Spontaneous symmetry breaking and topology in twisted bilayer graphene: the nature of the correlated insulating states and the quantum anomalous Hall effect

We theoretically study the nature of the correlated insulating states and the quantum anomalous Hall (QAH) effect in twisted bilayer graphene (TBG) at the magic angle. Using a generic Hartree-Fock theory applied to all the energy bands, both the experimentally observed correlated insulating states at +/-1/2 filling and the quantum anomalous Hall effect at 3/4 filling of the flat bands near magic angle can be successfully explained. Our results indicate that the correlated insulating states at +/-1/2 filling are states with co-existing valley coherent order and valley polarized order. When a hexagonal boron nitride (hBN) substrate is aligned with the TBG, the valley polarized states are energetically favored over the valley coherent states at +/-1/2 filling, giving rise to QAH insulating states with Chern numbers (C) +/-2 by virtue of the C2z symmetry breaking induced by the substrate. We propose that such valley polairzed QAH states would be strongly suppressed by weak in-plane magnetic fields. We further predict that, for the same valley polarization, the anomalous Hall conductivities are opposite for +/- p (p=1/2 or 3/4) fillings, which implies opposite hysteresis behaviors for the QAH states at the electron and hole fillings.