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 ${\sigma }_{xy}=-(2e^2/h)(2n+1)$ with $n=0,1,\dots$, 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.

It is shown that in 2 + 1 dimensions, a constant magnetic field is a strong catalyst of dynamical flavor symmetry breaking, leading to generating a fermion dynamical mass even at the weakest attractive interaction between fermions. The effect is illustrated in the Nambu-Jona-Lasinio model in a magnetic field. The low-energy effective action in this model is derived, and the thermodynamic properties of the model are established.

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.

Recent experiments have proven that the quasiparticles in graphene obey a Dirac equation. Here we show that microwaves are an excellent probe of their unusual dynamics. When the chemical potential is small, the intraband response can exhibit a cusp around zero frequency $\Omega$ and this unusual line shape changes to Drude-like by increasing the chemical potential $|\mu |$, with width linear in $\mu$. The interband contribution at $T=0$ is a constant independent of $\Omega$ with a lower cutoff at $2\mu$. Distinctly different behavior occurs if interaction-induced phenomena in graphene cause an opening of a gap $\Delta$. At a large magnetic field $B$, the diagonal and Hall conductivities at small $\Omega$ become independent of $B$ but remain nonzero and show a structure associated with the lowest Landau level. This occurs because in the Dirac theory the energy of this level, $E_0=\pm \Delta$, is field independent in sharp contrast to the conventional case.