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We will present a simple non-relativistic model to describe the low energy excitations of graphene. Our model is based on a deformation of the Heisen-berg algebra in such a way that the commutator of momenta is proportional to the pseudo-spin. We solve the Landau problem for the resulting Hamil-tonian, which reduces in the large mass limit, while keeping constant the Fermi velocity, to the usual linear one employed to describe these excitations as massless Dirac fermions. Extending this model to negative mass we re-produce the leading mass term in the low energy expansion of the dispersion relation for both nearest and next-to-nearest-neighbor interactions. Taking into account the contribution from both Dirac points, we evaluate the Hall conductivity with a zeta-function approach. The result is consistent with the anomalous integer quantum Hall e ect in graphene. The idea is to present also a short introduction to non-commutative quantum mechanics.
