Enhanced spin-orbit coupling and orbital moment in ferromagnets by electron correlations

In atomic physics, the Hund's rule states that largest spin and orbital state is realized due to interplay of spin-orbit coupling (SOC) Coulomb interactions. Here, we show in ferromagnetic solids effective SOC magnetic moment can be dramatically enhanced by a factor $1/[1\ensuremath{-}(2{U}^{\ensuremath{'}}\ensuremath{-}U\ensuremath{-}{J}_{H}){\ensuremath{\rho}}_{0}]$, where $U$ ${U}^{\ensuremath{'}}$ are on-site interaction within same orbitals between different orbitals, respectively, ${J}_{H}$ coupling, ${\ensuremath{\rho}}_{0}$ average density states. This obtained using two-orbital as well five-orbital Hubbard models with SOC. We also find polarization more favorable than polarization, being consistent experimental observations. The theory extended study fluctuations long-range interactions, applied understand giant Faraday effect nanogranules recent experiments. present paper provides fundamental basis for understanding enhancements interactions ferromagnets, which would have wide applications spintronics.