Symmetry, maximally localized Wannier states, and low energy model for the twisted bilayer graphene narrow bands

We build symmetry adapted maximally localized Wannier states, and construct the low energy tight binding model for the four narrow bands of the twisted bilayer graphene. We do so when the twist angle is commensurate, near the `magic' value, and the narrow bands are separated from the rest of the bands by energy gaps. On each layer and sublattice, every Wannier state has 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. Space group and the time reversal symmetries are realized locally. The corresponding tight binding model provides a starting point for studying the correlated many-body phases.