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.