  AA-stacked bilayer graphene supports Fermi circles in its bonding and
antibonding bands which coincide exactly, leading to symmetry-breaking in the
presence of electron-electron interactions. We analyze a continuum model of
this system in the Hartree-Fock approximation, using a self-consistently
screened interaction that accounts for the gap in the spectrum in the broken
symmetry state. The order parameter in the groundstate is shown to be of the
Ising type, involving transfer of charge between the layers in opposite
directions for different sublattices. We analyze the Ising phase transition for
the system, and argue that it continuously evolves into a Kosterlitz-Thouless
transition in the limit of vanishing interlayer separation $d$. The transition
temperature is shown to depend only on the effective spin stiffness of the
system even for $d>0$, and an estimate its value suggests the transition
temperature is of order a few degrees Kelvin.
