  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; this 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.
