Pseudo Landau level representation of twisted bilayer graphene: Band topology and implications on the correlated insulating phase

We propose that the electronic structure of twisted bilayer graphene (TBG) can be understood as Dirac fermions coupled with opposite pseudomagnetic fields generated by the moire pattern. The two low-energy flat bands from each monolayer valley originate from the two zeroth pseudo Landau levels of Dirac fermions under such opposite effective magnetic fields, which have opposite sublattice polarizations and carry opposite Chern numbers +/- 1, giving rise to helical edge states in the gaps below and above the low-energy bulk bands near the first magic angle. We argue that small Coulomb interactions would split the eightfold degeneracy (including valley and physical spin) of these zeroth pseudo Landau levels, and may lead to insulating phases with nonvanishing Chern numbers at integer fillings. Besides, we show that all the high-energy bands below or above the flat bands are also topologically nontrivial in the sense that for each valley the sum of their Berry phases is quantized as +/-pi. Such quantized Berry phases give rise to nearly flat edge states, which are dependent on truncations on the moire length scale. Our paper provides a complete and clear picture for the electronic structure and topological properties of TBG, and has significant implications on the nature of the correlated insulating phase observed in experiments.