Nematic and quantum valley Hall states in twisted bilayer-monolayer graphene systems

The experimental discovery of correlated insulating states in twisted bilayer graphene drives the moiré two-dimensional materials system into a thriving area of condensed matter physics. Here, we use unrestricted Hartree-Fock variational method both in the original basis and in the projected-flat-band basis to study the correlated insulator and quantum anomalous Hall (QAH) states twisted monolayer-bilayer graphene. Our mean-field calculations reveal that the system can show the QAH state with |C|=2 Chern number at the 1/4 and 3/4 fillings under electric displacement field as an orbital Chern insulator, which is consistent with recent experimental results. When the twisted monolayer-bilayer graphene is set at the 1/2 filling under about -0.4 V/nm electric displacement field, quantum valley Hall effect will emerge in this moiré system. There is spin polarization but no valley polarization in this phase. The distributions of order parameters in the momentum space reveal that the system at this phase breaks the C_3z rotational symmetry and orbital time-reversal symmetry. The quantum valley Hall effect features the valley-contrasting current loops circulating around the ABB region of the moiré pattern which shows the opposite orbital magnetization originated from different valleys.