Trigonal warping, satellite Dirac points, and multiple field tuned topological transitions in twisted double bilayer graphene

We show that the valley Chern number of low energy band in twisted double bilayer graphene can be tuned through two successive topological transitions, where direct bandgap closes, by changing electric field perpendicular to plane layers. The transitions with changes -3 and +1 explained formation three satellite Dirac points around central cone moir\'e Brillouin zone due presence trigonal warping. cones have opposite chirality cone. Considering overlap bands energy, which lead metallic states, we construct experimentally observable phase diagram system terms indirect anomalous Hall conductivity. while most intermediate becomes metallic, there is a narrow parameter regime transition insulating phases different quantized conductivity seen. systematically study effects variations model parameters on reveal importance particle-hole asymmetry warping constructing diagram. also effect interlayer tunneling this