  Bilayer graphene under a magnetic field has an octet of quasidegenerate
levels due to spin, valley, and orbital degeneracies. This zero-energy Landau
level is resolved into several incompressible states whose nature is still
elusive. We use a Hartree-Fock treatment of a realistic tight-binding four-band
model to understand the quantum ferromagnetism phenomena expected for integer
fillings of the octet levels. We include the exchange interaction with filled
Landau levels below the octet states. This Lamb-shift-like effect contributes
to the orbital splitting of the octet. We give phase diagrams as a function of
applied bias and magnetic field. Some of our findings are in agreement with
experiments. We discuss the possible appearance of phases with orbital
coherence.
