Orbital longitudinal magnetoelectric coupling in rhombohedral multilayer graphene

Magnetoelectric coupling enables the manipulation of magnetization by electric fields and polarization by magnetic fields. While typically found in heavy element materials with large spin-orbit coupling, recent experiments on rhombohedral-stacked pentalayer graphene have demonstrated a longitudinal magnetoelectric coupling (LMC) without spin-orbit coupling. Here, we develop a microscopic theory of LMC in layered quantum materials and identify how it is controlled by a "layer-space" quantum geometry. Focusing on rhombohedral multilayer graphene systems, we find that the interplay between LMC and valley-polarized order produces a butterfly shaped magnetic hysteresis controlled by out-of-plane electric field: a signature of LMC and a multiferroic valley order. Furthermore, we identify a nonlinear LMC in rhombohedral multilayer graphene under time-reversal symmetry, while the absence of centrosymmetry enables the generation of a second-order nonlinear electric dipole moment in response to an out-of-plane magnetic field. Our theoretical framework provides a quantitative understanding of LMC, as well as the emergent magnetoelectric properties of rhombohedral multilayer graphene.