  Bilayer graphene bears an eight-fold degeneracy due to spin, valley and layer
symmetry, allowing for a wealth of broken symmetry states induced by magnetic
or electric fields, by strain, or even spontaneously by interaction. We study
the electrical transport in clean current annealed suspended bilayer graphene.
We find two kind of devices. In bilayers of type B1 the eight-fold zero-energy
Landau level (LL) is partially lifted above a threshold field revealing an
insulating nu=0 quantum Hall state at the charge neutrality point (CNP). In
bilayers of type B2 the LL lifting is full and a gap appears in the
differential conductance even at zero magnetic field, suggesting an insulating
spontaneously broken symmetry state. Unlike B1, the minimum conductance in B2
is not exponentially suppressed, but remains finite with a value G < e^2/h even
in a large magnetic field. We suggest that this phase of B2 is insulating in
the bulk and bound by compressible edge states.
