Aperiodic Systems Research Articles

On aperiodicity theorems

In this paper, researched by A. T. Fuller (deceased) and written by E. I. Jury, the necessary and sufficient conditions for aperiodicity which require the roots of a real polynomial to be real, simple, and confined to a specified interval on the real axis are presented in terms of four theorems. These theorems require a positivity test of a certain auxiliary polynomial in the specified interval on the real axis. One of these theorems is a generalization of an earlier published paper by Meerov and Jury (1998). An application of these theorems occurs when the real roots are to be confined to the interval (0,1), a condition related to discrete-time aperiodicity. Another application is related to stable aperiodic continuous-time systems where the real roots are to be confined to (-,0). An example is presented to indicate the application for the discrete-time case. This paper also represents a generalization of Lipka Theorem (Lipka 1943).

Interfaces in quasicrystals: problems and prospects

We propose to discuss the importance of unified 6D description for understanding the geometrical aspects of interfaces in quasicrystalline and its related phases. These will in particular be presented in relation to the icosahedrally related phases. However, the viewpoints adopted will be sufficiently general to help discuss interfaces in any quasiperiodic systems. The consequence of 6D homophase/heterophase interfaces on 3D physical space will be considered by taking examples. These include RAs/IQC, IQC/DQC interfaces apart from the hypertwin and antiphase domain boundaries. It will be shown that such a description does not promise to solve the problems of energetics of interfaces in these systems. It will be concluded that 3D and hypercrystal structural description need to supplement each other for unravelling the underlying mechanism of growth of a particular interface during processing of these materials. The importance of HREM studies in understanding the interface structures in aperiodic systems will be clearly brought out.

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 $2\ifmmode\times\else\texttimes\fi{}2$ 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.

Energy correction for isolated impurities under periodic boundary conditions

The Coulomb energy of aperiodic systems was investigated. To treat completely isolated disorder in infinite systems, energy correction for a supercell method is presented. We discuss a definition of the correction term, and then consider a direct approach taking into account interactions between charge distribution and an indirect approach based on a multipole expansion. In test calculations for isotropic-charged, anisotropic-charged, and neutral impurities, impurity energies independent of supercell sizes were obtained. The present energy correction can be applied to arbitrary systems and is expected to realize more practical simulations for aperiodic systems.

Breakup of a dimer: a new approach to localization transition

Within the framework of tight binding models, aperiodic systems are mapped to a renormalized lattice with a dimer defect. In models exhibiting metal-insulator transition, the dimer acts like a resonant cavity and explains the existence of the ballistic transport in the system. The localization in the model can be attributed to the vanishing of the coupling between the two sites of the dimer. Our approach unifies Anderson transition and resonance transition and provides a new formulation to understand localization and its absence in aperiodic systems.

Un théorème central limite presque sûr à moments généralisés pour les rotations irrationnelles

In a recent work, we indicated another formulation of the Almost Sure Central Limit Theorem (A.S.C.L.T.), with series in place of averages, by showing that the property of the A.S.C.L.T. directly follows from the theory of orthogonal sums. For, we used the notion of quasi-orthogonal systems introduced earlier by R. Bellmann, and later developed by Kac–Salem–Zygmund. The main object of this paper is to prove a similar result for irrational rotations of the torus. We prove the existence of a generalized moment version of the A.S.C.L.T., with a speed of convergence. In our strategy, we use again the notion of quasi-orthogonal system, and purpose a Gaussian randomization technic, new at least in this context. The proof avoid notably the use of Volny's result on the existence of good Gaussian approximations in aperiodic dynamical systems, and should also permit to be able to treat problems of comparable nature, in particular in non-ergodic cases.

Elimination of the long-range dipole interaction in calculations with periodic boundary conditions

In this paper we consider the effect of the long-range Coulomb interactions in calculations of aperiodic systems with nonvanishing dipole moment in the unit cell using periodic boundary conditions. Specifically, the method of Makov and Payne [Phys. Rev. B 51, 4014 (1995)] for eliminating the effect of a net dipole moment in the cubic cells is re-examined, and we show that its success for cubic cells is rather accidental. Instead, we suggest another method that works well also in the case of simulation cells of arbitrary shape. Several examples are considered to illustrate the method.

