Connecting edge states in bilayer graphene nanoribbon with SSH ladder

We study edge states in AB-stacked bilayer graphene (BLG) ribbon in the simplest tight-binding model. Topological features of edge states when two layers ended with identical or distinct edge terminations (zigzag, bearded, armchair) are discussed. Existence of edge states near Fermi level are protected by chiral symmetry, whose topology is well captured by Su-Schrieffer-Heeger (SSH) ladder model. These zero-energy edge states can exist in the whole k_y region when two layers are zigzag and bearded, respectively. Winding number calculation shows a topological phase transition between two distinct non-trivial topological phases when crossing Dirac points. Additionally, we illustrate the importance of  stacking configurations of BLG ribbon since they can lead to unexpected edge states without protection from 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 origin of these states are explained by lattice struture of  SSH ladder near the decoupling point k_y=π.