  Bilayer graphene in a magnetic field hosts a variety of ordered phases built
from eight Landau levels close in energy to the neutrality point. These levels
are characterized by orbital $n=0,1$, valley $\xi=+,-$ and spin
$\sigma=\uparrow,\downarrow$; their relative energies depend strongly on the
Coulomb interaction, magnetic field, and interlayer bias. We treat interactions
at the Hartree-Fock level, including the effects of metallic gates, layer
separation, spatial extent of the $p_z$ orbitals, all Slonczewski-Weiss-McClure
tight-binding parameters, and pressure. We obtain the ground state as function
of the applied magnetic field, bias, and pressure. The gates, layer separation
and extent of the $p_z$ orbitals weaken the Coulomb interaction at different
length scales; these effects distort the phase diagram but do not change its
topology. However, previously-predicted continuous transitions become
discontinuous when all tight-binding parameters are included nonperturbatively.
We find that pressure increases the importance of the noninteracting scale with
respect to the Coulomb energy, which drives phase transitions to occur at lower
fields. This brings two orbitally polarized states not yet predicted or
observed into the experimentally accessible region of the phase diagram, in
addition to previously-identified valley-, spin-, and partially orbitally
polarized states.
