A new Hamiltonian model of the Fibonacci quasicrystal using non-local interactions: simulations and spectral analysis (original) (raw)

Methods for Calculating Empires in Quasicrystals

This paper reviews the empire problem for quasiperiodic tilings and the existing methods for generating the empires of the vertex configurations in quasicrystals, while introducing a new and more efficient method based on the cut-and-project technique. Using Penrose tiling as an example, this method finds the forced tiles with the restrictions in the high dimensional lattice (the mother lattice) that can be cut-and-projected into the lower dimensional quasicrystal. We compare our method to the two existing methods, namely one method that uses the algorithm of the Fibonacci chain to force the Ammann bars in order to find the forced tiles of an empire and the method that follows the work of N.G. de Bruijn on constructing a Penrose tiling as the dual to a pentagrid. This new method is not only conceptually simple and clear, but it also allows us to calculate the empires of the vertex configurations in a defected quasicrystal by reversing the configuration of the quasicrystal to its higher dimensional lattice, where we then apply the restrictions. These advantages may provide a key guiding principle for phason dynamics and an important tool for self error-correction in quasicrystal growth.

Quasiperiodic graphs: structural design, scaling and entropic properties

Arxiv preprint arXiv: …, 2012

A novel class of graphs, here named quasiperiodic, are constructed via application of the Horizontal Visibility algorithm to the time series generated along the quasiperiodic route to chaos. We show how the hierarchy of mode-locked regions represented by the Farey tree is inherited by their associated graphs. We are able to establish, via Renormalization Group (RG) theory, the architecture of the quasiperiodic graphs produced by irrational winding numbers with pure periodic continued fraction. And finally, we demonstrate that the RG fixed-point degree distributions are recovered via optimization of a suitably defined graph entropy.

Empires: The Nonlocal Properties of Quasicrystals

In Tech Open, 2019

In quasicrystals, any given local patch-called an emperor-forces at all distances the existence of accompanying tiles-called the empire-revealing thus their inherent nonlocality. In this chapter, we review and compare the methods currently used for generating the empires, with a focus on the cut-and-project method, which can be generalized to calculate empires for any quasicrystals that are projections of cubic lattices. Projections of non-cubic lattices are more restrictive and some modifications to the cut-and-project method must be made in order to correctly compute the tilings and their empires. Interactions between empires have been modeled in a game-of-life approach governed by nonlocal rules and will be discussed in 2D and 3D quasicrystals. These nonlocal properties and the consequent dynamical evolution have many applications in quasicrystals research, and we will explore the connections with current material science experimental research.

Non-Local Game of Life in 2D Quasicrystals

MDPI Crystals Journal , 2018

On a two-dimensional quasicrystal, a Penrose tiling, we simulate for the first time a game of life dynamics governed by non-local rules. Quasicrystals have inherently non-local order since any local patch, the emperor, forces the existence of a large number of tiles at all distances, the empires. Considering the emperor and its local patch as a quasiparticle, in this case a glider, its empire represents its field and the interaction between quasiparticles can be modeled as the interaction between their empires. Following a set of rules, we model the walk of life in different setups and we present examples of self-interaction and two-particle interactions in several scenarios. This dynamic is influenced by both higher dimensional representations and local choice of hinge variables. We discuss our results in the broader context of particle physics and quantum field theory, as a first step in building a geometrical model that bridges together higher dimensional representations, quasicrystals and fundamental particles interactions.

Quasicrystal Tilings in Three Dimensions and Their Empires

The projection method for constructing quasiperiodic tilings from a higher dimensional lattice provides a useful context for computing a quasicrystal's vertex configurations, frequencies, and empires (forced tiles). We review the projection method within the framework of the dual relationship between the Delaunay and Voronoi cell complexes of the lattice being projected. We describe a new method for calculating empires (forced tiles) which also borrows from the dualisation formalism and which generalizes to tilings generated projections of non-cubic lattices. These techniques were used to compute the vertex configurations, frequencies and empires of icosahedral quasicrystals obtained as a projections of the D 6 and Z 6 lattices to R 3 and we present our analyses. We discuss the implications of this new generalization.

The Nature of the Phonon Eigenstates in Quasiperiodic Chains (The Role of Fibonacci Lattices)

Iranian Journal of Physics Research, 2005

Using the forced oscillator method (FOM) and the transfer-matrix technique, we numerically investigate the nature of the phonon states and the wave propagation, in the presence of an external force, in the chains composed of Fibonacci lattices of type site, bond and mixing models, as the quasiperiodic systems. Calculating the Lyapunov exponent and the participation ratio, we also study the localization properties of phonon eigenstates in these chains. The focus is on the significant relationship between the transmission spectra and the nature of the phonon states. Our results show that in the presence of the Fibonacci lattices, at low and medium frequencies the spectra of the quasiperiodic systems are not much different from those of the periodic ones and the corresponding phonon eigenstates are extended. However, the numerical results of the calculations of the transmission coefficient ) (ω T , the inverse Lyapunov exponent 1 ) ( − ω Γ and the participation ratio ) (ω PR show that ...

Hierarchical structure of a one-dimensional quasiperiodic model

Physical Review B, 1988

A hierarchical structure of a Fibonacci chain is obtained exactly. This is done by transforming the original chain into another one in which the ne~hopping matrix elements and on-site energies are arranged hierarchically. An exact renormalization-group transformation is derived for the hierarchical chain.

On the geometry of ground states and quasicrystals for lattice systems

2008

We propose a geometric point of view to study the structure of ground states in lattice models, especially those with 'non-periodic long-range order' which can be seen as toy models for quasicrystals. In a lattice model, the configuration space is S Z d where S is a finite set, and Θ denotes action of the group Z d by translation or 'shift'. Given a shift-invariant potential Φ, ground states are none other than the shift-invariant probability measures supported on the set of ground configurations of that potential, i.e., those configurations with minimal specific energy. The ground states of Φ are supported by a multi-dimensional subshift of the d-dimensional 'full shift' (S Z d , Θ). This subshift may be minimal or not, uniquely ergodic or not, and with entropy zero or not. These three properties are different notions of order. For a finite-range potential, the subshift carrying its ground states configurations is a shift of finite type (SFT). The corresponding ground states are then naturally associated with the boundary a certain finite-dimensional convex polytope. This boundary becomes drastically different from d = 1 to d ≥ 2. This is because when d ≥ 2 there exist uniquely ergodic SFT's with no periodic configurations which can be seen as toy-models of quasicrystals. Here we construct such an example from the Penrose tiling. The framework we propose may help to investigate the stability of such models of quasicrystals when one perturbs the ad hoc potential one can always construct to have a given uniquely ergodic SFT as its ground state.