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

In mean-field-theory bilayer graphene's massive Dirac fermion model has a family of broken inversion symmetry ground states with charge gaps and flavor dependent spontaneous inter layer charge transfers. We use a lattice Hartree-Fock model to explore some of the physics which controls whether or not this type of broken symmetry state, which can be viewed as a pseudospin ferromagnet, occurs in nature. We find that inversion symmetry is still broken in the lattice model and estimate that transferred areal densities are $\sim 10^{-5}$ electrons per carbon atom, that the associated energy gaps are $\sim 10^{-2} {\rm eV}$, that the ordering condensation energies are $\sim 10^{-7} eV$ per carbon atom, and that the energy differences between competing orders at the neutrality point are $\sim 10^{-9} eV$ per carbon atom. We explore the quantum phase transitions induced by external magnetic fields and by externally controlled electric potential differences between the layers. We find, in particular, that in an external magnetic field coupling to spontaneous orbital moments favors broken time-reversal-symmetry states that have spontaneous quantized anomalous Hall effects. Our theory predicts a non monotonic behavior of the band gap at neutrality as a function of interlayer potential difference in qualitative agreement with recent experiments.