▎ 摘　　要

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 leading instabilities of the system as its temperature is lowered as a function of the range of the interaction. We discover that the system is unstable towards an antiferromagnetic phase for short ranges of the interaction and towards a nematic phase at long ranges, in agreement with previous studies. 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) similar to B-2. At higher fields, when omega(c) less than or similar to 2 Delta(0), we find that Delta(B) approximate to omega(c)/[ln(omega(c)/Delta(0)) + C], where C approximate to 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 nonuniversal, interaction-dependent, constant.