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In mean-field-theory bilayer graphene's massive Dirac fermion model has afamily of broken inversion symmetry ground states with charge gaps and flavordependent spontaneous inter layer charge transfers. We use a latticeHartree-Fock model to explore some of the physics which controls whether or notthis type of broken symmetry state, which can be viewed as a pseudospinferromagnet, occurs in nature. We find that inversion symmetry is still brokenin the lattice model and estimate that transferred areal densities are $\sim10^{-5}$ electrons per carbon atom, that the associated energy gaps are $\sim10^{-2} {\rm eV}$, that the ordering condensation energies are $\sim 10^{-7}eV$ per carbon atom, and that the energy differences between competing ordersat the neutrality point are $\sim 10^{-9} eV$ per carbon atom. We explore thequantum phase transitions induced by external magnetic fields and by externallycontrolled electric potential differences between the layers. We find, inparticular, that in an external magnetic field coupling to spontaneous orbitalmoments favors broken time-reversal-symmetry states that have spontaneousquantized anomalous Hall effects. Our theory predicts a non monotonic behaviorof the band gap at neutrality as a function of interlayer potential differencein qualitative agreement with recent experiments.

Lattice Theory of Pseudospin Ferromagnetism in Bilayer Graphene: Competing Orders and Interaction Induced Quantum Hall States