# Postponed

## Main Content

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. In general, knowledge of the magnetic properties of a single atom will not tell much about the magnetic properties of a material, and requires understanding the cooperative effects of many degrees of freedom, particularly the spin. Starting from the chemistry, "cooking" a single crystal with enough purity and without defects is already an enormous challenge. In addition, being able to "design" a material with the desired geometry and interactions is only possible in a few cases, usually using organic molecules as a fundamental building block. 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 another degree of complexity, and competition with other quantum phenomena: the Kondo effect. This competition is quite subtle and non-trivial, and can only be studied by numerical means. We have found that there is a critical distance at which the Kondo effect dominates, translating into a finite range for the RKKY interaction. We study this mechanism on different lattice geometries in 2 and 3 dimensions by introducing an exact mapping onto an effective one-dimensional problem that we can solve with the density matrix renormalization group method (DMRG). We show a clear and departure from the conventional RKKY theory, and important differences that can be attributed to the dimensionality and geometry. In particular, for dimension d>1, Kondo physics dominates even at short distances, while the ferromagnetic RKKY state is energetically unfavorable, which can have important implications in our understanding of heavy fermion materials and magnetic semiconductors.