Aperiodic Order Research Articles

Open Problems Column

This issue's Open Problem Column is by Stephen Fenner, Fred Green, and Steven Homer. It is titled: Fixed-Parameter Extrapolation and Aperiodic Order: Open Problems. It is not on xedparameter tractability. It is on a generalization of the geometric notion of convexity.

A tenfold twin of the CrB structure type

NiZr crystallized from an amorphous matrix or solidified from an undercooled melt exhibits a tenfold twinned microstructure, which is explained by an ideal twin model utilizing special geometric properties of the CrB structure type. The model is unique in several ways: (i) it contains no adjustable parameters other than a scaling factor accounting for the smallest interatomic distance; (ii) it features an irrational shift in the translational part of the twin operation; and (iii) it has many traits commonly observed for quasicrystals, connected to the occurrence of decagonal long-range orientational order, making NiZr the first experimental example of the recently introduced concept of {\bb Z}-module twinning. It is shown how these remarkable properties of the tenfold twin's structure model are related to one another and founded in number theory as well as in the mathematical theory of aperiodic order.

Edge modes of scattering chains with aperiodic order.

We study the scattering resonances of one-dimensional deterministic aperiodic chains of electric dipoles using the vectorial Green's matrix method, which accounts for both short- and long-range electromagnetic interactions in open scattering systems. We discover the existence of edge-localized scattering states within fractal energy gaps with characteristic topological band structures. Notably, we report and characterize edge-localized modes in the classical wave analogues of the Su-Schrieffer-Heeger (SSH) dimer model, quasiperiodic Harper and Fibonacci crystals, as well as in more complex Thue-Morse aperiodic systems. Our study demonstrates that topological edge-modes with characteristic power-law envelope appear in open aperiodic systems and coexist with traditional exponentially localized ones. Our results extend the concept of topological states to the scattering resonances of complex open systems with aperiodic order, thus providing an important step towards the predictive design of topological optical metamaterials and devices beyond tight-binding models.

A classification of aperiodic order via spectral metrics and Jarník sets

Given an$\unicode[STIX]{x1D6FC}>1$and a$\unicode[STIX]{x1D703}$with unbounded continued fraction entries, we characterize new relations between Sturmian subshifts with slope$\unicode[STIX]{x1D703}$with respect to (i) an$\unicode[STIX]{x1D6FC}$-Hölder regularity condition of a spectral metric, (ii) level sets defined in terms of the Diophantine properties of$\unicode[STIX]{x1D703}$, and (iii) complexity notions which we call$\unicode[STIX]{x1D6FC}$-repetitiveness,$\unicode[STIX]{x1D6FC}$-repulsiveness and$\unicode[STIX]{x1D6FC}$-finiteness—generalizations of the properties known as linear repetitiveness, repulsiveness and power freeness, respectively. We show that the level sets relate naturally to (exact) Jarník sets and prove that their Hausdorff dimension is$2/(\unicode[STIX]{x1D6FC}+1)$.

Aperiodic order and spherical diffraction, I: auto-correlation of regular model sets

We study uniform and non-uniform model sets in arbitrary locally compact second countable (lcsc) groups, which provide a natural generalization of uniform model sets in locally compact abelian groups as defined by Meyer and used as mathematical models of quasi-crystals. We then define a notion of auto-correlation for subsets of finite local complexitiy in arbitrary lcsc groups, which generalizes Hof's classical definition beyond the class of amenable groups, and provide a formula for the auto-correlation of a regular model set. Along the way we show that the punctured hull of an arbitrary regular model set admits a unique invariant probability measure, even in the case where the punctured hull is non-compact and the group is non-amenable. In fact this measure is also the unique stationary measure with respect to any admissible probability measure.

Spectra of Schreier graphs of Grigorchuk’s group and Schroedinger operators with aperiodic order

We study spectral properties of the Laplacians on Schreier graphs arising from Grigorchuk’s group acting on the boundary of the infinite binary tree. We establish a connection between the action of G on its space of Schreier graphs and a subshift associated to a non-primitive substitution and relate the Laplacians on the Schreier graphs to discrete Schroedinger operators with aperiodic order. We use this relation to prove that the spectrum of the anisotropic Laplacians is a Cantor set of Lebesgue measure zero. We also use it to show absence of eigenvalues both almost-surely and for certain specific graphs. The methods developed here apply to a large class of examples.

Fractional Transport of Photons in Deterministic Aperiodic Structures

The propagation of optical pulses through primary types of deterministic aperiodic structures is numerically studied in time domain using the rigorous transfer matrix method in combination with analytical fractional transport models. We demonstrate tunable anomalous photon transport, including the elusive logarithmic Sinai sub-diffusion in photonic systems for the first time. Our results are in excellent agreement with the scaling theory of transport in aperiodic media with fractal spectra, and additionally demonstrate logarithmic sub-diffusion in the presence of multifractality. Moreover, we establish a fruitful connection between tunable photon diffusion and fractional dynamics, which provides analytical insights into the asymptotic transport regime of optical media with deterministic aperiodic order. The demonstration of tunable sub-diffusion and logarithmic photon transport in deterministic aperiodic structures can open novel and fascinating scenarios for the engineering of wave propagation and light-matter interaction phenomena beyond the conventional diffusive regime.

