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The experimentally observed correlated insulating states and quantumanomalous Hall (QAH) effect in twisted bilayer graphene (TBG) have drawnsignificant attention. However, up to date, the specific mechanisms of theseintriguing phenomena are still open questions. Using a fully unrestrictedHartree-Fock variational method, we have explained the correlated insulatingstates and QAH effects at various integer fillings of the flat bands in TBG.Our results indicate that states breaking flavor (valley and spin) symmetriesare energetically favored at all integer fillings. In particular, thecorrelated insulating states at $\pm 1/2$ filling and at the charge neutralitypoint are all valley polarized sates which break $C_{2z}$ and time-reversal($\mathcal{T}$) symmetries, but preserves $C_{2z}\mathcal{T}$ symmetry. Suchvalley polarized states exhibit "moir\'e orbital antiferromagnetic ordering" onan emergent honeycomb lattice with compensating circulating current pattern inthe moir\'e supercell. Within the same theoretical framework, our calculationsindicate that the $C\!=\!\mp 1$ QAH states at $\pm 3/4$ fillings of themagic-angle TBG are spin and orbital ferromagnetic states, which emerge when astaggered sublattice potential is present. We find that the nonlocalness of theexchange interactions tend to enhance the bandwidth of the low-energy bands dueto the exchange-hole effect, which reduces the gaps of the correlated insulatorphases. The nonlocal exchange interactions also dramatically enhance the spinpolarization of the system, which significantly stabilize the orbital and spinferromagnetic QAH state at $3/4$ filling of TBG aligned with hexagonal boronnitride (hBN). We also predict that, by virtue of the orbital ferromagneticnature, the QAH effects at electron and hole fillings of hBN-aligned TBG wouldexhibit hysteresis loops with opposite chiralities.

Theories for the correlated insulating states and quantum anomalous Hall phenomena in twisted bilayer graphene