  We analyze competition between different ordered states in bilayer graphene
(BLG). Combining arguments based on SU(4) spin-valley flavor symmetry with a
mean field analysis, we identify the lowest energy state with the anomalous
Hall insulator (AHI). This state is an SU(4) singlet excitonic insulator with
broken time reversal symmetry, exhibiting quantized Hall effect in the absence
of external magnetic field. Applied electric field drives an Ising-type phase
transition, restoring time reversal symmetry. Applied magnetic field drives a
transition from the AHI state to a quantum Hall ferromagnet state. We estimate
energies of these states, taking full account of screening, and predict the
phase diagram.
