Impact of localized $4f$ magnetism on Berry curvature and the intrinsic anomalous Hall effect in CeRu${}_2$Si${}_2$

The intrinsic anomalous Hall effect (AHE), originating from Berry curvature, has been extensively studied in $f$-electron systems, including Ce-based compounds. However, how the magnetism of localized $f$ electrons affect the intrinsic AHE remains unclear. Here, we investigate this issue in CeRu${}_2$Si${}_2$ by calculating the intrinsic anomalous Hall conductivity (AHC) for several occupied Ce-$4f$ states using first-principles LDA+$U$ calculations combined with Wannier interpolation method. We find that the intrinsic AHC remains finite even when the Ce-$4f$ states are localized and do not contribute directly to the Fermi surface, and that its magnitude depends on which Ce-$4f$ state is occupied. We show that localized $4f$-electron magnetism induces exchange splitting in the conduction bands, generating Berry curvature in specific regions near the Fermi surface. This mechanism enables conduction bands that are derived from nonmagnetic orbitals and form the Fermi surface to contribute to the intrinsic AHE, suggesting a route to realizing large intrinsic AHE in $f$-electron materials.