  We study edge states in AB-stacked bilayer graphene (BLG) ribbon where the
Chern number of the corresponding two-dimensional (2D) 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 (non-dispersive bands near the Fermi level) are
states localized at the edge of graphene nanoribbon that only exists in certain
range of momentum $k_y$. Their existence near the Fermi level are protected by
the chiral symmetry with topology well described by coupled
Su-Schrieffer-Heeger (SSH) chains model, i.e., SSH ladder, based on the
bulk-edge correspondence of one-dimensional (1D) systems. These zero-energy
edge states can exist in the whole $k_y$ region when two layers have zigzag and
bearded edges, respectively. Winding number calculation shows a topological
phase transition between two distinct non-trivial topological phases when
crossing the Dirac points. Interestingly, we find the stacking configuration of
BLG ribbon is important since they 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 (NNN) interaction and interlayer bias are also discussed to fit the
realistic graphene materials, which suggest the robust topological features of
edge states in BLG systems.
