arXiv subjects

Michael Baake

Publications and source records attributed to Michael Baake.

At least 163 records · Page 9Linked to original sources

arXiv1999

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.

math.MG↗

arXiv1999

Reversing Symmetry Groups of Cat Maps

Toral automorphisms are widely used (discrete) dynamical systems, the perhaps most prominent example (in 2D) being Arnold's cat map. Given such an automorphism M, its symmetries (i.e. all automorphisms that commute with M) and reversing symmetries (i.e. all automorphisms that conjugate M into its inverse) can be determined by means of number theoretic tools. Here, the case of GL(2,Z) is presented and the possible (reversing) symmetry groups are completely classified. Extensions to affine mappings and to k-(reversing) symmetries (i.e. (reversing) symmetries of the k-th power of M), and applications to the projective group PGL(2,Z) and to trace maps (compare math.DS/9901124), are briefly discussed.

math.DS↗

arXiv1999

A note on palindromicity

Two results on palindromicity of bi-infinite words in a finite alphabet are presented. The first is a simple, but efficient criterion to exclude palindromicity of minimal sequences and applies, in particular, to the Rudin-Shapiro sequence. The second provides a constructive method to build palindromic minimal sequences based upon regular, generic model sets with centro-symmetric window. These give rise to diagonal tight-binding models in one dimension with purely singular continuous spectrum.

math-ph↗

arXiv1999

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) = x^2 + y^2 + z^2 - 2 x y z - 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.

math-ph↗

arXiv1999

Similarity submodules and root systems in four dimensions

Lattices and Z-modules in Euclidean space possess an infinitude of subsets that are images of the original set under similarity transformation. We classify such self-similar images according to their indices for certain 4D examples that are related to 4D root systems, both crystallographic and non-crystallographic. We encapsulate their statistics in terms of Dirichlet series generating functions and derive some of their asymptotic properties.

math.MG↗

arXiv1999

Some remarks on the visible points of a lattice

We comment on the set of visible points of a lattice and its Fourier transform, thus continuing and generalizing previous work by Schroeder and Mosseri. A closed formula in terms of Dirichlet series is obtained for the Bragg part of the Fourier transform. We compare this calculation with the outcome of an optical Fourier transform of the visible points of the 2D square lattice.

math-ph↗

arXiv1999

Non-periodic Ising quantum chains and conformal invariance

A while ago, Luck (J. Stat. Phys. 72 (1993) 417) investigated the critical behaviour of one-dimensional Ising quantum chains with couplings constants modulated according to general non-periodic sequences. In this short note, we take a closer look at the case where the sequences are obtained from (two-letter) substitution rules and at the consequences of Luck's results at criticality. They imply that only for a certain class of substitution rules the long-distance behaviour is still described by the c=1/2 conformal field theory of a free Majorana fermion as for the periodic Ising quantum chain, whereas the general case does not lead to a conformally invariant scaling limit.

cond-mat.stat-mech↗

arXiv1999

Ising spins on the labyrinth

We consider a zero-field Ising model defined on a quasiperiodic graph, the so-called Labyrinth tiling. Exact information about the critical behaviour is obtained from duality arguments and the subclass of models which yield commuting transfer matrices. For the latter, the magnetization is independent of the position and the phase transition between ordered and disordered phase belongs to the Onsager universality class. In order to obtain information about the generic case, we calculate the magnetization for a series of couplings by standard Monte-Carlo methods.

cond-mat.stat-mech↗

arXiv1999

Lee-Yang zeros for substitutional systems

Qualitative and quantitative information about critical phenomena is provided by the distribution of zeros of the partition function in the complex plane. We apply this idea to Ising models on non-periodic systems based on substitution. In 1D we consider the Thue-Morse chain and show that the magnetic field zeros are filling a Cantor subset of the unit circle, the gaps being related to a general gap labeling theorem. In 2D we study the temperature zeros of the Ising model on the Ammann-Beenker tiling. The use of corner transfer matrices allows an efficient calculation of the partition function for rather large patches which results in a reasonable estimate of the critical temperature.

