Renormalization Group for the Octagonal Quasi-Periodic Tiling (original) (raw)
Quantum dynamics in two- and three-dimensional quasiperiodic tilings
Physical Review B, 2002
We investigate the properties of electronic states in two and three-dimensional quasiperiodic structures : the generalized Rauzy tilings. Exact diagonalizations, limited to clusters with a few thousands sites, suggest that eigenstates are critical and more extended at the band edges than at the band center. These trends are clearly confirmed when we compute the spreading of energy-filtered wavepackets, using a new algorithm which allows to treat systems of about one million sites. The present approach to quantum dynamics, which gives also access to the low frequency conductivity, opens new perspectives in the analyzis of two and three-dimensional models.
Geometric study of a 2D tiling related to the octagonal quasiperiodic tiling
Journal De Physique, 1989
2014 Au moyen de trois tuiles, nous construisons un pavage quasipériodique du plan, que nous relions au quasicristal octogonal. Ainsi, nous montrons que les coordonnées des n0153uds peuvent être obtenues de deux manières différentes. Le facteur de structure est calculé exactement. Ce pavage qui possède « presque » une symétrie d'ordre huit, soulève la difficulté de la détermination pratique de la symétrie d'un quasicristal. Finalement, nous montrons comment construire une large classe de pavage du type de l'octogonal, à partir de ce nouveau pavage.
Quantum dynamics in high codimension tilings: From quasiperiodicity to disorder
Physical Review B, 2003
We analyze the spreading of wavepackets in two-dimensional quasiperiodic and random tilings as a function of their codimension, i.e. of their topological complexity. In the quasiperiodic case, we show that the diffusion exponent that characterizes the propagation decreases when the codimension increases and goes to 1/2 in the high codimension limit. By constrast, the exponent for the random tilings is independent of their codimension and also equals 1/2. This shows that, in high codimension, the quasiperiodicity is irrelevant and that the topological disorder leads in every case, to a diffusive regime, at least in the time scale investigated here. PACS numbers: 61.44.Br, 71.23.Ft It is now well established that quasiperiodic order has a strong influence on the quantum dynamics of wavepackets. Indeed, the nature of the eigenstates in quasiperiodic systems, which are neither spatially extended (as in periodic systems) nor localized (as in disordered systems) but critical, is often responsible for a sub-ballistic motion. Although most of the studies about this anomalous diffusion concern one-dimensional systems such as the Fibonacci or the Harper chain, there has also been a great interest for the, more physical, higher-dimensional ones . However, the parameters that determine the characteristics of the long time dynamics, such as the diffusion exponent β, remains misunderstood.
Statistical mechanics of dimers on quasiperiodic tilings
2021
Jerome Lloyd, 2, 3, ∗ Sounak Biswas, ∗ Steven H. Simon, S. A. Parameswaran, and Felix Flicker 4 Rudolf Peierls Centre for Theoretical Physics, Parks Road, Oxford OX1 3PU, United Kingdom School of Physics and Astronomy, University of Birmingham, Edgbaston Park Road, Birmingham, B15 2TT, United Kingdom Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße, Dresden, 01187, Germany School of Physics and Astronomy, Cardiff University, The Parade, Cardiff CF24 3AA, United Kingdom
Winding Numbers and Topology of Aperiodic Tilings
arXiv (Cornell University), 2021
We show that diffraction features of 1D quasicrystals can be retrieved from a single topological quantity, the Čech cohomology group, Ȟ1 ∼ = Z 2 , which encodes all relevant combinatorial information of tilings. We present a constructive way to calculate Ȟ1 for a large variety of aperiodic tilings. By means of two winding numbers, we compare the diffraction features contained in Ȟ1 to the gap labeling theorem, another topological tool used to label spectral gaps in the integrated density of states. In the light of this topological description, we discuss similarities and differences between families of aperiodic tilings, and the resilience of topological features against perturbations.
A geometrical approach of quasiperiodic tilings
Communications in Mathematical Physics, 1988
Tilings provide generalized frames of coordinates and as such they are used in different areas of physics. The aim of the present paper is to present a unified and systematic description of a class of tilings which have appeared in contexts as disconnected as crystallography and dynamical systems. The tilings of this class show periodic or quasiperiodic ordering and the tiles are related to the unit cube through affine transformations. We present a section procedure generating canonical quasiperiodic tilings and we prove that true tilings are indeed obtained. Moreover, the procedure provides a direct and simple characterization of quasiperiodicity which is suitable for tilings but which does not refer to Fourier transform.
Numerical investigation of electronic wave functions in quasiperiodic lattices
Journal of Physics: Condensed Matter, 1998
We study electronic eigenstates on quasiperiodic lattices using a tight-binding Hamiltonian in the vertex model. In particular, the two-dimensional Penrose tiling and the three-dimensional icosahedral Ammann-Kramer tiling are considered. Our main interest concerns the decay form and the self-similarity of the electronic wave functions, which we compute numerically for periodic approximants of the perfect quasiperiodic structure. In order to investigate the suggested power-law localization of states, we calculate their participation numbers and structural entropy. We also perform a multifractal analysis of the eigenstates by standard box-counting methods. Our results indicate a rather different behaviour of the two-and the three-dimensional systems. Whereas the eigenstates on the Penrose tiling typically show power-law localization, this was not observed for the icosahedral tiling.
