Theory of matching rules for the 3-dimensional Penrose tilings

We consider packings of the two Ammann rhombohedra used for tiling the three dimensional space. We define decorations for the facets of the rhombohedra. Using elementary algebraic topology, we prove that any tiling by these rhombohedra with matching decorations is a quasiperiodic Penrose tiling. The proof does not involve any reference to self similarity.     

Local rules for quasicrystals

The relationship of local ordering and long-range order is studied for quasicrystalline tilings of plane and space. Two versions of the concept of local rules are introduced: strong and weak. Necessary conditions of the existence of strong local rules are found. They are mainly reduced to the constraints for irrational numbers related to incommensurabilities of the quasicrystals. For planar quasicrystals the quadratic irrationalities play an important role. For three-dimensional quasicrystals not only quadratic but also cubic irrationalities are allowed. The existence of weak local rules is established for almost all two-dimensional quasicrystals based on quadratic irrationalities and for the three-dimensional quasicrystal having icosahedral symmetry.     

Self-similarity and scaling of wave function for binary quasiperiodic chains associated with quadratic irrationals

By establishing the correspondence between the substitution rule (aa ⁿb; ba) and the transformation on the value of =tan by1/(n+) in the cut-and-project (CP) method, it is proved that the necessary and sufficient conditions for a binary quasiperiodic (QP) sequence made by the CP method to be self-similar is that is a quadratic irrational (QI) number. And, vice versa, the necessary condition for a binary self-similar sequence generated by the substitution rule to be obtainable by the CP method is that the corresponding substitution rule can be rewritten as a simple composition of transformations with the type (aa ⁿb; ba). To illustrate some physical properties of the self-similar QP chains associated with QI numbers, we analyze the scaling behaviour of the wave function atE=0 for an off-diagnonal tight-binding model. It is shown that the wave function atE=0 grows at most by a power-law for the QP lattices, characterized by a special class of QI numbers. For the QP chains associated with general QI numbers, with the same logic, the typical scaling behaviour of the wave function is conjectured to be the same.     

Non-periodic tilings in 2-dimensions: 4- and 7-fold symmetries

By inflating basic rhombuses, with a self-similarity principle, non-periodic tiling of 2-d planes is possible with 4, 5, 6, 7, 8, … -fold symmetries. As examples, non-periodic tilings with crystallographically allowed 4-fold symmetry and crystallographically forbidden 7-fold symmetry are presented in detail. The computed diffraction patterns of these tilings are also discussed.     

Local aspects of superselection rules

We study a theory of short range forces in terms of local observable quantities; among the superselection structure determined by the algebra of all local observables, to each additive independent charge we associate local observables having a meaning analogous to the regularized integrals of charge density fields over a finite volume. Among other assumptions, we require that parastatistics are absent from the theories considered.     

Electronic properties of CuO 2 -planes: A DMFT study

We study one-particle spectra and the electronic band-structure of a CuO ₂ -plane within the three-band Hubbard model. The Dynamical Mean-Field Theory (DMFT) is used to solve the many particle problem. The calculations show that the optical gap is given by excitations from the lower Hubbard band into the so called Zhang-Rice singlet band. The optical gap turns out to be considerably smaller than the bare charge transfer energy () for a typical set of parameters, which is in agreement with experiment. We also investigate the dependence of the shape of the Fermi surface on the different hopping parameters t CuO and t OO. A value t OO / t CuO >0 leads to a Fermi surface surrounding the M point. Received 21 September 1998 and Received in final form 8 June 1999     

Local aspects of superselection rules. II

In a theory where the local observables are determined by local field algebras as the fixed points under a (a priori noncommutative) group of gauge transformations of the first kind, we show that, if the field algebras possess intermediate type I factors, we can construct observables having the meaning of local charge measurements, and local current algebras in the field algebras.     

Non-periodic tilings in 2-dimensions with 4, 6, 8, 10 and 12-fold symmetries

The two dimensional plane can be filled with rhombuses, so as to generate non-periodic tilings with 4, 6, 8, 10 and 12-fold symmetries. Some representative tilings constructed using the rule of inflation are shown. The numerically computed diffraction patterns for the corresponding tilings are also shown to facilitate a comparison with possible X-ray or electron diffraction pictures.     

Possible relationship between Cu-NQR frequencies and Cu valence states in CuO2-planes for high-T c superconductors

We report on⁶³Cu-NQR frequency ν_NQR measurements on various high-T _c superconductors. An empirical relationship betweenT _c and ν_NQR is presented. We attribute this finding to variations in the valence state of the in-plane Cu ions which, due to the unusual electric polarizability of the O^2? ions, depends sensitively on the Cu?O distance.     

An example of quasiperiodic motion on T4

According to a theorem of Ruelle and Takens, the occurrence of quasiperiodic motion on T⁴ should be unlikely. Here with the help of numerical calculations, we give a concrete example for which such motion is shown to exist and survive under small perturbations.     

The cutting rules for the 4-reggeon amplitude

We derive the s-channel cutting rules for the four-pomeron amplitude in the context of a ø³ weak coupling model.     

Selection rules for superstructures in thiourea,Rb2ZnBr4 and Rb2ZnCl4

This paper concerns the superstructures to which a modulated structure locks in. For thiourea, all observed rational values of $\text{γ (}\text{q}\text{= γ}\text{c}{}^1\text{)}$ belong to a certain type of fractions, namely - if we put $\text{γ =}\text{e}\text{v}$ - those with both u and v odd. In Rb₂ZnBr₄ and Rb₂ZnCl₄, published data point strongly to the existence of superstructures which so far had been regarded as incommensurate, but which turn out to yield values $\text{u}\text{v}$ also obeying the above parity conditions. Such conditions can be explained by a new hypothesis, viz. that the point group of the lock-in phase is identical to the three-dimensional part of the point group of the adjoining incommensurate phase.     

Iterative techniques for computing the linearized manifolds of quasiperiodic tori

We develop an iterative technique for computing the unstable and stable eigenfunctions of the invariant tori of diffeomorphisms. Using the approach of Jorba [Nonlinearity 14, 943 (2001)], the linearized equations are rewritten as a generalized eigenvalue problem. Casting the system in this light allows us to take advantage of the speed of eigenvalue solvers and create an efficient method for finding the first-order approximations to the invariant manifolds of the torus. We present a numerical scheme based on the power method that can be used to determine the behavior normal to such tori, and give some examples of the application of the method. We confirm the qualitative conclusions of the Melnikov calculations of Lomeli and Meiss [Nonlinearity 16, 1573 (2003)] for a volume-preserving mapping.