Truncated atomic plane wave method for the subband structure calculations of Moir\'e systems

We propose a highly efficient and accurate numerical scheme named Truncated Atomic Plane Wave (TAPW) method to determine the subband structure of Twisted Bilayer Graphene (TBG) inspired by BM model. Our method utilizes real space information of carbon atoms in the moir\'e unit cell and projects the full tight binding Hamiltonian into a much smaller subspace using atomic plane waves. We present accurate electronic band structures of TBG in a wide range of twist angles together with detailed moir\'e potential and screened Coulomb interaction at the first magic angle using our new method. Furthermore, we generalize our formalism to solve the problem of low frequency moir\'e phonons in TBG.