Aperiodic topological order in the domain configurations of functional materials

In numerous functional materials, such as steels, ferroelectrics and magnets, new functionalities can be achieved through the engineering of the domain structures, which are associated with the ordering of certain parameters within the material. The recent progress in imaging techniques with atomic-scale spatial resolution transformed our understanding of domain topology revealing that, along with simple stripe-like or irregularly shaped domains, intriguing vortex-type topological domain configurations also exist. In this Review, we present a new classification scheme of "ZmxZn domains with Zl vortices" for two-dimensional macroscopic domain structures with m directional variants and n translational antiphases. This classification, together with the concept of topological protection and topological charge conservation, can be applied to a wide range of materials, such as multiferroics, improper ferroelectrics, layered transition-metal dichalcogenides and magnetic superconductors, as we discuss through selected examples. The resulting topological considerations provide a new basis for the understanding of the formation, kinetics, manipulation and property optimization of domains and domain boundaries in functional materials.

Quo Vadis Quasicrystals?

This Special Issue aims at gaining a deeper understanding on the relationship between the underlying structural order and the resulting physical properties in aperiodic systems, including quasicrystalline and related complex metallic alloys, photonic quasicrystals, and other structures exhibiting long-range aperiodic order. This Special Issue contains 12 papers which highlight recent developments in quasiperiodic crystal structure, photonic quasicrystals and related optical devices, the intrinsic electrical, thermal, and mechanical properties of icosahedral and decagonal metallic alloys, and the nature of chemical bonding in intermetallic compounds, from a multidisciplinary perspective. In light of the results presented in the contributions collected in this Special Issue, we can confidently expect that new insights into the interdisciplinary science of quasicrystals will be gained in the years to come, providing a sharper picture of their structures and related physical properties, and spurring further progress in practical issues related to both materials engineering science and nanotechnology.

Growth and decay of a two‐dimensional oxide quasicrystal: High‐temperature in situ microscopy

The recently discovered two‐dimensional oxide quasicrystal (OQC) derived from BaTiO3 on Pt(111) is the first material in which a spontaneous formation of an aperiodic structure at the interface to a periodic support has been observed. Herein, we report in situ low‐energy electron microscopy (LEEM) studies on the fundamental processes involved in the OQC growth. The OQC formation proceeds in two steps via of an amorphous two‐dimensional wetting layer. At 1170 K the long‐range aperiodic order of the OQC develops. Annealing in O2 induces the reverse process, the conversion of the OQC into BaTiO3 islands and bare Pt(111), which has been monitored by in situ LEEM. A quantitative analysis of the temporal decay of the OQC shows that oxygen adsorption on bare Pt patches is the rate limiting step of this dewetting process.image

WHAT IS...Aperiodic Order?

This article provides a brief introductory account of the theory of aperiodic order.

Measures with locally finite support and spectrum

The goal of this paper is the construction of measures μ on R(n)enjoying three conflicting but fortunately compatible properties: (i) μ is a sum of weighted Dirac masses on a locally finite set, (ii) the Fourier transform μ f μ is also a sum of weighted Dirac masses on a locally finite set, and (iii) μ is not a generalized Dirac comb. We give surprisingly simple examples of such measures. These unexpected patterns strongly differ from quasicrystals, they provide us with unusual Poisson's formulas, and they might give us an unconventional insight into aperiodic order.

A Metric Characterisation of Repulsive Tilings

A tiling of \({\mathbb {R}}^d\) is repulsive if no r-patch can repeat arbitrarily close to itself, relative to r. This is a characteristic property of aperiodic order, for a non-repulsive tiling has arbitrarily large locally periodic patterns. We consider a non-periodic, repetitive tiling T of \({\mathbb {R}}^d\), with finite local complexity. From a spectral triple built on the discrete hull \(\varXi \) of T and its Connes distance, we derive two metrics \(d_{\mathrm{sup}}\) and \(d_{\mathrm{inf}}\) on \(\varXi \). We show that T is repulsive if and only if \(d_{\mathrm{sup}}\) and \(d_{\mathrm{inf}}\) are Lipschitz equivalent. This generalises the previous works on subshifts by J. Kellendonk, D. Lenz, and the author.

Neutron Laue and X-ray diffraction study of a new crystallographic superspace phase in n-nonadecane-urea.

