Monolayer graphite films, or graphene, have quasiparticle excitations that can be described by (2+1)-dimensional Dirac theory. We demonstrate that this produces an unconventional form of the quantized Hall conductivity sigma(xy) = -(2e2/h)(2n+1) with n = 0, 1, ..., which notably distinguishes graphene from other materials where the integer quantum Hall effect was observed. This unconventional quantization is caused by the quantum anomaly of the n=0 Landau level and was discovered in recent experiments on ultrathin graphite films.

Landau level quantization in graphene reflects the Dirac nature of its quasiparticles and has been found to exhibit an unusual integer quantum Hall effect. In particular, the lowest Landau level can be thought of as shared equally by electrons and holes, and this leads to characteristic behaviour of the magneto-optical conductivity as a function of frequency for various values of the chemical potential μ. Particular attention is paid to the optical spectral weight under various absorption peaks and its redistribution as μ is varied. We also provide results for magnetic field B as well as chemical potential sweeps at selected fixed frequencies, which can be particularly useful for possible measurements in graphene. Both diagonal and Hall conductivities are considered.

We consider the relationship between the tight-binding Hamiltonian of the twodimensional honeycomb lattice of carbon atoms with nearest neighbor hopping only and the 2 + 1 dimensional Hamiltonian of quantum electrodynamics which follows in the continuum limit. We pay particular attention to the symmetries of the free Dirac fermions including spatial inversion, time reversal, charge conjugation and chirality. We illustrate the power of such a mapping by considering the effect of the possible symmetry breaking which corresponds to the creation of a finite Dirac mass, on various optical properties. In particular, we consider the diagonal AC conductivity with emphasis on how the finite Dirac mass might manifest itself in experiment. The optical sum rules for the diagonal and Hall conductivities are discussed.

The analytical expressions for both diagonal and off-diagonal ac and dc conductivities of graphene placed in an external magnetic field are derived. These conductivities exhibit rather unusual behavior as functions of frequency, chemical potential, and applied field which is caused by the fact that the quasiparticle excitations in graphene are Dirac-like. One of the most striking effects observed in graphene is the odd integer quantum Hall effect. We argue that it is caused by the anomalous properties of the Dirac quasiparticles from the lowest Landau level. Other quantities such as Hall angle and Nernst signal also exhibit rather unusual behavior, in particular when there is an excitonic gap in the spectrum of the Dirac quasiparticle excitations.