Abstract: Electron-electron interaction effects on the graphene honeycomb lattice, and its AB stacked bilayer, will be compared. While there are no low temperature weak coupling instabilities of interacting massless Dirac fermions in 2D, such instabilities are unavoidable for two parabolically touching bands. We use weak-coupling renormalization group as well as strong-coupling expansion to determine the dominant ordering tendency for spinless and spin 1/2 fermions on the bilayer for models with different microscopic interactions. We find that for spinless fermions on the honeycomb bilayer the broken symmetry state is typically a gapped insulator with either broken inversion or broken time-reversal symmetry, with a quantized anomalous Hall effect (i.e., either a layer polarized state or an anomalous quantum Hall state). Additionally, a tight-binding model with nearest-neighbor hopping and nearest-neighbor repulsion is studied in weak and strong couplings and in each regime a gapped phase with inversion symmetry breaking is found. In the strong-coupling limit, the ground-state wave function can be constructed for vanishing in-plane hopping but finite interplane hopping, which explicitly displays the broken inversion symmetry and a finite difference between the number of particles on the two layers. For spin-1/2 fermions the resulting instabilities are studied as a function of the range of the electron-electron repulsion. For longer range interactions (several tens of lattice spacings) the dominant ordering tendency is towards an electronic nematic, while for short range repulsion (of order a lattice spacing as in a repulsive Hubbard model) the leading instability is found towards a Neel antiferromagnet.
 Oskar Vafek and Kun Yang, PRB 81, 041401 (2010). (Physics 3, 1 (2010))
 Oskar Vafek, PRB 82, 205106 (2010)