In atomic physics, the Hund's rule states that the largest spin and orbital state is realized due to the interplay of spin-orbit coupling (SOC) and Coulomb interactions. Here, we show that in ferromagnetic solids the effective SOC and the orbital magnetic moment can be dramatically enhanced by a factor of $1/[1-(2U^{\prime }-U-J_H){\rho }_0]$, where $U$ and $U^{\prime }$ are the on-site Coulomb interaction within the same orbitals and between different orbitals, respectively, $J_H$ is the Hund's coupling, and ${\rho }_0$ is the average density of states. This factor is obtained by using the two-orbital as well as five-orbital Hubbard models with SOC. We also find that the spin polarization is more favorable than the orbital polarization, being consistent with experimental observations. The theory is also extended to study the spin fluctuations and long-range Coulomb interactions, and can be applied to understand the enhanced orbital magnetic moment and giant Faraday effect in ferromagnetic nanogranules in recent experiments. This present paper provides a fundamental basis for understanding the enhancements of SOC and orbital moment by Coulomb interactions in ferromagnets, which would have wide applications in spintronics.

• Received 20 May 2021
• Revised 19 January 2023
• Accepted 30 January 2023

DOI:https://doi.org/10.1103/PhysRevB.107.104407