  Tight-binding calculations predict that the AA-stacked graphene bilayer has
one electron and one hole conducting bands, and that the Fermi surfaces of
these bands coincide. We demonstrate that as a result of this degeneracy, the
bilayer becomes unstable with respect to a set of spontaneous symmetry
violations. Which of the symmetries is broken depends on the microscopic
details of the system. We find that antiferromagnetism is the more stable order
parameter. This order is stabilized by the strong on-site Coulomb repulsion.
For an on-site repulsion energy typical for graphene systems, the
antiferromagnetic gap can exist up to room temperatures.
