  We take a two-step theoretical approach to study magnetism of rare earth
quasicrystals by considering Ising spins on quasiperiodic tilings, coupled via
RKKY interactions. First, we compute RKKY interactions from a tight-binding
Hamiltonian defined on the two-dimensional quasiperiodic tilings. We find that
the magnetic interactions are frustrated and strongly dependent on the local
environment. This results in the formation of clusters with strong bonds at
certain patterns of the tilings that repeat quasiperiodically. Second, we
examine the statistical mechanics of Ising spins with these RKKY interactions,
using extensive Monte Carlo simulations. Although models that have frustrated
interactions and lack translational invariance might be expected to display
spin glass behaviour, we show that the spin system has a phase transition to
low-temperature states with long-range quasiperiodic magnetic order.
Additionally, we find that in some of the systems spin clusters can fluctuate
much below the ordering temperature.
