Two-dimensional Penrose Tiling Research Articles

Hyperuniformity classes of quasiperiodic tilings via diffusion spreadability.

Hyperuniform point patterns can be classified by the hyperuniformity scaling exponent α>0, that characterizes the power-law scaling behavior of the structure factor S(k) as a function of wave number k≡|k| in the vicinity of the origin, e.g., S(k)∼|k|^{α} in cases where S(k) varies continuously with k as k→0. In this paper, we show that the spreadability is an effective method for determining α for quasiperiodic systems where S(k) is discontinuous and consists of a dense set of Bragg peaks. It has been shown in [Phys. Rev. E 104, 054102 (2021)10.1103/PhysRevE.104.054102] that, for media with finite α, the long-time behavior of the excess spreadability S(∞)-S(t) can be fit to a power law of the form t^{-(d-α)/2}, where d is the space dimension, to accurately extract α for the continuous case. We first transform quasiperiodic and limit-periodic point patterns into two-phase media by mapping them onto packings of identical nonoverlapping disks, where space interior to the disks represents one phase and the space in exterior to them represents the second phase. We then compute the spectral density χ[over ̃]_{_{V}}(k) of the packings, and finally compute and fit the long-time behavior of their excess spreadabilities. Specifically, we show that the excess spreadability can be used to accurately extract α for the one-dimensional (1D) limit-periodic period-doubling chain (α=1) and the 1D quasicrystalline Fibonacci chain (α=3) to within 0.02% of the analytically known exact results. Moreover, we obtain a value of α=5.97±0.06 for the two-dimensional Penrose tiling and present plausible theoretical arguments strongly suggesting that α is exactly equal to six. We also show that, due to the self-similarity of the structures examined here, one can truncate the small-k region of the scattering information used to compute the spreadability and obtain an accurate value of α, with a small deviation from the untruncated case that decreases as the system size increases. This strongly suggests that one can obtain a good estimate of α for an infinite self-similar quasicrystal from a modestly sized finite sample. The methods described here offer a simple and general procedure to characterize accurately the large-scale translational order present in quasicrystalline and limit-periodic media in any space dimension that are self-similar. Moreover, the scattering information extracted from these two-phase media encoded in χ[over ̃]_{_{V}}(k), can be used to estimate their physical properties, such as their effective dynamic dielectric constants, effective dynamic elastic constants, and fluid permeabilities.

Exploiting Anyonic Behavior of Quasicrystals for Topological Quantum Computing

The concrete realization of topological quantum computing using low-dimensional quasiparticles, known as anyons, remains one of the important challenges of quantum computing. A topological quantum computing platform promises to deliver more robust qubits with additional hardware-level protection against errors that could lead to the desired large-scale quantum computation. We propose quasicrystal materials as such a natural platform and show that they exhibit anyonic behavior that can be used for topological quantum computing. Different from anyons, quasicrystals are already implemented in laboratories. In particular, we study the correspondence between the fusion Hilbert spaces of the simplest non-abelian anyon, the Fibonacci anyons, and the tiling spaces of the one-dimensional Fibonacci chain and the two-dimensional Penrose tiling quasicrystals. A concrete encoding on these tiling spaces of topological quantum information processing is also presented by making use of inflation and deflation of such tiling spaces. While we outline the theoretical basis for such a platform, details on the physical implementation remain open.

Constructing of Penrose tiling by means of the fractal of five-pointed stars

Fractal i0w(1b) (in author’s notation) of five-pointed stars is considered. The initiator of the fractal is a five-pointed star. When building the fractal, the prefractals of previous step are situated so as their centers coincide with the vertices of a five-pointed star. This star is called a “generalized star.” The size of the generalized star increases τ times at each step, where τ ≈ 1.618 is the golden mean. The orientation of the generalized star reverses at each step. A comparison of the internal parts of the fractal with the layers of the Wieringa roof (layers of Penrose tiling) shows their similarity. Computer simulation confirms that the two-dimensional Penrose tiling can be constructed from a set of four types of fractal i0w(1b).

Novel properties of frustrated low-dimensional magnets with pentagonal symmetry

We consider a new type of frustrated spin network with pentagonal loops and long-range quasiperiodic structural order. Five-fold loops can be expected to occur naturally in quasicrystals, and experimental studies of icosahedral alloys show manifestations of local five-fold symmetry in a variety of different physical contexts. Our model considers classical spins placed on the vertices of a subtiling of the two-dimensional Penrose tiling, and interacting with nearest neighbours via antiferromagnetic bonds. The ground state of this fractal system has a complex magnetic structure, and is studied analytically as well by Monte Carlo simulation on finite clusters.

