Uwe Grimm

Trace and antitrace maps for aperiodic sequences: Extensions and applications

We study aperiodic systems based on substitution rules by means of a transfer-matrix approach. In addition to the well-known trace map, we investigate the so-called “antitrace” map, which is the corresponding map for the difference of the off-diagonal elements of the 2x2 transfer matrix. The antitrace maps are obtained for various binary, ternary, and quaternary aperiodic sequences, such as the Fibonacci, Thue-Morse, period-doubling, Rudin-Shapiro sequences, and certain generalizations. For arbitrary substitution rules, we show that not only trace maps, but also antitrace maps exist. The dimension of our antitrace map is r(r+1)/2, where r denotes the number of basic letters in the aperiodic sequence. Analogous maps for specific matrix elements of the transfer matrix can also be constructed, but the maps for the off-diagonal elements and for the difference of the diagonal elements coincide with the antitrace map. Thus, from the trace and antitrace map, we can determine any physical quantity related to the global transfer matrix of the system. As examples, we employ these dynamical maps to compute the transmission coefficients for optical multilayers, harmonic chains, and electronic systems.

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.

Homometric model sets and window covariograms

Two Delone sets are called homometric when they share the same autocorrelation or Patterson measure. A model set Λ within a given cut and project scheme is a Delone set that is defined through a window W in internal space. The autocorrelation measure of Λ is a pure point measure whose coefficients can be calculated via the so-called covariogram of W. Two windows with the same covariogram thus result in homometric model sets. On the other hand, the inverse problem of determining Λ from its diffraction image ultimately amounts to reconstructing W from its covariogram. This is also known as Matheron’s covariogram problem. It is well studied in convex geometry, where certain uniqueness results have been obtained in recent years. However, for non-convex windows, uniqueness fails in a relevant way, so that interesting applications to the homometry problem emerge. We discuss this in a simple setting and show a planar example of distinct homometric model sets.

Bravais colourings of planar modules with N-fold symmetry

Abstract The first step in investigating colour symmetries for periodic and aperiodic systems is the determination of all colouring schemes that are compatible with the symmetry group of the underlying structure, or with a subgroup of it. For an important class of colourings of planar structures, this mainly combinatorial question can be addressed with methods of algebraic number theory. We present the corresponding results for all planar modules with N-fold symmetry that emerge as the rings of integers in cyclotomic fields with class number one. The counting functions are multiplicative and can be encapsulated in Dirichlet series generating functions, which turn out to be the Dedekind zeta functions of the corresponding cyclotomic fields.

Noncollinear magnetic order in quasicrystals.

Based on Monte Carlo simulations, the stable magnetization configurations of an antiferromagnet on a quasiperiodic tiling are derived theoretically. The exchange coupling is assumed to decrease exponentially with the distance between magnetic moments. It is demonstrated that the superposition of geometric frustration with the quasiperiodic ordering leads to a three-dimensional noncollinear antiferromagnetic spin structure. The structure can be divided into several ordered interpenetrating magnetic supertilings of different energy and characteristic wave vector. The number and the symmetry of subtilings depend on the quasiperiodic ordering of atoms.

Spectral and topological properties of a family of generalised Thue-Morse sequences

The classic middle-thirds Cantor set leads to a singular continuous measure via a distribution function that is known as the Devils staircase. The support of the Cantor measure is a set of zero Lebesgue measure. Here, we discuss a class of singular continuous measures that emerge in mathematical diffraction theory and lead to somewhat similar distribution functions, yet with significant differences. Various properties of these measures are derived. In particular, these measures have supports of full Lebesgue measure and possess strictly increasing distribution functions. In this sense, they mark the opposite end of what is possible for singular continuous measures. For each member of the family, the underlying dynamical system possesses a topological factor with maximal pure point spectrum, and a close relation to a solenoid, which is the Kronecker factor of the system. The inflation action on the continuous hull is sufficiently explicit to permit the calculation of the corresponding dynamical zeta funct...

Kinematic diffraction from a mathematical viewpoint

Abstract Mathematical diffraction theory is concerned with the analysis of the diffraction image of a given structure and the corresponding inverse problem of structure determination. In recent years, the understanding of systems with continuous and mixed spectra has improved considerably. Simultaneously, their relevance has grown in practice as well. In this context, the phenomenon of homometry shows various unexpected new facets. This is particularly so for systems with stochastic components. After the introduction to the mathematical tools, we briefly discuss pure point spectra, based on the Poisson summation formula for lattice Dirac combs. This provides an elegant approach to the diffraction formulas of infinite crystals and quasicrystals. We continue by considering classic deterministic examples with singular or absolutely continuous diffraction spectra. In particular, we recall an isospectral family of structures with continuously varying entropy. We close with a summary of more recent results on the diffraction of dynamical systems of algebraic or stochastic origin.

Kinematic diffraction is insufficient to distinguish order from disorder

Diffraction methods are at the heart of structure determination of solids. While Bragg-like scattering (pure point diffraction) is a characteristic feature of crystals and quasicrystals, it is not straightforward to interpret continuous diffraction intensities, which are generally linked to the presence of disorder. However, based on simple model systems, we demonstrate that it may be impossible to draw conclusions on the degree of order in the system from its diffraction image. In particular, we construct a family of one-dimensional binary systems which cover the entire entropy range but still share the same purely diffuse diffraction spectrum.

Aperiodic Ising quantum chains

Some years ago, Luck proposed a relevance criterion for the effect of aperiodic disorder on the critical behaviour of ferromagnetic Ising systems. In this article, we show how Lucks criterion can be derived within an exact renormalization scheme for Ising quantum chains with coupling constants modulated according to substitution rules. Lucks conjectures for this case are confirmed and refined. Among other outcomes, we give an exact formula for the correlation length critical exponent for arbitrary two-letter substitution sequences with marginal fluctuations of the coupling constants.

Level-spacing distributions of planar quasiperiodic tight-binding models

We study statistical properties of energy spectra of two-dimensional quasiperiodic tight-binding models. Taking into account the symmetries of models defined on various finite approximants of quasiperiodic tilings, we find that the underlying universal level-spacing distribution is given by the Gaussian orthogonal random matrix ensemble. Our data allow us to see the difference to the Wigner surmise. In particular, our result differs from the critical level-spacing distribution observed at the metal-insulator transition in the three-dimensional Anderson model of disorder.