$\textit{Ab initio}$ four-band Wannier tight-binding model for generic twisted graphene systems

The newly realized twisted graphene systems such as twisted bilayer graphene (TBG), twisted double bilayer graphene (TDBG), and twisted trilayer graphene (TTG) have attracted widespread theoretical attention. Therefore, a simple and accurate model of the systems is of vital importance for the further study. Here, we construct the symmetry-adapted localized Wannier functions and the corresponding $\textit{ab initio}$ minimal two-valley four-band effective tight-binding models for generic twisted graphene systems with small twist angle. Such two-valley model evades the Wannier obstruction caused by the fragile topology in one-valley model. The real space valley operator is introduced to explicitly describe the valley $U_{v}\left(1\right)$ symmetry. Each symmetry-adapted Wannier orbital shows a peculiar three-peak form with its maximum at AA spots and its center at AB or BA spots. An extended Hubbard model is also given and the related parameters are presented explicitly. We provide an approach to systematically build the Wannier tight-binding model for generic twisted graphene systems. Our model provides a firm basis for further study of the many-body effects in these systems.