title: 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.
authors.affiliation: Department of Applied Physics, East China Jiaotong University, Nanchang 330013, People's Republic of China. Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Canada.
authors.forename: Zheng-Fang
authors.initials: ZF
authors.lastname: Liu
authors.forename: Qing-Ping
authors.initials: QP
authors.lastname: Wu
authors.forename: Ai-Xi
authors.initials: AX
authors.lastname: Chen
authors.forename: Xian-Bo
authors.initials: XB
authors.lastname: Xiao
authors.forename: Nian-Hua
authors.initials: NH
authors.lastname: Liu
authors.forename: Guo-Xing
authors.initials: GX
authors.lastname: Miao