The physics of nonlocal exchange interactions in graphene sheets is treated within a π-orbital tight-binding model using a Hartree-Fock approximation and Coulomb interactions modified at short distances by lattice effects and at large distances by dielectric screening. The strong nonlocality of exchange effects in systems with isolated band crossings at energies close to the Fermi level leads to renormalization of Fermi velocity and eventually to broken symmetry states for strong enough interactions. We show the role played by lattice scale details of the effective Coulomb interaction at neutrality point in determining the character of broken symmetry states at zero field and in the quantum Hall regime.

For zero field we analyze the renormalization of the Fermi velocity of graphene, in particular the logarithmic divergence of the band dispersion, in relationship with the slow decay of the off diagonal density matrix in real space, originated by the Fermi point structure of the bands in graphene. In our full Brillouin zone analysis we are able to obtain the next order correction to the velocity enhancement that cannot be obtained in the continuum model due to the arbitrariness in the choice of the low energy cutoff in momentum space. We further discuss the relevance of non-local exchange in driving instabilities and obtain a phase diagram for broken symmetry solutions as a function of the onsite potential strength and the Coulomb tail screened by the dielectric medium. We show how the broken symmetry phases are sensitive to the particular way the number of nearest neighbors for the interactions are included.

For the quantum Hall phase at finite magnetic field we give an explanation for the nature of the insulating broken symmetry nu=0 quantum Hall states. The strong Landau level mixing due to the lattice scale details of the Coulomb interaction as well as the special valley-sublattice equivalence of the anomalous Landau levels in graphene leads to the preference of density wave solutions over spin polarized ferromagnetic solutions. These insulating states have a gap in the bulk and do not have current carrying edge states. The AF spin density wave solutions are energetically favored over the charge density wave counterpart when the onsite repulsion is sufficiently strong in comparison with the long range Coulomb tail. The transition from AF to the spin polarized ferromagnetic phase is expected to happen in a continuous fashion when the Zeeman coupling energy overcomes the exchange energy preference of the density wave states. This crossover depends on the specific details of the onsite repulsion U and the Coulomb tail.

References:
`Enhancement of non-local exchange near isolated band-crossings in graphene'  Jeil Jung and A. H. MacDonald, Phys. Rev. B 84, 085446 (2011). http://link.aps.org/doi/10.1103/PhysRevB.84.085446
`Theory of the Magnetic-Field-Induced Insulator in Neutral Graphene' Jeil Jung and A. H. MacDonald, Phys. Rev. B 80, 235417 (2009). http://link.aps.org/doi/10.1103/PhysRevB.80.235417