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

Abstract

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.