Trigonal Warping and anomalous valley-Hall conductivity in twisted double bilayer graphene

We calculate the topological phase diagrams of both ABAB and ABBA stacked twisted double bilayer graphene as a function of both twist angles and perpendicular potential. The two transitions with Chern number changes of −3 and +1 can be explained by the splitting of the band crossing point into three satellite cones around a central Dirac cone in the moiré Brillouin zone due to the presence of trigonal warping. Considering the overlap of the bands in energy, which leads to metallic states, we construct the experimentally observable phase diagram of the system in terms of the indirect bandgap and the anomalous valley-Hall conductivity. We show that while most of the intermediate phase becomes metallic, there is a narrow parameter regime where the transition through three insulating phases with different quantized valley Hall conductivity can be seen.