What Sets the Magic Angle in Twisted Bi-Layer Graphene?

We study the energy spectrum of bi-layer graphene at commensurate twist angles within the tight-binding description. We start with the smallest possible number of Moire reciprocal lattice vectors in the Brillouin zone of a single graphene sheet given by the prime numbers p = 7, 19, and 37. It is pointed out that the characteristic polynomial for the energy spectrum is governed by the cyclic group Z_p, which is a simple group. The chiral limit is taken, where inter-sheet hopping between A sites only and between B sites only of the two honeycomb lattices is turned off. The dispersion in momentum of the remaining hopping matrix elements between the A sites in one honeycomb lattice and the B sites in the other honeycomb lattice is neglected, and it is replaced by an effective constant matrix element, w₁. This approximation is valid in the low-energy limit, at small Moire Brillouin zones. We find numerically that flat low-energy bands appear at a magic effective matrix element w₁*. The magic effective matrix element w₁* for A/B inter-sheet hopping is thereby determined per commensurate twist angle θ_p. Comparison of these results with experimentally determined magic angles in twisted bi-layer graphene will be made.