  A bosonic field theory is derived for the tunable edge magnetism at graphene
zigzag edges. The derivation starts from an effective fermionic theory for the
interacting graphene edge states, derived previously from a two-dimensional
interacting tight-binding model for graphene. The essential feature of this
effective model, which gives rise to the weak edge magnetism, is the
momentum-dependent non-local electron-electron interaction. It is shown that
this momentum-dependence may be treated by an extension of the bosonization
technique, and leads to interactions of the bosonic fields. These interactions
are reminiscent of a \phi^4 field theory. Focussing onto the regime close to
the quantum phase transition between the ferromagnetic and the paramagnetic
Luttinger liquid, a semiclassical interpretation of the interacting bosonic
theory is given. Furthermore, it is argued that the universal critical behavior
at the quantum phase transition between the paramagnetic and the ferromagnetic
Luttinger liquid is governed by a small number of terms in this theory, which
are accessible by quantum Monte-Carlo methods.