Self-assembly-assisted growth of two-dimensional Penrose tiling

An algorithm for the growth of two-dimensional Penrose tiling, based on symmetry operations on a seed rhombus, is discussed and demonstrated. Independent of empirical matching rules, as suggested by Penrose [Bull. Inst. Math. Appl. 10 266 (1974)], and also overcoming the scaling-down operation, as proposed by Ramachandrarao et al. (Acta Crystallogr. A 47 210 (1991)], the present algorithm follows a mechanism akin to self-assembly and can be continued ad infinitum.

Structure model for icosahedral quasicrystals which consists of three types of unit cells without glue atoms

By using new decoration for the two-dimensional Penrose tiling, we construct a τ 4-scaling three-dimensional icosahedral structure consisting of three types of zonohedra; a triacontahedron, a star-like sixty-face rhombohedron and a rhombic icosahedron. The present 3D model extends only in a layer perpendicular to the five-fold axis of icosahedral symmetry with a finite thickness along the axis and there are few overlaps between the zonohedra. We calculate the electron diffraction pattern for the present model by putting atoms at all the vertices of the three types of zonohedral units. Reasonable agreement with the experimental results for i-AlMn phase is obtained.

Elasticity of a one-dimensional tiling model and its implication to the phason elasticity of quasicrystals

A one-dimensional tiling model with matching rule energy (antiferromagnetic Ising Hamiltonian) is studied. We present an analytic study of a transition from the unlocked phase, where free energy is proportional to the square gradient of the perp-space field $[f\ensuremath{\sim}(\ensuremath{\partial}{w)}^{2}],$ to the locked phase $(f\ensuremath{\sim}|\ensuremath{\partial}w|)$ in perp-space elasticity. The phase diagram and the temperature dependence of the elastic constant in the unlocked phase show similarity with the two-dimensional Penrose tiling. The results imply that the unlocking transition of a two-dimensional Penrose tiling model is related to the disordering transition in a one-dimensional Ising model.

Real-space renormalization group approach to the Potts model on the two-dimensional Penrose tiling

A one-step real-space renormalization group (RSRG) transformation is used to study the q-state Potts model on the two-dimensional (2D) Penrose tiling (PT). The critical exponents of the correlation length ν( q) in the region between q=1 and q=4 are obtained. The bond percolation (BP) threshold and Ising critical temperature in the isotropic case are also obtained. The comparison of the results with previous results on the PT and the exact results on square lattice (SQL) indicates that the universal classes of q-state Potts model on PT and SQL may be the same.

New perspectives on forbidden symmetries, quasicrystals, and Penrose tilings.

Quasicrystals are solids with quasiperiodic atomic structures and symmetries forbidden to ordinary periodic crystals-e.g., 5-fold symmetry axes. A powerful model for understanding their structure and properties has been the two-dimensional Penrose tiling. Recently discovered properties of Penrose tilings suggest a simple picture of the structure of quasicrystals and shed new light on why they form. The results show that quasicrystals can be constructed from a single repeating cluster of atoms and that the rigid matching rules of Penrose tilings can be replaced by more physically plausible cluster energetics. The new concepts make the conditions for forming quasicrystals appear to be closely related to the conditions for forming periodic crystals.

On the grey Penrose tiling

The construction of a two-dimensional and two-colour Penrose tiling known as a grey Penrose tiling is accomplished by using an algorithm comprising the geometrical operations akin to normal crystallographic operations. This method rules out the involvement of selection-making steps based on empirical matching rules as proposed by Li, Dubois and Kuo in a recent publication.

The real-space renormalization group and generating function for Penrose lattices

We present a real-space renormalization group (RG) approach to study the electronic properties of the two-dimensional Penrose tilings. This approach is based on a recursive evaluation of the generating function which gives the density of states (DOS) of the system. The RG equations are derived in terms of the inflation rules of the Penrose tilings and the calculated DOS matches the results derived by purely numerical methods.

A simple algebra and geometry in the two-dimensional Penrose tiling

A simple algebra and geometry in the two-dimensional Penrose tiling

Frustrated classical XY spin systems in the two-dimensional Penrose tiling

Magnetic ground states of frustrated classical XY spin systems in a finite two-dimensional Penrose tiling are determined by Monte Carlo simulations using the simulated annealing method. The interactions are limited to the third neighbours ( J 1, J 2, J 3) and the next nearest neighbour interaction is assumed to be antiferromagnetic. If frustration is low, the ground state is antiferromagnetic. If J 1 or J 3 is positive, increasing the absolute value of the negative exchange integral ( J 3 or J 1, respectively) which increases the local frustration does not break up the magnetic order but only broadens the two peaks of the spin angular distribution. There is evidence for a Kosterlitz-Thouless transition, as in periodic systems, in the thermal variation of the specific heat. This transition temperature is found to be independent of the nearest neighbour interaction, J 1, but varies linearly with the third neighbour one, J 3.

Local electronic properties of two-dimensional Penrose tilings: A renormalization-group approach.

