RKKY Interaction in Graphene Landau Levels

We consider the problem of RKKY coupling of classical magnetic moments on graphene when the conduction electrons are in Landau level states. This can be due to a real applied field, or to non-uniform strain. The problem cannot be fully handled in the usual perturbative approach due to the large degeneracy of the Landau levels. For zero chemical potential and two impurities, we demonstrate that four bound states break away from zero energy, with energies dependent of the relative orientations of the spins, two of which are below the Fermi energy. In strained graphene, these two filled states create an effective spin-spin interaction (∝J_sd) in addition to the standard RKKY (∝J²_sd). Moreover, this contribution is active on only one of the graphene sublattices. At the mean-field level, this sublattice has stronger ferromagnetic coupling than the other, while the two sublattices are anti-ferromagnetically coupled. This can potentially give rise to a ferrimagnetic phase with a broken O(3) symmetry. By comparison, the RKKY interaction of dilute impurities in unstrained graphene in a real magnetic field (perpendicular to sample) allows for a canted antiferromagnetic state with broken U(1) symmetry, and an accompanying Kosterlitz-Thouless transition.