Conventional group analysis of twisted bilayer graphene within the tight-binding framework

The symmetry of twisted bilayer graphene (tBG) has been extensively studied in the continuum model but somewhat overlooked in the tight-binding (TB) framework. In contrast to the continuum model, which requires operators for both sublattice and layer spaces, the TB framework relies solely on operators in real space. This paper begins by discussing the symmetries of the TB Hamiltonian and the single-valley TB Hamiltonian of tBG. Subsequently, the band structures with specific irreducible representations (irreps) are obtained by diagonalizing the irrep-dependent Hamiltonian constructed through projection operators. We also investigate the impact of symmetry on intervalley coupling and the influence of a non-zero mass term on the C2zT symmetry of the single-valley Hamiltonian. To understand the irrep change as the wavevector moves inside the Brillouin zone and the irrep change after considering the intervalley coupling or introducing a non-zero mass term at a fixed wavevector, we derive two kinds of compatibility relationships. The methodology presented here can be applied to other layered materials because the basis functions we employ are generic and unique to twisted bilayers.