The proximity-induced couplings in graphene due to the vicinity of a ferromagnetic insulator are analyzed. We combine general symmetry principles and simple tight-binding descriptions to consider different orientations of the magnetization. We find that, in addition to a simple exchange field, a number of other terms arise. Some of these terms act as magnetic orbital couplings, and others are proximity-induced spin-orbit interactions. The couplings are of similar order of magnitude, and depend on the orientation of the magnetization. A variety of phases, and anomalous Hall effect regimes, are possible.Introduction. Electrons in graphene placed in proximity to a ferromagnet experience an induced exchange field that modifies spin-transport properties. The exchange field results from virtual electrons hopping between the graphene layer and the ferromagnet. For instance, on an yttrium-iron garnet (YIG) substrate, this leads to magnetoresistance, spin-currentto-charge-current conversion [1], and spin precession from induced ferromagnetism [1][2][3]. Furthermore, in addition to induced ferromagnetism, we can also expect induced spin-orbit coupling (SOC) in graphene because atoms in the YIG substrate have non-negligible SOC [1]. The YIG ferromagnetism breaks time-reversal symmetry and thus gives rise to orbital couplings in the graphene layer which are not invariant under time inversion. We can expect similar effects on graphene in proximity to an EuO ferromagnet [4][5][6][7]. Heavy atoms near a graphene layer can, in general, induce non-trivial spin-orbit couplings [8], and these effects have been experimentally confirmed [9][10][11][12]. With such experimental platforms now available, we can attempt to engineer proximity-induced interactions in graphene for spintronics applications [13][14][15].In the present work, we analyze the general interactions induced in a graphene layer by proximity to a ferromagnetic insulator with a significant SOC. We give a classification of the possible terms allowed by lattice symmetries, and present simple models which allow us to determine the relative strengths of the various terms. The resulting electronic structure of graphene depends sensitively on the balance between these couplings, and can exhibit a number of interesting topological features. We discuss the main features of the different possible phases.The model. For simplicity, we consider the interaction between a graphene sheet and the top layer of a magnetic substrate. While this approach does not allow us to make quantitative numerical predictions for the strength of the induced couplings in graphene, this is sufficient for a general analysis of the type of perturbations and their relative strengths. It is worth noting that the couplings between graphene and a substrate depend exponentially on the distance between the two systems, so that a quantitative theoretical study is, in any case, extremely challenging. We further assume that the magnetic atoms form a lattice commensurate with the graphene layer, and neglect the ef...
