**
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
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