Quantum fluctuations and excitations in antiferromagnetic quasicrystals

We study the effects of quantum fluctuations and the excitation spectrum for the antiferromagnetic Heisenberg model on a two-dimensional quasicrystal, by numerically solving linear spin-wave theory on finite approximants of the octagonal tiling. Previous quantum Monte Carlo results for the distribution of local staggered magnetic moments and the static spin structure factor are reproduced well within this approximate scheme. Furthermore, the magnetic excitation spectrum consists of magnonlike low-energy modes, as well as dispersionless high-energy states of multifractal nature. The dynamical spin structure factor, accessible to inelastic neutron scattering, exhibits linear-soft modes at low energies, self-similar structures with bifurcations emerging at intermediate energies, and flat bands in high-energy regions. We find that the distribution of local staggered moments stemming from the inhomogeneity of the quasiperiodic structure leads to a characteristic energy spread in the local dynamical spin susceptibility, implying distinct nuclear magnetic resonance spectra, specific for different local environments.

Figure 1

(Color online) Finite approximant of the octagonal tiling with 239 sites (thin lines), along with a superimposed inflated 41 sites approximant (thick lines) of rescaled edge length by a factor of $\lambda =1+\sqrt{2}$. For the 41 sites approximant the vertices of one of the two sublattices are dressed with a disk to exhibit the bipartite nature of the octagonal tiling.

Figure 2

(Color online) Finite-size scaling of the ground-state staggered magnetization $m_s$ and the lowest excitation gap $\Delta$ for the $S=1∕2$ Heisenberg model on the octagonal tiling in linear spin-wave theory. The inset shows the finite-size scaling of the ground-state energy $E_0$ for the same system.

Figure 3

(Color online) Dependence of the local staggered magnetization on the coordination number $z$ for all sites in the 1393 sites approximant of the octagonal tiling within linear spin-wave theory (LSWT) of the $S=1∕2$ Heisenberg model. For reference, results of quantum Monte Carlo (QMC) simulations of the same system are also shown (Ref. 10). The inset exhibits the hierarchy of the local staggered magnetization of the $z=8$ sites, grouped according to the value of $z^{\prime }$ under a deflation transformation. Numbers next to symbols give the value of $z^{″}$ for $z^{\prime }=8$ sites under further deflation.

Figure 4

Intensity plot of the static longitudinal magnetic structure factor $S^{\Vert }\left(\mathbf{k}\right)$ (left), and the integrated dynamical spin structure factor $S^{\perp }\left(\mathbf{k}\right)$ (right) for the $S=1∕2$ Heisenberg antiferromagnet on the 1393 sites approximant of the octagonal tiling. For both panels, the relative intensity is denoted by a linear gray scale ranging between zero (white) and maximum intensity (black).

Figure 5

(Color online) Frequency dependence of quantum fluctuations to the staggered magnetization $\delta m_s\left(\omega \right)$ for the $S=1∕2$ Heisenberg model on the octagonal tiling, averaged separately over sites with coordination numbers $z=3$ and 8. The inset shows $\delta m_s\left(\omega \right)$ for $z=5$, averaged separately over sites with a small and large staggered magnetization, respectively.

Figure 6

(Color online) Excitation spectra of the Heisenberg antiferromagnet on finite approximants of the octagonal tiling of different sizes within linear spin-wave theory. The inset shows the density of state $\rho \left(\omega \right)$ for the largest approximant.

Figure 7

(Color online) Cumulative density of states $N\left(\omega \right)$ of the Heisenberg antiferromagnet on finite approximants of the octagonal tiling of different sizes within linear spin-wave theory. The inset exhibits the quadratic scaling of $N\left(\omega \right)$ at low energies for the largest approximant, indicative of a linear low-energy density of states.

Figure 8

(Color online) Frequency-dependent inverse participation ratio $I\left(\omega \right)$ for the $S=1∕2$ Heisenberg antiferromagnet on finite approximants of the octagonal tiling of different sizes, obtained within linear spin-wave theory. Dashed lines indicate a finite-size scaling as $N_s^{-1}$, expected for extended states. The inset shows linear spin-wave results for $I\left(\omega \right)$ on square lattices of different sizes.

Figure 9

Intensity plot of the dynamical spin structure factor $S(\mathbf{k},\omega )$ for fixed $\omega ∕JS=0$ (left) and $\omega ∕JS=1.8$ (right) for the $S=1∕2$ Heisenberg antiferromagnet on the 1393 sites approximant of the octagonal tiling. For both panels, the relative intensity is denoted by a linear gray scale ranging between zero (white) and maximum intensity (black).

Figure 10

Intensity plot of the dynamical spin structure factor $S(\mathbf{k},\omega )$ for fixed $\omega ∕JS=2.9$ (left) and $\omega ∕JS=3.0$ (right) for the $S=1∕2$ Heisenberg antiferromagnet on the 1393 sites approximant of the octagonal tiling. For both panels, the relative intensity is denoted by a linear gray scale ranging between zero (white) and maximum intensity (black).

Figure 11

Intensity plot of the dynamical spin structure factor $S(\mathbf{k},\omega )$ for fixed $\omega ∕JS=4.6$ (left) and $\omega ∕JS=6.4$ (right) for the $S=1∕2$ Heisenberg antiferromagnet on the 1393 sites approximant of the octagonal tiling. For both panels, the relative intensity is denoted by a linear gray scale ranging between zero (white) and maximum intensity (black).

Figure 12

Frequency dependence of the momentum-integrated dynamical spin structure factor $S^{\perp }\left(\omega \right)$ for the $S=1∕2$ Heisenberg antiferromagnet on the octagonal tiling.

Figure 13

Intensity plot of the dynamical spin structure factor $S(\mathbf{k},\omega )$ along the (5,2) momentum space direction (left) and along the line $(2.9\pi ,k_y)$ (right) for the $S=1∕2$ Heisenberg antiferromagnet on the 1393 sites approximant of the octagonal tiling. For both panels, the relative intensity is denoted by a linear gray scale ranging between zero (white) and maximum intensity (black).

Figure 14

Intensity plot of the dynamical spin structure factor $S(\mathbf{k},\omega )$ along the (1,0) momentum space direction (left) and the (3,2) direction (right) for the $S=1∕2$ Heisenberg antiferromagnet on the 1393 sites approximant of the octagonal tiling. For both panels, the relative intensity is denoted by a linear gray scale ranging between zero (white) and maximum intensity (black).

Figure 15

(Color online) Imaginary part of the local dynamical spin susceptibility ${\chi }_{\text{local}}^{″}(i,\omega )$ for the $S=1∕2$ Heisenberg antiferromagnet on octagonal tiling, averaged separately over sites with coordination numbers $z=3\text{–}8$. The left inset shows ${\chi }_{\text{local}}^{″}(i,\omega )$ for the fivefold sites, averaged separately over sites with a small and large staggered magnetization, respectively. The right inset shows ${\chi }_{\text{local}}^{″}(i,\omega )$ for the eightfold sites, averaged separately over sites with different behavior under deflation transformation, grouped according to the value of $z^{\prime }$.