Real space representation of topological system: twisted bilayer graphene as an example

We construct a Wannier basis for twisted bilayer graphene that is projected only from the Bloch functions of the twisted bilayer flat bands. The point symmetries including $C_3$ and $C_{2} mathcal{T}$ symmetries act locally on the Wannier functions while the Wannier function charge density is strongly peaked at the triangular sites and becomes fully sublattice-polarized in the chiral limit. The Wannier functions have a power-law tail, due to the topological obstruction, but most of the charge density is concentrated within one unit cell so that the on-site local Coulomb interaction is much larger than the further neighbor interactions and in general the Hamiltonian parameters may be accurately estimated from a modest number of Wannier functions. One exception is the momentum space components of the single-particle Hamiltonian, where because of the topological obstruction convergence is non-uniform across the Brillouin zone. We observe, however, that mixed position and momentum space representations may be used to avoid this difficulty in the context of quantum embedding methods. Our work provides a new route to study systems with topological obstruction and paves the way for the future investigation of correlated states in twisted bilayer graphene, including studies of non-integer fillings and temperature dependence.