The amorphous iron-germanium system (a-FexGe1-x) lacks long-range structural order and hence lacks a meaningful Brillouin zone. The magnetization of a-FexGe1-x is well explained by the Stoner model for Fe concentrations x above the onset of magnetic order around x=0.4, indicating that the local order of the amorphous structure preserves the spin-split density of states of the Fe-3d states sufficiently to polarize the electronic structure despite k being a bad quantum number. Measurements reveal an enhanced anomalous Hall resistivity ρxyAH relative to crystalline FeGe; this ρxyAH is compared to density-functional theory calculations of the anomalous Hall conductivity to resolve its underlying mechanisms. The intrinsic mechanism, typically understood as the Berry curvature integrated over occupied k states but shown here to be equivalent to the density of curvature integrated over occupied energies in aperiodic materials, dominates the anomalous Hall conductivity of a-FexGe1-x (0.38≤x≤0.61). The density of curvature is the sum of spin-orbit correlations of local orbital states and can hence be calculated with no reference to k space. This result and the accompanying Stoner-like model for the intrinsic anomalous Hall conductivity establish a unified understanding of the underlying physics of the anomalous Hall effect in both crystalline and disordered systems.

Emergent phenomena at polar-nonpolar oxide interfaces have been studied intensely in pursuit of next-generation oxide electronics and spintronics. Here we report the disentanglement of critical thicknesses for electron reconstruction and the emergence of ferromagnetism in polar-mismatched LaMnO_{3}/SrTiO_{3} (001) heterostructures. Using a combination of element-specific x-ray absorption spectroscopy and dichroism, and first-principles calculations, interfacial electron accumulation, and ferromagnetism have been observed within the polar, antiferromagnetic insulator LaMnO_{3}. Our results show that the critical thickness for the onset of electron accumulation is as thin as 2 unit cells (UC), significantly thinner than the observed critical thickness for ferromagnetism of 5 UC. The absence of ferromagnetism below 5 UC is likely induced by electron overaccumulation. In turn, by controlling the doping of the LaMnO_{3}, we are able to neutralize the excessive electrons from the polar mismatch in ultrathin LaMnO_{3} films and thus enable ferromagnetism in films as thin as 3 UC, extending the limits of our ability to synthesize and tailor emergent phenomena at interfaces and demonstrating manipulation of the electronic and magnetic structures of materials at the shortest length scales.