Understanding Magnetism is a complex undertaking: it relies on our knowledge of the exact position of magnetic ions in a crystal and their interactions. More important, at its core, this is fundamentally a quantum problem and requires understanding the cooperative effects of many degrees of freedom. In the past decade, we have witnessed enormous progress in experiments that consist of placing magnetic atoms at predetermined positions on substrates, and building magnetic nanostructures, one atom at a time. The electrons in the substrate mediate the interactions between the spins, and scanning tunneling microscopy allows one to study their properties.

In order to understand these interactions, we rely on a theory developed decades ago by Ruderman, Kittel, Kasuya, and Yosida, dubbed "RKKY Theory", which applies when the spins are classical. The quantum nature of the electronic spin introduces more complexity, and competition with another quantum phenomena: the Kondo effect. The combined effect is non-trivial, and can only be studied by numerical means. I will describe this effect by introducing an exact mapping onto an effective one-dimensional problem that we can solve with the density matrix renormalization group method (DMRG). I will also show that for dimension d>1, Kondo physics dominates even at short distances, while the ferromagnetic RKKY state is energetically unfavorable. This may have important implications for our understanding of heavy fermion materials and magnetic semiconductors.