Effect of Spin-Orbit Coupling on Kondo Phenomena in $f^7$-Electron Systems

In order to promote our basic understanding on the Kondo behavior recently observed in europium compounds, we analyze an impurity Anderson model with seven $f$ electrons at an impurity site by employing a numerical renormalization group method. The local part of the model consists of Coulomb interactions among $f$ electrons, spin-orbit coupling $\lambda$, and crystalline electric field (CEF) potentials, while we consider the hybridization $V$ between local $f$ electrons and single-band conduction electrons with $a_{\rm u}$ symmetry. For $\lambda=0$, we observe the underscreening Kondo behavior for appropriate values of $V$, characterized by the entropy change from $\ln 8$ to $\ln 7$, in which one of seven $f$ electrons is screened by conduction electrons. When $\lambda$ is increased, we obtain two types of behavior depending on the values of $V$. For large $V$, we find the entropy release of $\ln 7$ at low temperatures, determined by the level splitting energy due to the hybridization. For small $V$, we also observe the entropy change from $\ln 8$ to $\ln 2$ by the level splitting due to the hybridization, but at low temperatures, $\ln 2$ entropy is found to be released, leading to the Kondo effect. We emphasize that the Kondo behavior for small $V$ is observed for realistic values of $\lambda$ in the order of $0.1$ eV. We also discuss the effect of CEF potentials and the multipole properties in the Kondo behavior found in this paper.