Helical Network Model of the Twisted Graphene Bilayer

The electronic structure of a graphene bilayer is altered by applying a transverse electric field, which induces an energy gap, and also by applying a small twist in the orientation of one layer relative to its neighbor, which creates a periodic moiré pattern. When both perturbations are applied, the result is periodic moiré pattern containing large insulating regions with local AB and BA stacking. These regions can be characterized by opposite valley Chern numbers and are separated by domain walls with helical electronic states. To describe the low-energy part of the pattern band structure we introduce a new network model that can be considered as a relative of the Chalker-Coddington model. The band structure of the network is gapless and periodic and contains bands that touch at Dirac points. The network density of states vanishes at these Dirac points and also has a set of strong maxima corresponding to van Hove singularities associated with saddle points in the pattern band structure. The predictions of our model can in principle be tested via a scanning tunneling microscopy experiments.