Hyperuniformity and anti-hyperuniformity in one-dimensional substitution tilings

This work considers the scaling properties characterizing the hyperuniformity (or anti-hyperuniformity) of long-wavelength fluctuations in a broad class of one-dimensional substitution tilings. A simple argument is presented which predicts the exponent α governing the scaling of Fourier intensities at small wavenumbers, tilings with α > 0 being hyperuniform, and numerical computations confirm that the predictions are accurate for quasiperiodic tilings, tilings with singular continuous spectra and limit-periodic tilings. Quasiperiodic or singular continuous cases can be constructed with α arbitrarily close to any given value between −1 and 3. Limit-periodic tilings can be constructed with α between −1 and 1 or with Fourier intensities that approach zero faster than any power law.

Quasicrystal Tilings in Three Dimensions and Their Empires

The projection method for constructing quasiperiodic tilings from a higher dimensional lattice provides a useful context for computing a quasicrystal’s vertex configurations, frequencies, and empires (forced tiles). We review the projection method within the framework of the dual relationship between the Delaunay and Voronoi cell complexes of the lattice being projected. We describe a new method for calculating empires (forced tiles) which also borrows from the dualisation formalism and which generalizes to tilings generated projections of non-cubic lattices. These techniques were used to compute the vertex configurations, frequencies and empires of icosahedral quasicrystals obtained as a projections of the D 6 and Z 6 lattices to R 3 and we present our analyses. We discuss the implications of this new generalization.

RKKY interactions and magnetic structure of rare-earth quasicrystals

We study the structure of the RKKY interactions and the corresponding low-temperature behaviour of magnetic moments for quasiperiodic tilings. The alignment of magnetic moments in rare-earth quasicrystals remains a fundamental open problem despite the continuous effort since the discovery of this material class. We compute the RKKY interactions between the localized magnetic moments by means of a continued fraction expansion of the Green's function of the conduction electrons. Thus, our approach takes the structure of the critical electronic wave functions into account. The results show the emergence of strongly coupled spin clusters while the inter-cluster coupling is significantly weaker. Monte Carlo simulations reveal with decreasing temperature first the freezing of spins within the clusters followed by the freezing of the clusters. Thus, the low-temperature phase behaves has similarities to a cluster spin glass which is in good agreement with previous experimental findings.