▎ 摘　　要

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.