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.

Figure 1

Portion of the Penrose tiling.

Figure 2

(Color online) Local staggered magnetization plotted vs coordination number $z$ as obtained by QMC (red) and by LSW theory (blue).

Figure 3

(Color online) Two out of the four perpendicular space projected domains of the Penrose tiling, with a color coding of the sites according to the value of the local staggered magnetization determined by linear spin-wave theory.

Figure 4

A two-level Heisenberg star showing the central spin and its $z$ nearest neighbors and $z{z}^{\prime }$ next-nearest neighbors. In the example shown, $z=6$ and $z^{\prime }=4$.

Figure 5

(Color online) Staggered magnetization as predicted by Eq. (2) as a function of $z$ for different $z^{\prime }$ values. The points indicate the value of $z^{\prime }$ computed (see text) for sites of the Penrose tiling.