A real-space renormalization-group (RSRG) scheme is developed to study the local electronic properties of the two-dimensional Penrose tilings. The RSRG transformations are derived in terms of the inflation rule of the Penrose tilings. Two basic transformations ${\mathit{T}}_{\mathrm{\ensuremath{\alpha}}}$ and ${\mathit{T}}_{\mathrm{\ensuremath{\beta}}}$ are introduced, of which ${\mathit{T}}_{\mathrm{\ensuremath{\alpha}}}$ transforms a rhombus (R) Penrose tiling to a kite-dart (KD) Penrose tiling, while ${\mathit{T}}_{\mathrm{\ensuremath{\beta}}}$ transforms a KD Penrose tiling to a R Penrose tiling. Suitable combinations of them give rise to two transformations ${\mathit{T}}_{1}$ and ${\mathit{T}}_{2}$, of which ${\mathit{T}}_{1}$ or ${\mathit{T}}_{2}$ transforms a R or KD Penrose tiling to the same kind of quasiperiodic tiling that is scaled by a factor of ${\mathrm{\ensuremath{\tau}}}^{2}$, in which \ensuremath{\tau}=(\ensuremath{\surd}5 +1)/2 is the golden mean. By infinite iterations of ${\mathit{T}}_{1}$ or ${\mathit{T}}_{2}$, we obtain the local Green's functions and the local densities of states (LDOS) at the key sites of the R or KD Penrose tilings. The LDOS at the key sites of the R Penrose tilings are numerically calculated and are consistent with the results obtained by others.

An algorithm to construct quasilattices and study percolation on them

This paper reports algorithms for: (a) computing the coordinates of the vortices in two-dimensional Penrose tilings of arbitrary size; (b) Monte Carlo calculation of the threshold probability and cluster statistics in percolation on such tilings (quasilattices). Results of implementing these algorithms into programmes are given relative to maximum cluster density ( M), mean cluster size ( S), and the finite-size scaling plots for M and S obtained directly from the cumulative distribution function of clusters.

Statistics of 'worms' in Penrose tilings

Certain structural aspects of two-dimensional Penrose tilings are studied using de Bruijn's pentagrid picture. The authors discuss the statistics of hexagons, decagons and 'worms' (sequences of adjacent hexagons bounded by two decagons). They show that within the discrete framework considered here, phason modes and structural transformation modes are located along particular 'worms' and they derive the spatial distribution of the latter.

Faceting and roughening in quasicrystals.

The question whether quasicrystal shapes should be faceted is studied in a simple model of quasicrystalline order. At T=0, the model is proved to yield a completely faceted equilibrium shape in both two and three dimensions. At T>0, an interface model is derived for a two-dimensional Penrose tiling. By mapping it onto a one-dimensional quasiperiodic Schr\"odinger equation, we show that the roughness exponent varies continuously with T at low T.

Electronic properties of a Penrose tiling lattice in a magnetic field

The one-electron energy spectrum of a two-dimensional Penrose tiling lattice in a uniform magnetic field is calculated as a function of magnetic fields with a tight-binding Hamiltonian. The calculated results show the following remarkable features characteristic of the Penrose lattice. (1) The density of states in a magnetic field has a central peak with zero width at the zero energy. It is shown that the zero-energy states correspond to the ring states in which the wavefunction has a non-vanishing amplitudes only at the sites on a ring-like region around the origin. (2) The energy levels coalesce into Landau type levels and the boundary states due to the finite size effects based on a fixed boundary condition appear in the gap region between Landau levels. (3) The magnetic field dependence of the energy spectrum has a repeated pattern of self-similarity with the golden mean ratio of two successive periods.

Magnetic phase structure on the Penrose lattice

The Ising model on a two-dimensional Penrose tiling is studied by means of the Migdal-Kadanoff scheme. This approximate renormalization method closely follows the inflation rules of the tiling, which are easily described in terms of Robinson triangles, and lead to the consideration of four types of nearest neighbor couplings. The ferromagnetic phase transition is similar to the usual one encountered on periodic lattices. When the couplings have both signs, the presence of frustration without randomness yields a fairly intricate phase diagram, essentially made up of two regions with a very complicated border. Region I consists of “quasiferromagnetic” models, which exhibit long-range order below some finite critical temperature. The models of region II are paramagnetic at nonzero (low) temperature, but may become ordered (reen-trant phases) in a higher temperature range.

A Quasicrystalline Decagonal Phase in Al-Fe-Ce

ABSTRACTA number of dispersed phases in Al-Fe-Ce alloys, varying structurally and chemically, have been observed by TEM. Of particular interest were rod-like particles which showed five-fold symmetry and irrational spacings. These dispersoids contain linear features, a periodic, columnar structure parallel to the five-fold axis, aperiodic spacings perpendicular to the five-fold axis and were coherent in one direction with the Al matrix. The evidence indicates a two-dimensional Penrose tiling with a columnar structure, possibly a stacking of eclipsed pentagonal units.