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.

Figure 1

A finite patch of the perfect Penrose tiling.

Figure 2

Local environments in the Penrose tiling.

Figure 3

(Color online) (a) Integrated density of states $N\left(E\right)$ versus $E$ (expressed in units of $J$) calculated for three consecutive approximants (black dots, $N=644$; red dots, $N=1686$; gray, $N=4414$). (b) Low energy tail of $N\left(E\right)$ versus $E^2$ (the straight line is a fit to the data).

Figure 4

Density of states calculated for the Taylor 6 approximant ($N=11556$ sites).

Figure 5

[(a)–(e)] Weight fractions $f_n$ (see text) as a function of $E$ for $z=n$ $(n=3,\dots ,7)$, as computed for the Taylor 5 approximant ($N=4414$ sites).

Figure 6

The average value of ${\psi }_i{\left(E\right)}^2$ per site for each $z$ plotted against $E$ for states in the bottom of the spectrum.

Figure 7

Intensity plots representing the probability ${\psi }_i{\left(E\right)}^2$ for different energies $E$ (a darker shade corresponds to a higher probability): (a) $E=6.469$, (b) $E=3.0000$, (c) $E=0.8887$, and (d) $E=0.1692$. The solutions have been obtained for the Taylor 5 approximant ($N=4414$ sites).

Figure 8

Intensity plots in perpendicular space representing the probability ${\psi }_i{\left(E\right)}^2$ for different energies $E$ (a darker shade corresponds to a higher probability): (a) $E=6.469$, (b) $E=3.0000$, (c) $E=0.8887$, and (d) $E=0.1692$.

Figure 9

(Color online) IPR plotted on a logarithmic scale as a function of the energy for three successive Taylor approximants.

Figure 10

Some examples of $f\left(\alpha \right)$ plots for small energy eigenstates in three Taylor approximants.

Figure 11

The scaling of ground state energy of the Taylor approximants with system size.

Figure 12

(Color online) A portion of a Taylor approximant with vertices colored according to the value of the on site magnetization [magnetization values: red (small), green (intermediate), and blue (highest)].

Figure 13

Averages and standard deviations of the local staggered magnetization as a function of $z$ for $N=4414$ [LSW theory (dashed line) and QMC (continuous line)].

Figure 14

(Color online) Probability distribution of the local staggered magnetizations for the five different values of $z$ $(N=11556)$.

Figure 15

System size dependence of the average local staggered magnetization for each coordination number $z$ (lines indicate the fit to power law $N^{-1∕2}$).

Figure 16

(Color online) Figure 12 represented in perpendicular space. Two planes are shown (corresponding to $z_{\perp }=1,2$) [magnetization values: red (small), green (intermediate), and blue (highest)].

Figure 17

Two-tier Heisenberg star.

Figure 18

(Color online) $m_s(z,z^{\prime })$ plotted as a function of $z$ (see text) for selected values of $z^{\prime }$ (as given in Table ).

Figure 19

Intensity plot of the static longitudinal magnetic structure factor $S^{\parallel }\left(\mathbf{k}\right)$ for the $N=4414$ Taylor approximant. The highest intensity peaks are shown in the figure at positions that are indicated by circles (squares) for the magnetic (nonmagnetic) structures. The relative intensity is denoted by a linear gray scale ranging between zero (white) and maximum intensity (black).

Figure 20

(Color online) (Upper) Portion of the tiling showing vertices colored differently, according to coordination number $z$ (lower). The same tiling after projection into perpendicular space: (a) the plane $z_{\perp }=1$ and (b) the plane $z_{\perp }=2$.

Figure 21

Taylor ${\tau }^3$ approximant ($N=644$ sites).