Quantum Antiferromagnetism in Quasicrystals

The antiferromagnetic Heisenberg model is studied on a two-dimensional bipartite quasiperiodic lattice. Using the stochastic series expansion quantum Monte Carlo method, the distribution of local staggered magnetic moments is determined on finite square approximants with up to 1393 sites, and a nontrivial inhomogeneous ground state is found. A hierarchical structure in the values of the moments is observed which arises from the self-similarity of the quasiperiodic lattice. The computed spin structure factor shows antiferromagnetic modulations that can be measured in neutron scattering and nuclear magnetic resonance experiments. This generic model is a first step towards understanding magnetic quasicrystals such as the recently discovered Zn-Mg-Ho icosahedral structure.

Figure 1

Approximants of the octagonal tiling. (a) Deflation from 239 (thin lines) to 41 (thick lines) sites. (b) These six “Heisenberg stars” are the local building blocks.

Figure 2

Dependence of the staggered magnetization on the coordination number $z$. Inset: finite-size extrapolation of the average staggered magnetization of the octagonal tiling.

Figure 3

Dependence of the local approximation of the staggered magnetization on the coordination number $z$. The points on the dashed line are exact values obtained for Heisenberg stars. Inset: spatial variation of the local approximation to the local staggered magnetization where the radii of the circles correspond to the size of local approximation.

Figure 4

Hierarchy of the local staggered magnetization of the $z=8$ sites in the 1393 site tiling, grouped according to the value of $z^{\prime }$ under the deflation transformation. Numbers on top of the symbols give the value of $z^{\prime \prime }$ for the $z^{\prime }=8$ sites under a further deflation transformation.

Figure 5

(a),(b) Simple Bragg scattering, and (c),(d) antiferromagnetic superstructure in the octagonal tiling, extracted from quantum Monte Carlo data on the 1393 site approximant.