Existence of only delocalized eigenstates in the electronic spectrum of the Thue–Morse lattice

We present an analytical method for finding all the electronic eigenfunctions and eigenvalues of the aperiodic Thue–Morse lattice. We prove that this system supports only extended electronic states which is a very unusual behavior for this class of systems, and so far as we know, this is the only example of a quasiperiodic or aperiodic system in which critical or localized states are totally absent in the spectrum. Interestingly, we observe that the symmetry of the lattice leads to the existence of degenerate eigenstates and all the eigenvalues excepting the four global band edges are doubly degenerate. We show exactly that the Landauer resistivity is zero for all the degenerate eigenvalues and it scales as ∼L 2 (L= system size ) at the global band edges. We also find that the localization length ξ diverges in the limit of infinite system size.

A Closer Look at Chaotic Advection in the Stratosphere. Part I: Geometric Structure

The relevance of chaotic advection to stratospheric mixing and transport is addressed in the context of (i) a numerical model of forced shallow-water flow on the sphere, and (ii) a middle-atmosphere general circulation model. It is argued that chaotic advection applies to both these models if there is suitable large-scale spatial structure in the velocity field and if the velocity field is temporally quasi-regular. This spatial structure is manifested in the form of ‘‘cat’s eyes’’ in the surf zone, such as are commonly seen in numerical simulations of Rossby wave critical layers; by analogy with the heteroclinic structure of a temporally aperiodic chaotic system the cat’s eyes may be thought of as an ‘‘organizing structure’’ for mixing and transport in the surf zone. When this organizing structure exists, Eulerian and Lagrangian autocorrelations of the velocity derivatives indicate that velocity derivatives decorrelate more rapidly along particle trajectories than at fixed spatial locations (i.e., the velocity field is temporally quasi-regular). This phenomenon is referred to as Lagrangian random strain.

Quantum Point Contacts for Neutral Atoms

We show that the conductance of neutral atoms through a tightly confining waveguide constriction is quantized in units of lambda_dB^2/pi, where lambda_dB is the de Broglie wavelength of the incident atoms. Such a constriction forms the atom analogue of an electron quantum point contact and is an example of quantum transport of neutral atoms in an aperiodic system. We present a practical constriction geometry that can be realized using a microfabricated magnetic waveguide, and discuss how a pair of such constrictions can be used to study the quantum statistics of weakly interacting gases in small traps.

Formalism for the computation of the RKKY interaction in aperiodic systems

A new formalism to investigate RKKY interaction in aperiodic materials is presented. Particularly useful in the tight-binding scheme, it is used in this paper to probe the role of local environment and long range quasiperiodic potential on this mesoscopic coupling. Interesting features are revealed and discussed within the context of anomalous localization and transport

A new blind receiver for downlink DS-CDMA communications

A linear receiver is proposed for downlink DS-CDMA communications over unknown frequency-selective fading channels. The new receiver exploits the fact that all synchronized downlink signals go through the same channel and recovers the desired signal with a constrained channel equalizer followed by a despreader. Such a scheme allows the receiver to operate blindly in a time varying environment for both periodic CDMA and aperiodic CDMA systems.

Invariance principles and Gaussian approximation for strictly stationary processes

We show that in any aperiodic and ergodic dynamical system there exists a square integrable process ( f ∘ T i ) (f\circ T^{i}) the partial sums of which can be closely approximated by the partial sums of Gaussian i.i.d. random variables. For ( f ∘ T i ) (f\circ T^{i}) both weak and strong invariance principles hold.

Multiple-scattering x-ray-absorption fine-structure Debye-Waller factor calculations

An efficient local equation-of-motion method is introduced for calculations of the mean-square half-path-length fluctuations ${\ensuremath{\sigma}}_{j}^{2}$ in multiple-scattering x-ray-absorption fine-structure Debye-Waller factors in aperiodic systems. Given a few local force constants, the method yields ${\ensuremath{\sigma}}_{j}^{2}$ via projected densities of modes or via the displacement-displacement correlation function in real time, over a few vibration cycles. The calculation scales linearly with the system size and does not rely on any symmetry considerations. Sample applications are presented for crystalline Cu and Ge, and zinc tetraimidazole.

Aperiodic tilings of the hyperbolic plane by convex polygons

Several aperiodic hyperbolic tiling systems consisting of a single convex tile are constructed.

Comparative study of the critical behavior in one-dimensional random and aperiodic environments

We consider cooperative processes (quantum spin chains and random walks) in one-dimensional fluctuating random and aperiodic environments characterized by fluctuating exponents omega>0. At the critical point the random and aperiodic systems scale essentially anisotropically in a similar fashion: length (L) and time (t) scales are related as t ~ log^{1/omega}. Also some critical exponents, characterizing the singularities of average quantities, are found to be universal functions of omega, whereas some others do depend on details of the distribution of the disorder. In the off-critical region there is an important difference between the two types of environments: in aperiodic systems there are no extra (Griffiths)-singularities.