Sub-diffusive electronic states in octagonal tiling
Journal of Physics: Conference Series, 2017
We study the quantum diffusion of charge carriers in octagonal tilings. Our numerical results show a power law decay of the wave-packet spreading, L(t) ∝ t β , characteristic of critical states in quasicrystals at large time t. For many energies states are sub-diffusive, i.e. β < 0.5, and thus conductivity increases when the amount of defects (static defects and/or temperature) increases.
Physica A: Statistical Mechanics and its Applications, 1992
We generate a quasiperiodic, dodecagonally symmetric tiling of the plane by squares and equilateral triangles embedded in a higher-dimensional periodic structure. Starting from a 4D lattice frequently used for the embedding of dodecagonal structures, we iteratively construct an acceptance domain (AD) for a quasiperiodic dodecagonal point set which proves to be the vertex set of a square-triangle tiling. It turns out that our procedure leads to fractally bounded ADS but leaves enough freedom to generate several different local isomorphism classes.
On the geometry of ground states and quasicrystals for lattice systems
arXiv: Mathematical Physics, 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.
Critical wave functions and a Cantor-set spectrum of a one-dimensional quasicrystal model
Physical Review B, 1987
The electronic properties of a tight-binding model which possesses two types of hopping matrix element (or on-site energy) arranged in a Fibonacci sequence are studied. The wave functions are either self-similar (fractal) or chaotic and show "critical" (or "exotic") behavior. Scaling analysis for the self-similar wave functions at the center of the band and also at the edge of the band is performed. The energy spectrum is a Cantor set with zero Lebesque measure. The density of states is singularly concentrated with an index ae which takes a value in the range [cte'", az'"]. The fractal dimensions f(ae) of these singularities in the Cantor set are calculated. This function f(ae) represents the global scaling properties of the Cantor-set spectrum.
Equilibrium quasicrystal phase of a Penrose tiling model
Physical Review B, 1990
A two-dimensional rhombus tiling model with a matching-rule-based energy is analyzed using real-space renormalization-group methods and Monte Carlo simulations. The model spans a range from T=O quasiperiodic crystal (Penrose tiling) to a random-tiling quasicrystal at high temperatures. A heuristic picture for the disordering of the ground-state quasiperiodicity at low temperatures is proposed and corroborated with exact and renormalization-group calculations of the phason elastic energy, which shows a linear dependence on the strain at T=O but changes to a quadratic behavior at T&0 and sufficiently small strain. This is further supported by the Monte Carlo result that phason fluctuations diverge logarithmically with system size for all T &0, which indicates the presence of quasi-long-range translational order in the system, meaning algebraically decaying correlations. A close connection between the rhombus tiling model and the general surface-roughening phenomena is established. Extension of the results to three dimensions and their possible implication to experimental systems is also addressed.
Random-tiling quasicrystal in three dimensions
Physical Review Letters, 1990
A three-dimensional random-tiling icosahedral quasicrystal is studied by a Monte Carlo simulation. The hypothesis of long-range positional order in the system is confirmed through analysis of the finitesize scaling behavior of phason fluctuations and Fourier peak intensities. By investigating the diffuse scattering we determine the phason stiffness constants. A finite-size scaling form for the Fourier intensity near an icosahedral reciprocal wave vector is proposed.
Relating diffraction and spectral data of aperiodic tilings: Towards a Bloch theorem
Journal of Geometry and Physics, 2021
The purpose of this paper is to show the relationship in all dimensions between the structural (diffraction pattern) aspect of tilings (described by Čech cohomology of the tiling space) and the spectral properties (of Hamiltonians defined on such tilings) defined by K-theory, and to show their equivalence in dimensions ≤ 3. A theorem makes precise the conditions for this relationship to hold. It can be viewed as an extension of the "Bloch Theorem" to a large class of aperiodic tilings. The idea underlying this result is based on the relationship between cohomology and K-theory traces and their equivalence in low dimensions.
Tight-binding models in a quasiperiodic optical lattice
Acta Physica Polonica Series a
We report on a two-dimensional quasiperiodic structure with eight-fold symmetry obtained by trapping atoms in an optical potential, for appropriately tuned experimental parameters. We briefly describe the geometrical properties of the structure, and comment on the tight-binding models for particles moving in this potential, closely related to models which have been studied on the well-known quasiperiodic octagonal (or Ammann-Beenker) tiling.
Group-theoretical analysis of aperiodic tilings from projections of higher-dimensional lattices Bn
Acta crystallographica. Section A, Foundations and advances, 2015
A group-theoretical discussion on the hypercubic lattice described by the affine Coxeter-Weyl group Wa(Bn) is presented. When the lattice is projected onto the Coxeter plane it is noted that the maximal dihedral subgroup Dh of W(Bn) with h = 2n representing the Coxeter number describes the h-fold symmetric aperiodic tilings. Higher-dimensional cubic lattices are explicitly constructed for n = 4, 5, 6. Their rank-3 Coxeter subgroups and maximal dihedral subgroups are identified. It is explicitly shown that when their Voronoi cells are decomposed under the respective rank-3 subgroups W(A3), W(H2) × W(A1) and W(H3) one obtains the rhombic dodecahedron, rhombic icosahedron and rhombic triacontahedron, respectively. Projection of the lattice B4 onto the Coxeter plane represents a model for quasicrystal structure with eightfold symmetry. The B5 lattice is used to describe both fivefold and tenfold symmetries. The lattice B6 can describe aperiodic tilings with 12-fold symmetry as well as a...