Flat Bands and Bad Valley Quantum Numbers in Magic-Angle Twisted Bi-Layer Graphene

We study a nearest-neighbor electron hopping model for the electronic structure of bi-layer graphene at commensurate twist angles. The commensurate twisted bi-layers studied here coincide with the continuum approach that is based on the Dirac nodes in momentum space [1]. Such graphene bi-layers, in particular, show twist centers at Bernal stacks, as well as at the centers of honeycomb cells. The hamiltonian of the nearest-neighbor hopping model is constructed in momentum space. Degenerate perturbation theory at the M points in between the corners of the Moire first Brillouin zone finds the emergence of flat electronic bands at a magic twist angle. This effect is a result of level repulsion. Degenerate perturbation theory at both the M points and at the corners of the Moire first Brillouin zone further finds that the two graphene valleys in momentum space are approximately good quantum numbers. Direct numerical calculations of the energy spectrum of the nearest-neighbor hopping hamiltonian are consistent with the previous analytical results. The numerical calculations also find, however, that the two valley quantum numbers can mix in the flat electronic bands that emerge at the magic twist angle. The implications of such bad valley quantum numbers on the correlated electron states that may exist in magic-angle twisted bi-layer graphene will be discussed.

[1] R. Bistritzer and A.H. MacDonald, Proc. Natl. Acad. Sci. (USA) 108 (30), 12237 (2011).