Penrose quantum antiferromagnet

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 fivefold 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.