Exact diagonalization study of the tunable edge magnetism in graphene

The tunable magnetism at graphene edges with lengths of up to 48 unit cells is analyzed by an exact diagonalization technique. For this we use a generalized interacting one-dimensional model which can be tuned continuously from a limit describing graphene zigzag edge states with a ferromagnetic phase, to a limit equivalent to a Hubbard chain, which does not allow ferromagnetism. This analysis sheds light onto the question why the edge states have a ferromagnetic ground state, while a usual one-dimensional metal does not. Essentially, we find that there are two important features of edge states: (a) Umklapp processes are completely forbidden for edge states, which allows a spin-polarized ground state; (b) the strong momentum dependence of the effective interaction vertex for edge states gives rise to a regime of partial spin-polarization and a second-order phase transition between a standard paramagnetic Luttinger liquid and ferromagnetic Luttinger liquid.