cond-mat.stat-mech↗

arXiv1999

Symmetries and reversing symmetries of trace maps

A (discrete) dynamical system may have various symmetries and reversing symmetries, which together form its so-called reversing symmetry group. We study the set of 3D trace maps (obtained from two-letter substitution rules) which preserve the Fricke-Vogt invariant I(x,y,z). This set of dynamical systems forms a group G isomorphic with the projective linear (or modular) group PGL(2,Z). For such trace maps, we give a complete characterization of the reversing symmetry group as a subgroup of the group A of all polynomial mappings that preserve I(x,y,z).

math.DS↗

arXiv1999

A guide to mathematical quasicrystals

This introductory survey deals with mathematical and physical properties of discrete structures such as point sets and tilings. The emphasis is on proper generalizations of concepts and ideas from classical crystallography. In particular, we focus on their interplay with various physically motivated equivalence concepts such as local indistinguishability and local equivalence. Various discrete patterns with non-crystallographic symmetries are described in detail, and some of their magic properties are introduced. This perfectly ordered world is augmented by a brief introduction to the stochastic world of random tilings.

math-ph↗

arXiv1999

Limit-(quasi)periodic point sets as quasicrystals with p-adic internal spaces

Model sets (or cut and project sets) provide a familiar and commonly used method of constructing and studying nonperiodic point sets. Here we extend this method to situations where the internal spaces are no longer Euclidean, but instead spaces with p-adic topologies or even with mixed Euclidean/p-adic topologies. We show that a number of well known tilings precisely fit this form, including the chair tiling and the Robinson square tilings. Thus the scope of the cut and project formalism is considerably larger than is usually supposed. Applying the powerful consequences of model sets we derive the diffractive nature of these tilings.

math-ph↗

arXiv1998

Coordination Sequences and Critical Points

Coordination sequences of periodic and quasiperiodic graphs are analysed. These count the number of points that can be reached from a given point of the graph by a number of steps along its bonds, thus generalising the familiar coordination number which is just the first member of this series. A possible application to the theory of critical phenomena in lattice models is outlined.

cond-mat.stat-mech↗

arXiv1998

The Entropy of Square-Free Words

Finite alphabets of at least three letters permit the construction of square-free words of infinite length. We show that the entropy density is strictly positive and derive reasonable lower and upper bounds. Finally, we present an approximate formula which is asymptotically exact with rapid convergence in the number of letters.

math-ph↗

arXiv1998

Symmetry Structure of the Elser-Sloane Quasicrystal

The 4D quasicrystal of Elser and Sloane, obtained from the root lattice E8 by the cut-and-project method, can be parametrized by the points of an 8D torus. This allows for an explicit analysis of its point and inflation symmetry structure.

cond-mat↗

arXiv1998

Invariant Submodules and Semigroups of Self-Similarities for Fibonacci Modules

The problem of invariance and self-similarity in Z-modules is investigated. For a selection of examples relevant to quasicrystals, especially Fibonacci modules, we determine the semigroup of self-similarities and encapsulate the number of similarity submodules in terms of Dirichlet series generating functions.

math-ph↗

arXiv1998

Multi-Component Model Sets and Invariant Densities

Model sets (also called cut and project sets) are generalizations of lattices, and multi-component model sets are generalizations of lattices with colourings. In this paper, we study self-similarities of multi-component model sets. The main point may be simply summarized: whenever there is a self-similarity, there are also naturally related density functions. As in the case of ordinary model sets, we show that invariant densities exist and that they produce absolutely continuous invariant measures in internal space, these features now appearing in matrix form. We establish a close connection between the theory of invariant densities and the spectral theory of matrix continuous refinement operators.

math-ph↗

arXiv1998

Self-Similarities and Invariant Densities for Model Sets

Model sets (also called cut and project sets) are generalizations of lattices. Here we show how the self-similarities of model sets are a natural replacement for the group of translations of a lattice. This leads us to the concept of averaging operators and invariant densities on model sets. We prove that invariant densities exist and that they produce absolutely continuous invariant measures in internal space. We study the invariant densities and their relationships to diffraction, continuous refinement operators, and Hutchinson measures.

math-ph↗