Final Report Summary - SPINSINQUASICRYSTALS (Magnetic Moments in Geometrically Frustrated Systems with Quasiperiodic Order and Disorder) Quasicrystals with their unusual atomic structure have rather exotic physical properties, and so far we lack a good theoretical understanding for most of them. During the project we worked on models to get a better insight into the magnetic properties of quasicrystals. An important question in this context is how their rather exotic electronic properties (e.[*]g[/*]. pseudogap at the Fermi energy and multifractal electronic states) influences the magnetism in this material class. We focused on studying theoretical models for rare-earth quasicrystals. This is a common type of magnetic quasicrystals which has well-defined local moments at concentrations of 5-10% interacting via long-range magnetic interactions (so-called RKKY interactions) mediated by the conduction electrons. To model these systems we applied a two-step theoretical approach:First, we designed an improved numerical method to compute the form of the RKKY interactions using a tight-binding Hamiltonian defined on quasiperiodic tilings. We found that the coupling between pairs of magnetic moments depends not only on their distance but also varies strongly with the position on the tiling. Although we find ferromagnetic and antiferromagnetic bonds as in periodic systems, the magnetic interactions do not show a well-defined spatial period with a Fermi wave vector as they do in crystalline systems. In a second step, we studied the magnetic properties of quasiperiodic systems with these RKKY interactions using extensive Monte Carlo simulations. For all systems we found the emergence of strongly-coupled spin clusters with weak inter-cluster coupling on certain patterns of the tiling and the formation of long-range magnetic order at low temperature. This order persists across a range of models, including different tilings, choices of magnetic site and value of the Fermi energy. Moreover, we find a finite domain wall energy per unit length in the ordered states. For this reason, quasi-periodic magnetic order can be expected to be robust against perturbations that lead only to small changes in RKKY interactions. We also applied finite size scaling to show that the critical behaviour is consistent with the two-dimensional Ising universality class.The formation of strongly coupled clusters and their fluctuations even at very low temperatures appears to be consistent with experimental observations. In contrast, the nature of the ordering transition in the model shows clear differences to experiments, which typically show a spin-glass-like freezing of the magnetic moments at low temperatures. However, disorder of the atomic sites is very common in quasicrystals. Hence, a special focus was given to the study of disorder phenomena in these systems, which have been hardly addressed in the research yet. To model this site disorder we add a random potential to the conduction electrons. Repeating the calculations for the RKKY interactions and the Monte Carlo simulations, we found that long-range magnetic order in quasiperiodic systems is destroyed at a finite disorder strength. For these systems we also do not find an increase in the domain wall energy with unit length which indicates a similarity to the experimentally observed spin-glass behaviour.