  We study the realization in a model of graphene of the phenomenon whereby the
tendency of gauge-field mediated interactions to break chiral symmetry
spontaneously is greatly enhanced in an external magnetic field. We prove that,
in the weak coupling limit, and where the electron-electron interaction
satisfies certain mild conditions, the ground state of charge neutral graphene
in an external magnetic field is a quantum Hall ferromagnet which spontaneously
breaks the emergent U(4) symmetry to U(2)XU(2).
  We argue that, due to a residual CP symmetry, the quantum Hall ferromagnet
order parameter is given exactly by the leading order in perturbation theory.
On the other hand, the chiral condensate which is the order parameter for
chiral symmetry breaking generically obtains contributions at all orders. We
compute the leading correction to the chiral condensate. We argue that the
ensuing fermion spectrum resembles that of massive fermions with a vanishing
U(4)-valued chemical potential. We discuss the realization of parity and charge
conjugation symmetries and argue that, in the context of our model, the charge
neutral quantum Hall state in graphene is a bulk insulator, with vanishing
longitudinal conductivity due to a charge gap and Hall conductivity vanishing
due to a residual discrete particle-hole symmetry.
