Simple derivation of moire-scale continuous models for twisted bilayer graphene

We provide a formal derivation of a reduced model for twisted bilayer graphene (TBG) from Density Functional Theory. Our derivation is based on a variational approximation of the TBG Kohn-Sham Hamiltonian and asymptotic limit techniques. In contrast with other approaches, it does not require the introduction of an intermediate tight-binding model. The so-obtained model is similar to that of the Bistritzer-MacDonald (BM) model but contains additional terms. Its parameters can be easily computed from Kohn-Sham calculations on single-layer graphene and untwisted bilayer graphene with different stackings. It allows one in particular to estimate the parameters wAA and wAB of the BM model from first principles. The resulting numerical values, namely wAA = wAB <^> 126 meV for the experimental interlayer mean distance are in good agreement with the empirical values wAA = wAB = 110 meV obtained by fitting to experimental data. We also show that if the BM parameters are set to wAA = wAB <^> 126 meV, the BM model is an accurate approximation of our reduced model.