Spin waves and local magnetizations on the Penrose tiling

We consider a Heisenberg antiferromagnet on the Penrose tiling, a quasiperiodic system having an inhomogeneous Néel-ordered ground state. Spin wave energies and wave functions are studied in the linear spin wave approximation. A linear dispersion law is found at low energies, as in other bipartite antiferromagnets, with an effective spin wave velocity lower than that in the square lattice. Spatial properties of eigenmodes are characterized in several different ways. At low energies, eigenstates are relatively extended and show multifractal scaling. At higher energies, states are more localized and, depending on the energy, confined to sites of a specified coordination number. The ground state energy of this antiferromagnet and local staggered magnetizations are calculated. Perpendicular space projections are presented in order to show the underlying simplicity of this “complex” ground state. A simple analytical model, the two-tier Heisenberg star, is presented to explain the staggered magnetization distribution in this antiferromagnetic system.