Edge state in AB-stacked bilayer graphene and its correspondence with the Su-Schrieffer-Heeger ladder

Abstract

We study edge states in AB-stacked bilayer graphene (BLG) ribbon where the Chern number of the corresponding two-dimensional bulk Hamiltonian is zero. The existence and topological features of edge states when two layers ended with the same or different edge terminations (zigzag, bearded, armchair) are discussed. The edge states (nondispersive bands near the Fermi level) are states localized at the edge of graphene nanoribbon that only exists in a certain range of momentum ky. Their existence near the Fermi level are protected by the chiral symmetry with topology well described by the coupled Su-Schrieffer-Heeger (SSH) chains model, i.e., the SSH ladder, based on the bulk-edge correspondence of one-dimensional systems. These zero-energy edge states can exist in the whole ky region when two layers have zigzag and bearded edges, respectively. A winding number calculation shows a topological phase transition between two distinct nontrivial topological phases when crossing the Dirac points. Interestingly, we find the stacking configuration of the BLG ribbon is important since it can lead to unexpected edge states without protection from the chiral symmetry both near the Fermi level in armchair-armchair case and in the gap within bulk bands that are away from Fermi level in the general case. The influence of interlayer next-nearest-neighbor interactions and interlayer bias are also discussed to fit the realistic graphene materials, which suggest the robust topological features of edge states in BLG systems.