Graphene is a zero gap semiconductor with a very large Fermi velocity and low energy bands made of p-orbitals. Therefore, the influence of correlation effects and spin-orbit coupling on the electronic properties is not expected to be large. In this talk I show how this is not the case in graphene nanostructures and I discuss two exotic electronic phases that arise in graphene zigzag nanoribbons. One of them has persistent charge currents driven by magnetic order and spin-orbit coupling [1]. The other is a half-metal with vanishing total magnetic moment, whose metallic character arises only when Coulomb repulsion between the electrons are taken into account, in dramatic contrast with the usual interaction-driven metal-to-insulator transition. Possible experimental realization of the predicted phases is discussed.
[1] D. Soriano, J. Fernández-Rossier, Phys. Rev. B 82, 161302 (2010)
