Noncollinear magnetic phases and edge states in graphene quantum Hall bars

Application of a perpendicular magnetic field to charge neutral graphene is expected to result in a variety of broken symmetry phases, including antiferromagnetic, canted, and ferromagnetic. All these phases open a gap in bulk but have very different edge states and noncollinear spin order, recently confirmed experimentally. Here we provide an integrated description of both edge and bulk for the various magnetic phases of graphene Hall bars making use of a noncollinear mean field Hubbard model. Our calculations show that, at the edges, the three types of magnetic order are either enhanced (zigzag) or suppressed (armchair). Interestingly, we find that preformed local moments in zigzag edges interact with the quantum spin Hall like edge states of the ferromagnetic phase and can induce backscattering.

Figure 1

(a) Noninteracting spectrum of a quantum Hall armchair ribbon, and (b) scheme of the magnetism developed when interactions, off-plane, and in-plane fields are present. (c) Magnetism in the ribbon, (d) magnetic order parameters, and (e) band structures as the in-plane field increases, showing a pure edge insulator to metal transition.

Figure 2

(a) Scheme of the two different gaps (edge and bulk) observed in the band structure for an armchair ribbon, and (b) their evolution with an increasing in-plane field. (c) Bulk magnetic order parameters obtained in the calculation, and (d) scheme representing the phase transition in the order parameter space. (e) Dependence of the AF gap in absence of in-plane field in a quantum Hall armchair bar as a function of the off-plane field, and (f) the electron-electron interaction.

Figure 3

(a) Energy bands for the AF, CAF, and FM in a zigzag ribbon and (b) magnetic order parameters across the width. (c) Band structure of a zigzag ribbon marking the two different kinds of edge states. (d) Band structure and (e) scheme of a zigzag ribbon in the ferromagnetic regime with an excited magnetic moment on one edge.

Figure 4

Quarter filling phase. (a) and (b) Energy bands, (c) magnetic order parameters, $\mathcal{N}$ and $\mathcal{M}$, as a function of position, and (d) sublattice resolved magnetic moments.