Moiré commensurability and the quantum anomalous Hall effect in twisted bilayer graphene on hexagonal boron nitride

The quantum anomalous Hall (QAH) effect is sometimes observed in twisted bilayer graphene (tBG) when it nearly aligned with an encapsulating hexagonal boron nitride (hBN) layer. We propose that the appearance or absence of QAH individual devices could be related to commensurability between graphene/graphene and graphene/hBN moir\'e patterns. identify a series points $(\theta_{\rm GG},\theta_{\rm GBN})$ twist-angle space at which two patterns are commensurate, allowing band theory applied, show Chern numbers this case sensitive rigid in-plane hBN displacement. Given property, we argue likely only i) twist-angle-pair close enough commensurate point yield supermoir\'e pattern sufficiently long length scale, ii) has percolating topologically non-trivial phase. For twist angles far from commensurability, layer acts as source disorder can destroy effect. Our proposal explain number current experimental observations. Further studies test more directly suggested.