RKKY Interactions in Graphene Landau Levels

We study RKKY interactions for magnetic impurities on graphene in situations where the electronic spectrum is in the form of Landau levels. Two such situations are considered: non-uniformly strained graphene, and graphene in a real magnetic field. RKKY interactions are enhanced by the lowest Landau level, which is shown to form electron states binding with the spin impurities and add a strong non-perturbative contribution to pairwise impurity spin interactions when their separation $R$ no more than the magnetic length. Beyond this interactions are found to fall off as $1/R^3$ due to perturbative effects of the negative energy Landau levels. Based on these results, we develop simple mean-field theories for both systems, taking into account the fact that typically the density of states in the lowest Landau level is much smaller than the density of spin impurities. For the strain field case, we find that the system is formally ferrimagnetic, but with very small net moment due to the relatively low density of impurities binding electrons. The transition temperature is nevertheless enhanced by them. For real fields, the system forms a canted antiferromagnet if the field is not so strong as to pin the impurity spins along the field. The possibility that the system in this latter case supports a Kosterlitz-Thouless transition is discussed

Quantum Geometric Exciton Drift Velocity

We show that the dipole moment of an exciton is uniquely determined by the quantum geometry of its eigenstates, and demonstrate its intimate connection with a quantity we call the dipole curvature. The dipole curvature arises naturally in semiclassical dynamics of an exciton in an electric field, adding a term additional to the anomalous velocity coming from the Berry's curvature. In a uniform electric field this contributes a drift velocity akin to that expected for excitons in crossed electric and magnetic fields, even in the absence of a real magnetic field. We compute the quantities relevant to semiclassical exciton dynamics for several interesting examples of bilayer systems with weak interlayer tunneling and Fermi energy in a gap, where the exciton may be sensibly described as a two-body problem. These quantities include the exciton dispersion, its dipole curvature, and Berry's curvature. For two gapped-graphene layers in a vanishing magnetic field, we find the dipole curvature vanishes if the layers are identical, but may be non-zero when the layers differ. We further analyze examples in the presence of magnetic fields, allowing us to examine cases involving graphene, in which a gap is opened by Landau level splitting. Heterostructures involving TMDs are also considered. In each case the dipole and/or Berry's curvatures play out differently. In some cases, the lowest energy exciton state is found to reside at finite momentum, with interesting possibilities for Bose condensation. We also find situations in which the dipole curvature increases monotonically with exciton momentum, suggesting that the quantum geometry can be exploited to produce photocurrents from initially bound excitons with electric fields. We speculate on further possible effects of the semiclassical dynamics in geometries where the constituent layers are subject to the same or different electric fields.Comment: 27 pages, 14 figure