Phase separation in a spin density wave state of twisted bilayer graphene

Twisted bilayer graphene at the so-called magic twist angle $\theta\sim1^\circ$ is theoretically studied. In the absence of interaction between electrons, the system under study is characterized by four almost degenerate flat bands near the Fermi level. The electron-electron interaction lifts this degeneracy and stabilizes a certain order parameter in the system. We assume that the arising order parameter corresponds to a spin density wave. The evolution of such spin density wave state upon doping is analyzed. It is shown that, in the doping range where this order parameter exists, the homogeneous state of the system can be unstable to phase separation. Namely, the doping dependence of the chemical potential is nonmonotonic, which is consistent with recent experiments. Phases in the inhomogeneous state are characterized by an even number ($\nu=0,\,\pm2,\,\pm4$) of electrons per supercell. This allows explaining some features in the behavior of the conductivity of the doped system.