ as long as the Fermi energy is within the bulk gap. This greatly facilitates the realization of the topological magnetoelectric effect proposed by Qi {\it et al.} (Phys. Rev. B\textbf{78}, 195424 (2008)) with the surface magnetization direction being controlled by the simultaneous application of magnetic and electric fields .

### References

Surface-Quantized Anomalous Hall Current and the Magnetoelectric Effect
in Magnetically Disordered Topological Insulators
*K. Nomura and N. Nagaosa*
Phys. Rev. Lett. 106, 166802 (2011) pdf

## Quantum Hall Effect of Massless Dirac Fermions in a Vanishing Magnetic Field

The effect of strong long-range disorder on the quantization of the Hall conductivity
$\sigma_{xy}$ in graphene is studied numerically.
It is shown that increasing Landau-level mixing progressively destroys all plateaus
in $\sigma_{xy}$ except the plateaus at $\sigma_{xy}=\mp e^2/2h$ (per valley and per spin).
The critical state at the charge-neutral Dirac point is robust to strong disorder
and belongs to the universality class of the conventional plateau transitions
in the integer quantum Hall effect. We propose that the breaking of time-reversal symmetry
by ripples in graphene can realize this quantum critical point in a vanishing magnetic field.

### References

Quantum Hall Effect of Massless Dirac Fermions in a Vanishing Magnetic Field
*K. Nomura, S. Ryu, M. Koshino, C. Mudry, A. Furusaki*
Phys. Rev. Lett. 100, 246806 (2008) pdf

## Topological delocalization of massless Dirac fermions

The beta function of a two-dimensional massless Dirac Hamiltonian subject to
a random scalar potential, which e.g., underlies the theoretical description of graphene,
is computed numerically. Although it belongs to, from a symmetry standpoint,
the two-dimensional symplectic class,
the beta function monotonically increases with decreasing $g$.
We also provide an argument based on the spectral flows under twisting boundary conditions,
which shows that none of states of the massless Dirac Hamiltonian can be localized.

### References

Topological delocalization of massless Dirac fermions
*K. Nomura, M. Koshino, S. Ryu*
Phys. Rev. Lett. 99, 146806 (2007) pdf

## Massless Dirac Fermion Transport in Graphene

Motivated by recent graphene transport experiments,
we have undertaken a numerical study of the conductivity of
disordered two-dimensional massless Dirac fermions.
Our results reveal distinct differences between the cases of
short-range and Coulomb randomly distributed scatterers.
We speculate that this behavior is related to the
Boltzmann transport theory prediction of dirty-limit behavior
for Coulomb scatterers.

### References

Quantum Transport of Massless Dirac Fermions
*K. Nomura and A. H. MacDonald*
Phys. Rev. Lett. 98, 076602 (2007) pdf

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