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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 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.