When the twist angle of a bilayer graphene is commensurate and near the `magic' value, there are four narrow bands near the neutrality point, each two-fold spin degenerate. These bands are separated from the rest of the bands by energy gaps.
In the first part of the talk, the method for microscopic construction of symmetry adapted maximally localized Wannier states for the four narrow bands of the twisted bilayer graphene will be presented. The subtleties associated with `obstruction' and the way around it will be discussed. On each layer and sublattice, every such Wannier state is found to have three peaks near the triangular moire lattice sites. However, each Wannier state is localized and centered around a site of the honeycomb lattice that is dual to the triangular moire lattice, with space group and the time reversal symmetries locally realized. The corresponding tight binding model is also constructed.
In the second part of the talk, I will identify states favored by electron-electron Coulomb interactions projected onto the Wannier basis. At the filling of two electrons/holes per moire unit cell (1/4-filling), such interactions favor an insulating SU(4) ferromagnet. The one-body kinetic terms select the ground state in which the two valleys with opposite spins are equally mixed, with vanishing magnetic moment per particle. We also find extended excited states, the gap to which decreases in parallel magnetic field. This makes such state a candidate for the experimentally observed insulator at 1/4-filling which is seen to be less insulating at large parallel magnetic field. An insulating stripe SU(4) ferromagnetic phase is favored at one electron/hole per moire unit cell. If spin ferromagnetic, such a state would be enhanced by the parallel magnetic field in agreement with the experiment at 1/8-filling.
 Jian Kang and Oskar Vafek, Phys. Rev. X 8, 031088 (2018).
 Jian Kang and Oskar Vafek, arXiv:1810.08642.