$-m$ on the $B$ sublattice, then a mass term $m\sigma_z$ is induced in valley $K$ while a mass term $-m\sigma_z$ is induced in valley $K'$.[@foot2]

The anomalous Hall effect {#four}
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In the previous section I demonstrated that there is a gap in the spectrum and that a low energy description in terms massive Dirac fermions is possible. The gap has several important consequences. Firstly, low energy electrons can be localized in selected regions of a graphene sample by inducing a gap in regions where electrons are to be excluded. The gap can be controlled by electromagnetic fields that are smooth on the scale of the graphene lattice constant, and there is no need to break the sub-lattice inversion symmetry.[@Zhou07] Secondly, since one has control over the sign of the mass, it is possible to produce a sample in which the sign of the mass differs in different spatial regions. At the interface between two such regions, chiral edge states are expected to appear.[@Kane05] Thirdly, as is the case in Ref. , an electric field $\bm{E}=E\hat{y}$ applied to the bulk of the sample (the $y$-direction is chosen arbitrarily) will produce a quantized current $j_x=\sigma_{xy}E$ in the direction perpendicular to it. 