Aperiodic composite crystals present long-range order without translational symmetry. These materials may be described as the intersection in three dimensions of a crystal which is periodic in a higher-dimensional space. In such materials, symmetry breaking must be described as structural changes within these crystallographic superspaces. The increase in the number of superspace groups with the increase in the dimension of the superspace allows many more structural solutions. This is illustrated in n-nonadecane-urea, revealing a fifth higher-dimensional phase at low temperature.

Thermal Emission Control via Bandgap Engineering in Aperiodically Designed Nanophotonic Devices.

Aperiodic photonic crystals can open up novel routes for more efficient photon management due to increased degrees of freedom in their design along with the unique properties brought about by the long-range aperiodic order as compared to their periodic counterparts. In this work we first describe the fundamental notions underlying the idea of thermal emission/absorption control on the basis of the systematic use of aperiodic multilayer designs in photonic quasicrystals. Then, we illustrate the potential applications of this approach in order to enhance the performance of daytime radiative coolers and solar thermoelectric energy generators.

The role of fractal aperiodic order in the transmittance, conductance and electronic structure of graphene-based systems

In this work we investigate the peculiar tunneling characteristics of Dirac electrons through quasi-regular or aperiodic single-layer graphene structures. In particular, we have implemented the aperiodic order to the barriers and potential wells arranged according to the Cantor sequence. We deal with two types of quasi-regular systems: the first one Electrostatic Cantor Graphene Structures (ECGSs), structures formed with electrostatic potentials, and the second one Substrate Cantor Graphene Structures (SCGSs), obtained by alternating substrates that can open and non-open, such as SiC and SiO2, an energy bandgap on graphene. We have used the transfer matrix method to compute the transmission probability, linear-regime conductance and spectrum of bound states of both systems, which are calculated for different combinations of the fundamental parameters: starting width, sequence generation number, electron incident energy and electron incident angle. The transmission spectra of ECGSs and SCGSs show self-similar patterns, which are more remarkable when the generation number increases. The conductance of ECGSs becomes smoother as the generation increases. Furthermore, in the case of SCGSs the conductance exhibits self-similar features as a function of the generation number. In general, the conductance of both systems presents an oscillatory behaviour that can be described directly by means of the bound states.

Spectral Approximation for Quasiperiodic Jacobi Operators

Quasiperiodic Jacobi operators arise as mathematical models of quasicrystals and in more general studies of structures exhibiting aperiodic order. The spectra of these self-adjoint operators can be quite exotic, such as Cantor sets, and their fine properties yield insight into the associated quantum dynamics, that is, the one-parameter unitary group that solves the time-dependent Schrodinger equation. Quasiperiodic operators can be approximated by periodic ones, the spectra of which can be computed via two finite dimensional eigenvalue problems. Since long periods are necessary for detailed approximations, both computational efficiency and numerical accuracy become a concern. We describe a simple method for numerically computing the spectrum of a period-K Jacobi operator in O(K 2) operations, then use the algorithm to investigate the spectra of Schrodinger operators with Fibonacci, period doubling, and Thue–Morse potentials.

Special Issue: Aperiodic order and disorder

Special Issue: Aperiodic order and disorder

Phonon and phasons : from incommensurate phases to quasicrystals

Aperiodic crystals are long range ordered phases, which lack translational symmetry. They encompass incommensurately modulated phases, incommensurate composites phases and quasicrystals (1). Whereas their atomic structure is now well understood, even for the case of quasicrystals, the understanding of their physical properties remains a challenging problem. In particular, because of the aperiodic long range order, the lattice dynamics present a specific behavior. In particular, it can be shown theoretically that besides phonon, a supplementary excitation exits in all aperiodic phases named phason. Phason modes arise from the degeneracy of the free energy of the system with respect to a phase shift and are always diffusive modes (1). After a general introduction on the different class of aperiodic crystals, we will illustrate experimental results on phason modes. We will in particular demonstrate that these phason modes lead to a flexibility of the structure that might have important consequences for physical properties. We will also discuss their importance for the understanding of stabilizing mechanisms that lead to the long-range aperiodic order.

Aperiodic crystals and beyond

Following Dan Shechtman's discovery of quasicrystals in 1982, the realm of crystallography has been extended to include structures that lack translational periodicity. While periodic crystals can be modelled as decorations of lattices, aperiodic crystals require more general discrete structures such as point sets or tilings in space for the description of their structure. For a mathematical introduction to the field of aperiodic order, we refer to the recent monograph [1]. Interesting examples are obtained by projections from higher-dimensional lattices, leading to model sets which have pure Bragg diffraction, though the Bragg peaks are, in general, dense in space. All symmetries that have been experimentally observed in quasicrystals can be reproduced in this way, and some of the resulting structures are standard examples of tilings that are frequently used in quasicrystal modelling, such as the famous Penrose tiling. But there exists a plethora of ordered structures beyond cut and project sets, some of which have even weirder properties. After a general introduction to aperiodically ordered structures, a couple of examples of such systems are briefly described, offering a glimpse at the largely unexplored world of order beyond (aperiodic) crystals.