Exotic 4f Correlated Electronic States of Ferromagnetic Kondo Lattice Compounds ReRh₆Ge₄ (Re=Ce, Ho, Er, Tm)

CeRh₆Ge₄ stands out as the first stoichiometric metallic compound with a ferromagnetic quantum critical point, thereby garnering significant attention. Ferromagnetic Kondo lattice compounds ReRh₆Ge₄ (Re=Ce, Ho, Er, Tm) have been systematically investigated with density functional theory incorporating Coulomb interaction U and spin-orbital coupling. We determined the magnetic easy axis of CeRh₆Ge₄ is within the ab plane, which is in agreement with previous magnetization measurements conducted under external magnetic field and muSR experiments. We also predicted the magnetic easy axes for the other three compounds. For TmRh₆Ge₄, the magnetic easy axis aligns along the c axis, thus preserving the C₃ rotational symmetry of the c axis. Especially, there are triply degenerate nodal points along the Γ-A direction in the band structure including spin-orbital coupling. A possible localized to itinerant crossover is revealed as $4f$ electrons increase from CeRh₆Ge₄ to TmRh₆Ge₄. Specifically, the $4f$ electrons of TmRh₆Ge₄ contribute to the formation of a large Fermi surface, indicating their participation in the conduction process. Conversely, the $4f$ electrons in HoRh₆Ge₄, ErRh₆Ge₄ and CeRh₆Ge₄ remain localized, which result in smaller Fermi surfaces for these compounds. These theoretical investigations on electronic structure and magnetic properties shed deep insight into the unique nature of $4f$ electrons, providing critical predictions for subsequent experimental studies.