Band Structure and Topological Properties of Graphene in a Superlattice   Spin Exchange Field
  We analyze the energy spectrum of graphene in the presence of spin-orbit coupling and a unidirectionally periodic Zeeman field, focusing on the stability and location of Dirac points it may support. It is found that the Dirac points at the $K$ and $K'$ points are generically moved to other locations in the Brillouin zone, but that they remain present when the Zeeman field $\vec{\Delta}(x)$ integrates to zero within a unit cell. A large variety of locations for the Dirac points is shown to be possible: when $\vec\Delta \parallel \hat{z}$ they are shifted from their original locations along the direction perpendicular to the superlattice axis, while realizations of $\vec\Delta(x)$ that rotate periodically move the Dirac points to locations that can reflect the orbit of the rotating electron spin as it moves through a unit cell. When a uniform Zeeman field is applied in addition to a periodic $\vec\Delta \parallel \hat{z}$ integrating to zero, the system can be brought into a metallic, Dirac semimetal, or insulating state, depending on the direction of the uniform field. The latter is shown to be an anomalous quantum Hall insulator. 
Luis Brey, A.R. Carvalho and H.A. Fertig
cond-mat.mes-hall cond-mat.mtrl-sci