Penrose Quantum Antiferromagnet

TL;DR

This paper investigates the ground state properties of the spin-1/2 Heisenberg antiferromagnet on the Penrose tiling, revealing how local magnetizations vary with coordination number and symmetry, using spin wave theory and quantum Monte Carlo methods.

Contribution

It provides the first detailed analysis of quantum antiferromagnetism on the Penrose tiling, highlighting the inhomogeneous ground state and the influence of local geometry.

Findings

Local staggered magnetizations depend on coordination number z.

Magnetizations are minimized at sites with five-fold symmetry.

The inhomogeneous ground state can be represented using perpendicular space.

Abstract

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.