Real-space multiple-scattering calculation and interpretation of x-ray-absorption near-edge structure

A self-consistent real-space multiple-scattering (RSMS) approach for calculations of x-ray-absorption near-edge structure (XANES) is presented and implemented in an ab initio code applicable to arbitrary aperiodic or periodic systems. This approach yields a quantitative interpretation of XANES based on simultaneous, self-consistent-field (SCF) calculations of local electronic structure and x-ray absorption spectra, which include full multiple scattering from atoms within a small cluster and the contributions of high-order MS from scatterers outside that cluster. In addition, the code includes a SCF estimate of the Fermi energy and an account of orbital occupancy and charge transfer. We also present a qualitative, scattering-theoretic interpretation of XANES. Sample applications are presented for cubic BN, ${\mathrm{UF}}_{6},$ Pu hydrates, and distorted ${\mathrm{PbTiO}}_{3}.$ Limitations and various extensions are also discussed.

Random and aperiodic quantum spin chains: A comparative study

According to the Harris-Luck criterion the relevance of a fluctuating interaction at the critical point is connected to the value of the fluctuation exponent omega. Here we consider different types of relevant fluctuations in the quantum Ising chain and investigate the universality class of the models. At the critical point the random and aperiodic systems behave similarly, due to the same type of extreme broad distribution of the energy scales at low energies. The critical exponents of some averaged quantities are found to be a universal function of omega, but some others do depend on other parameters of the distribution of the couplings. In the off-critical region there is an important difference between the two systems: there are no Griffiths singularities in aperiodic models.

On aperiodicity robustness

In this paper, the aperiodicity condition of a system's characteristic equation is first formulated for both continuous and discrete systems. It states that the roots of the characteristic equation have to be real, distinct, and negative for the continuous case, and real, distinct, and in the interval [0,1) for the discrete case. The coefficients of these equations are perturbed for an initially aperiodic system such that, in the limit, aperiodicity is violated. The values of the perturbed coefficients are determined and thus the robustness conditions are obtained. This approach to the topic is quite different from those that have appeared in the literature which first assume that the initial perturbation limits are known. Such approaches are quite similar to that first discussed by Kharitonov for stability problems. In situations where the initial perturbation limits are not known, the approach presented in this paper is a valid alternative. The solution presented in this paper is based on two formulations. The first is based on the critical aperiodicity conditions while the second is based on a theorem known in old literature which states that if the system is initially aperiodic, then Ψ (s)=[ F′(s)]² - F(s)F′′(s), where F′(s)= dF(s)/ds and F′′(s)= d²F(s)/ds², should have no real roots. Here F(s) is the continuous system's characteristic equation. Now, in the limit when the above condition is violated, one obtains the robustness limits ζ. Both approaches give the same values. To obtain similar robustness limits for the discrete case, we can use either the linear fractional transformation z=s/(s-1) which transforms the segment [0,1) in the z-plane onto the negative real axis in the s-plane and use the above conditions, or directly formulate the aperiodicity conditions in the z-plane. Both are discussed in this paper. An example is presented to illustrate the above formulations for obtaining the robustness conditions.

K-groups associated with substitution minimal systems

Two ordered Bratteli diagrams can be constructed from an aperiodic substitution minimal dynamical system. One, the proper diagram, has a single maximal path and a single minimal path and the Vershik map on the path space can be extended homeomorphically to a map conjugate to the substitution system. The other, the improper diagram, encodes the substitution more naturally but often has many maximal and minimal paths and no continuous compact dynamics. This paper connects the two diagrams by considering theirK 0-groups, obtaining the equation $$K_0 (Proper) = K_0 (Improper)/Q \oplus \mathbb{Z}^\nu $$ whereQ and ν can be determined from the combinatorial properties of the substitution. This allows several examples of substitution sequences to be distinguished at the level of strong orbit equivalence. A final section shows that every dimension group with unit which is a stationary limit of ℤ n groups can be represented as aK 0 group of some substitution minimal system. Also every stationary proper minimal ordered Bratteli diagram has a Vershik map which is either Kakutani equivalent to ad-adic system or is conjugate to a substitution minimal system. The equation above applies to a much wider class which includes those minimal transformations which can be represented as a path-sequence dynamical system on a Bratteli diagram with a uniformly bounded number of vertices in each level.