Band structure and topological properties of twisted double bilayer graphene

We study the electronic band structure and the topological properties of the twisted double bilayer graphene, or a pair of AB-stacked bilayer graphenes rotationally stacked on top of each other. We consider two different arrangements, AB-AB and AB-BA, which differ in the relative orientation. For each system, we calculate the energy band and the valley Chern number using the continuum Hamiltonian. We show that the AB-AB and the AB-BA have similar band structures, while the Chern numbers associated with the corresponding bands are completely different. In the absence of the perpendicular electric field, in particular, the AB-AB system is a trivial insulator when the Fermi energy is in a gap, while the AB-BA is a valley Hall insulator. Also, the lowest electron and hole bands of the AB-AB are entangled by the symmetry protected band touching points, while they are separated in the AB-BA. In both cases, the perpendicular electric field immediately opens an energy gap at the charge neutral point, where the electron branch becomes much narrower than the hole branch, due to the significant electron-hole asymmetry.