Intrinsic Magnetoelectric Hall Effect from Layer-Orbital Quantum Geometry

Intrinsic Hall effects, such as the anomalous Hall effect, originate from the orbital quantum geometry of Bloch states. However, in layered materials, the combined action of out-of-plane electric and magnetic fields couples to layer polarization and orbital moment, generating a mixed layer-orbital quantum geometry in field-dressed Bloch states. We show that this geometry produces an intrinsic magnetoelectric Hall effect that is bilinear in the electric and magnetic fields. The response is scattering-time independent and can arise in nonmagnetic systems without spin-orbit coupling. Its origin lies in interband coherence involving layer polarization and orbital moment, leading to a finite, non-quantized Hall response that persists in the band gap. The Hall coefficient is odd under gate reversal and tracks layer polarization. A symmetry analysis identifies the classes of layered materials that host this effect. As a representative realization, we demonstrate the effect in rhombohedral pentalayer graphene, where the conductivity reaches values of order $0.05\,e^2/h$. These results establish mixed layer-orbital quantum geometry as a mechanism for intrinsic magnetoelectric Hall transport and a direct probe of layer-resolved quantum geometry in Bloch bands.