  The Penrose tiling is a perfectly ordered two dimensional structure with
fivefold symmetry and scale invariance under site decimation. Quantum spin
models on such a system can be expected to differ significantly from more
conventional structures as a result of its special symmetries. In one
dimension, for example, aperiodicity can result in distinctive quantum
entanglement properties. In this work, we study ground state properties of the
spin-1/2 Heisenberg antiferromagnet on the Penrose tiling, a model that could
also be pertinent for certain three dimensional antiferromagnetic
quasicrystals. We show, using spin wave theory and quantum Monte Carlo
simulation, that the local staggered magnetizations strongly depend on the
local coordination number z and are minimized on some sites of five-fold
symmetry. We present a simple explanation for this behavior in terms of
Heisenberg stars. Finally we show how best to represent this complex
inhomogeneous ground state, using the "perpendicular space" representation of
the tiling.
