Lattice model for the Coulomb interacting chiral limit of magic-angle twisted bilayer graphene: Symmetries, obstructions, and excitations

We revisit the localized Wannier state description of the twisted bilayer graphene, focusing on the chiral limit. We provide a simple method for constructing such two-dimensional exponentially localized---yet valley polarized---Wannier states, centered on the sites of the honeycomb lattice, paying particular attention to maintaining all the unobstructed symmetries. This includes the unitary particle-hole symmetry, and the combination of ${C}_{2}\mathcal{T}$ and the chiral particle-hole symmetry. The ${C}_{2}\mathcal{T}$ symmetry alone remains topologically obstructed and is not represented in a simple site-to-site fashion. We also analyze the gap and the dispersion of single particle and single hole excitations above a strong coupling ground state at integer fillings, which we find to be dominated by the on-site and the nearest-neighbor terms of a triangular lattice hopping model, with a minimum at the center of the moir\'e Brillouin zone. Finally, we use the insight gained from this real-space description to understand the dependence of the gap and the effective mass on the range of the screened Coulomb interaction.