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 ferromagnet with an approximate SU(4) spin-valley symmetry. 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.
 Jian Kang and Oskar Vafek, Phys. Rev. X 8, 031088 (2018).
 Jian Kang and Oskar Vafek, arXiv:1810.08642.