Edge states and quantum phase transition in graphene under in-plane effective exchange fields.

We investigated the edge states and quantum phase transition in graphene under an in-plane effective exchange field. The result shows that the combined effects of the in-plane effective exchange field and a staggered sublattice potential can induce zero-energy flat bands of edge states. Such flat-band edge states can evolve into helical-like ones in the presence of intrinsic spin-orbit coupling, with a unique spin texture. We also find that the bulk energy gap induced by the spin-orbit coupling and staggered sublattice potential can be closed and reopened with the in-plane effective exchange field, and the reopened bulk gap can be even larger than that induced by only the spin-orbit coupling and staggered sublattice potential, which is different from the case of an out-of-plane effective exchange field. The calculated spin-dependent Chern numbers suggest that the bulk gap closing and reopening is accompanied by a quantum phase transition from a trivial insulator phase across a metal phase into a spin-dependent quantum Hall phase.