Dieter Joseph

TRACE MAPS, INVARIANTS, AND SOME OF THEIR APPLICATIONS

Trace maps of two-letter substitution rules are investigated with special emphasis on the underlying algebraic structure and on the existence of invariants. We illustrate the results with the generalized Fibonacci chains and show that the well-known Fricke character I(x, y, z)=x2+y2+z2−2xyz−1 is not the only type of invariant that can occur. We discuss several physical applications to electronic spectra including the gap-labeling theorem, to kicked two-level systems, and to the classical 1D Ising model with non-commuting transfer matrices.

Root lattices and quasicrystals

It is shown that root lattices and their reciprocals might serve as the right pool for the construction of quasicrystalline structure models. All noncrystallographic symmetries observed so far are covered in minimal embedding with maximal symmetry.

Periodic clustering in the spectrum of quasiperiodic Kronig-Penney models

Abstract The continuous Schrodinger equation is discussed for the Fibonacci chain and its generalizations and compared to the tight-binding approximation. For Kronig-Penney like models, the resulting pseudo spectrum of the well-known trace map has Cantor like structures, but a subclass of models additionally shows periodic clustering with respect to the wave number k . The clusters appear at the zeros of the invariant of the trace map as a function of k . From a matrix generalization of the trace map we compute the forward scattering of the chain and find the same periodic clustering. We briefly discuss how these results extend to more general non-periodic examples.

THE ROOT LATTICE D4 AND PLANAR QUASILATTICES WITH OCTAGONAL AND DODECAGONAL SYMMETRY

Quasiperiodic patterns with eight- and twelvefold symmetry are presented which share the root lattice D4, i.e., the 4-D face-centered hypercubic lattice, for their minimal embedding in four-space. We derive the patterns by means of the dualization method and investigate key properties like vertex configurations, local deflation/inflation symmetries and kinematic diffraction. The generalized point symmetries (and thus the Laue group) of these patterns are the dihedral groups d8 and d12, respectively, which share a common subgroup, d4. We introduce a contiunous one-parameter rotation between the two phases which leaves this subgroup invariant. This should prove useful for modelling alloys like V15Ni10Si where quasicrystalline phases with eight- and twelvefold symmetry coexist.

Boundary conditions, entropy and the signature of random tilings

We investigate the influence of boundary conditions on the results of Monte Carlo simulations of 2D random tilings. Looking at the fluctuations in internal space we compare fixed and periodic boundaries. We find that fixed boundary conditions will lead to a different random tiling ensemble with reduced finite-size entropy density in comparison with periodic boundary conditions. As a by-product, we derive improved estimates for the elastic constants of the octagonal rhombus-tiling ensemble. Finally, we introduce a robust tool, also suitable for the analysis of experimental data, to distinguish between quasiperiodic and random-tiling models.

Diffusion in 2D Quasi-Crystals

Self-diffusion induced by phasonic flips is studied in an octagonal model quasi-crystal. To determine the temperature dependence of the diffusion coefficient, we apply a Monte Carlo simulation with specific energy values of local configurations. We compare the results of the ideal quasi-periodic tiling and a related periodic approximant and comment on possible implications to real quasi-crystals.

Averaged shelling for quasicrystals

The shelling of crystals is concerned with counting the number of atoms on spherical shells of a given radius and a fixed centre. Its straight-forward generalization to quasicrystals, the so-called central shelling, leads to non-universal answers. As one way to cope with this situation, we consider shelling averages over all quasicrystal points. We express the averaged shelling numbers in terms of the autocorrelation coefficients and give explicit results for the usual suspects, both perfect and random.

The Schur rotation as a simple approach to the transition between quasiperiodic and periodic phases

The recently observed transition of CrNiSi from the octahedral to a cubic phase has a natural mathematical counterpart in terms of a one-parameter Schur rotation. The authors present the corresponding tilings and the diffraction patterns of delta -scatterers at vertex positions for a series of rotation angles. These angles show up as a length scaling in the patterns and provide a measurable order parameter. They comment on the rational reductions and give two further examples, one on a connection between octagonal and dodecagonal patterns and one on a transition between icosahedral and primitive or body-centred cubic phases.

The growth of entropically stabilized quasicrystals

We introduce a growth model for entropically stabilized quasicrystals. The dominating feature of this model is a fluctuating growth front which enables growth near equilibrium with small phason components. We summarize the results obtained for 2D and give a first presentation of 3D calculations.

Self-organized criticality on quasiperiodic graphs

Self-organized critical models are used to describe the 1/f-spectra of rather different physical situations like snow avalanches, noise of electric currents, luminosities of stars or topologies of landscapes. The prototype of the SOC-models is the sandpile model of Bak, Tang and Wiesenfeld (Phys. Rev. Lett. 59, 381 (1987)). We implement this model on non-periodic graphs where it can become either isotropic or anisotropic and compare its properties with the periodic counterpart on the square lattice.