Electron-electron interactions and the phase diagram of a graphene bilayer

We study the effects of long and short-range electron-electron interactions in a graphene bilayer. Using a variational wave function technique we show that in the presence of long-range Coulomb interactions the clean bilayer is always unstable to electron and hole pocket formation with a finite ferromagnetic polarization. Furthermore, we argue that short-range electron-electron interactions lead to a staggered orientation of the ordered ferromagnetic moment in each layer (that is, $c$-axis antiferromagnetism). We also comment on the effects of doping and trigonal distortions of the electronic bands.