Charge distribution and spin textures in magic-angle twisted bilayer graphene

The spin density wave existing on the background of the inhomogeneous charge distribution is examined as a possible ground state of the magic-angle twisted bilayer graphene. When interactions are not included, the spectrum of the material has four (eight if spin is taken into account) almost flat almost degenerate bands. Interactions break down the degeneracy forming an order parameter which is usually assumed to be a spin density wave with a preset spin structure. Here, a possible charge density wave contribution to the order parameter; i.e., an inhomogeneous distribution of the charge density within a twisted graphene supercell is taken into account. The spin structure of the order parameter is calculated self-consistently. It is found that the density wave order is stable in the whole doping range from $-4$ to $+4$ extra electrons per supercell. The spin texture changes from collinear at zero doping to almost coplanar at finite doping. The density wave order shows nematic distortion when we dope the system. It is demonstrated that the local spin magnetization in energy units is much stronger than the charge density variation, unless doping exceeds three extra electrons or holes per supercell.