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We take a two-step theoretical approach to study magnetism of rare earthquasicrystals by considering Ising spins on quasiperiodic tilings, coupled viaRKKY interactions. First, we compute RKKY interactions from a tight-bindingHamiltonian defined on the two-dimensional quasiperiodic tilings. We find thatthe magnetic interactions are frustrated and strongly dependent on the localenvironment. This results in the formation of clusters with strong bonds atcertain patterns of the tilings that repeat quasiperiodically. Second, weexamine the statistical mechanics of Ising spins with these RKKY interactions,using extensive Monte Carlo simulations. Although models that have frustratedinteractions and lack translational invariance might be expected to displayspin glass behaviour, we show that the spin system has a phase transition tolow-temperature states with long-range quasiperiodic magnetic order.Additionally, we find that in some of the systems spin clusters can fluctuatemuch below the ordering temperature.

Long-range Magnetic Order in Models for Rare Earth Quasicrystals