Anomalous Hall Response and Real-Space Current Correlations in Rhombohedral Graphene

We investigate the anomalous Hall effect (AHE) and magnetization observed in rhombohedral graphene through real-space calculations which lead to new insights. Our calculations reveal no equilibrium loop currents of the type associated with the Haldane model of a quantum anomalous Hall insulator. Instead, the AHE arises in this instance from interference between electronic paths forming oriented closed Wilson loops of current operators in the presence of valley polarization. We show some examples of these Wilson loops. Using the fact that the added-carrier charge density resides dominantly in the top and bottom layers, we develop a real-space expansion of the surface Hamiltonian that reproduces the Hall response of the full model with good accuracy. We also predict that a smooth local potential induces equilibrium circulating currents whose strength depends on both Berry curvature and the orbital moments of states at the Fermi surface.