In $TmB_4$, localized electrons with a large magnetic moment interact with metallic electrons in boron-derived bands. We examine the nature of $TmB_4$ using full-relativistic ab-initio density functional theory calculations, approximate tight-binding Hamiltonian results, and the development of an effective Kondo-Ising model for this system. Features of the Fermi surface relating to the anisotropic conduction of charge are discussed. The observed magnetic moment $sim 6 , mu_B$ is argued to require a subtle crystal field effect in metallic systems, involving a flipped sign of the effective charges surrounding a Tm ion. The role of on-site quantum dynamics in the resulting Kondo-Ising type impurity model are highlighted. From this model, elimination of the conduction electrons will lead to spin-spin (RKKY-type) interaction of Ising character required to understand the observed fractional magnetization plateaus in $TmB_4$.
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