  We map out the possible ordered states in bilayer graphene at the neutrality
point by extending the previous renormalization group treatment of many-body
instabilities to finite temperature, trigonal warping and externally applied
perpendicular electric field. We were able to analytically determine all
outcomes of the RG flow equations for the 9 four-fermion coupling constants.
While the full phase diagram exhibits a rich structure, we confirm that when
forward scattering dominates, the only ordering tendency with divergent
susceptibility at finite temperature is the nematic. At finite temperature this
result is stable with respect to small back and layer imbalance scattering;
further increasing their strength leads to the layer antiferromagnet. We also
determine conditions for other ordered states to appear and compare our results
to the special cases of attractive and repulsive Hubbard models where exact
results are available.
