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

We examine the coexisting spin and charge density waves as a possible ground state of the magic-angle twisted bilayer graphene. When interactions are not included, the spectrum of the material has 4 (8 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 we take into account a possible charge density wave contribution to the order parameter, that is, inhomogeneous distribution of the charge density within a twisted graphene supercell. We also calculate self-consistently the spin structure of the order parameter. We find 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. We demonstrate that the local spin magnetization is much stronger than the charge density variation, unless the doping exceeds $3$ extra electrons or holes per supercell.