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 one-dimensional periodically modulated 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 (Delta) over right arrow (x) integrates to zero within a unit cell. A large variety of locations for the Dirac points is shown to be possible: when (Delta) over right arrow parallel to (z) over cap they are shifted from their original locations along the direction perpendicular to the superlattice axis, while realizations of (Delta) over right arrow (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 (Delta) over right arrow parallel to (z) over cap 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.