Flat Topological Bands with Bad Valley Quantum Numbers in Twisted Bi-Layer Graphene

We study a nearest-neighbor electron hopping model for twisted bi-layer graphene (TBLG) in order to reveal the topological nature of the four electron bands that exist about charge neutrality[1]. TBLG with comensurate twist angles are assumed that show simple moire patterns, with no subcells. The K valley in one sheet of graphene and the K' valley in the other sheet of graphene are degenerate at the corners of the moire Brillouin zone[1], K_M and K'_M. Degenerate perturbation theory finds eigenstates near K_M that wind once on the A sublattice of the K valley, superposed with B sites of the K' valley that wind once about K_M in the same sense. A direct calculation of the Wilson loop finds that such components of the flat electron bands in TBLG carry positive unit topological winding number. We shall introduce perpendicular magnetic field into the continuum model that results from such degenerate perturbation theory in order to see how time-reversal symmetry breaking affects the mixing of valley quantum numbers near K_M. The previous calculation was performed in the chiral limit, with no inter-sheet hopping between A sites and A sites, nor between B sites and B sites. We will move off the chiral limit by turning on such inter-sheet electron hopping, and compute the Wilson loop numerically. We expect to confirm the unit Chern numbers of the four topological bands about charge neutrality that were found in the chiral limit.

[1] J.P. Rodriguez, "Flat Electron Bands with Bad Valley Quantum Numbers in Twisted Bi-Layer Graphene", arXiv:2410.18594 .