  We extend previous analyses of fermions on a honeycomb bilayer lattice via
weak-coupling renormalization group (RG) methods with extremely short-range and
extremely long-range interactions to the case of finite-range interactions. In
particular, we consider different types of interactions including screened
Coulomb interactions, much like those produced by a point charge placed either
above a single infinite conducting plate or exactly halfway between two
parallel infinite conducting plates. Our considerations are motivated by the
fact that, in some recent experiments on bilayer graphene there is a single
gate while in others there are two gates, which can function as the conducting
planes and which, we argue, can lead to distinct broken symmetry phases. We map
out the phases that the system enters as a function of the range of the
interaction. We discover that the system enters an antiferromagnetic phase for
short ranges of the interaction and a nematic phase at long ranges, in
agreement with previous work. While the antiferromagnetic phase results in a
gap in the spectrum, the nematic phase is gapless, splitting the quadratic
degeneracy points into two Dirac cones each. We also consider the effects of an
applied magnetic field on the system in the antiferromagnetic phase via
variational mean field theory. At low fields, we find that the
antiferromagnetic order parameter, Delta(B)-Delta(0) \sim B^2. At higher
fields, when omega_c > 2*Delta_0, we find that
Delta(B)=omega_c/[ln(omega_c/Delta(0))+C], where C=0.67 and omega_c=eB/m^*c. We
also determine the energy gap for creating electron-hole excitations in the
system, and, at high fields, we find it to be a*omega_c+2*Delta(B), where a is
a non-universal, interaction-dependent, constant.
