  We study the magnetic properties of graphene edges and graphene/graphane
interfaces under the influence of electrostatic gates. For this, an effective
low-energy theory for the edge states, which is derived from the Hubbard model
of the honeycomb lattice, is used. We first study the edge state model in a
mean-field approximation for the Hubbard Hamiltonian and show that it
reproduces the results of the extended 2D lattice theory. Quantum fluctuations
around the mean-field theory of the effective one-dimensional model are treated
by means of the bosonization technique in order to check the stability of the
mean-field solution. We find that edge magnetism at graphene/graphane
interfaces can be switched on and off by means of electrostatic gates. We
describe a quantum phase transition between an ordinary and a ferromagnetic
Luttinger liquid - a realization of itinerant one-dimensional ferromagnetism.
This mechanism may provide means to experimentally discriminate between edge
magnetism or disorder as the reason for a transport gap in very clean graphene
nanoribbons.
