Observation of log-periodic oscillations in the quantum dynamics of electrons on the one-dimensional Fibonacci quasicrystal (original) (raw)

Electronic spectra of quasi-regular Fibonacci systems: Analysis of simple 1D models

Microelectronics Journal, 2005

The electronic spectra of quasi-regular systems grown following the Fibonacci sequence are investigated via simple one-dimensional tight-binding, one-band models. Different models are considered and the influence of the model parameters and the number of atoms entering the different blocks on the electronic spectrum are discussed.

Electronic energy spectra of square and cubic Fibonacci quasicrystals

Philosophical Magazine, 2008

Understanding the electronic properties of quasicrystals, in particular the dependence of these properties on dimension, is among the interesting open problems in the field of quasicrystals. We investigate an off-diagonal tight-binding hamiltonian on the separable square and cubic Fibonacci quasicrystals. We use the well-studied singular-continuous energy spectrum of the 1-dimensional Fibonacci quasicrystal to obtain exact results regarding the transitions between different spectral behaviors of the square and cubic quasicrystals. We use analytical results for the addition of 1d spectra to obtain bounds on the range in which the higher-dimensional spectra contain an absolutely continuous component. We also perform a direct numerical study of the spectra, obtaining good results for the square Fibonacci quasicrystal, and rough estimates for the cubic Fibonacci quasicrystal.

Quasiperiodic Krönig-Penney Model on a Fibonacci Superlattice

We study the propagation of electrons in a one-dimensional Krönig-Penney model on a Fibonacci superlattice. We consider two different models, one with ô-function potentials and the other with finite potential barriers. The nature of the eigenstates, the integrated density of states and inverse localization length, as well as transmission and reflectance are investigated. In contrast to periodic arrangements, we observe a self-similar structure in the wavenumber or energy dependence of the physical quantities exhibiting scaling properties and multifractal behavior.

Spectral and Quantum Dynamical Properties of the Weakly Coupled Fibonacci Hamiltonian

Communications in Mathematical Physics, 2011

We consider the spectrum of the Fibonacci Hamiltonian for small values of the coupling constant. It is known that this set is a Cantor set of zero Lebesgue measure. Here we study the limit, as the value of the coupling constant approaches zero, of its thickness and its Hausdorff dimension. We prove that the thickness tends to infinity and, consequently, the Hausdorff dimension of the spectrum tends to one. We also show that at small coupling, all gaps allowed by the gap labeling theorem are open and the length of every gap tends to zero linearly. Moreover, for a sufficiently small coupling, the sum of the spectrum with itself is an interval. This last result provides a rigorous explanation of a phenomenon for the Fibonacci square lattice discovered numerically by Even-Dar Mandel and Lifshitz. Finally, we provide explicit upper and lower bounds for the solutions to the difference equation and use them to study the spectral measures and the transport exponents. Contents

Electronic energy spectra and wave functions on the square Fibonacci tiling

Philosophical Magazine, 2006

We study the electronic energy spectra and wave functions on the square Fibonacci tiling, using an off-diagonal tight-binding model, in order to determine the exact nature of the transitions between different spectral behaviors, as well as the scaling of the total bandwidth as it becomes finite. The macroscopic degeneracy of certain energy values in the spectrum is invoked as a possible mechanism for the emergence of extended electronic Bloch wave functions as the dimension changes from one to two.

Specific heat spectra of non-interacting fermions in a quasiperiodic ladder sequence

Physics Letters A, 2008

We compute the specific heat spectra of non-interacting fermions whose energy spectrum was obtained from a quasiperiodic ladder sequence (Fibonacci and Rudin-Shapiro type), mimicking a DNA molecule model. The specific heat is calculated from their underlying multi-fractal energy spectrum, considering several values of energy densities. Comparisons are made with a real DNA sequence, namely the human chromosome 22 (Ch22).

Electronic spectra of one-dimensional nano-quasi-periodic systems under bias

Superlattices and Microstructures, 2010

We investigate the properties of the energy spectra of quantum one-dimensional nano-quasi-crystals in the presence of external electric fields. These systems are modelled by means of finite sequences, ordered according to a Fibonacci rule which constituted of two blocks A (constant potentials of different heights defined on finite intervals) and B (delta potentials of different intensities). We use the electric field ability of producing Stark ladders in periodic systems to obtain well separated energy levels and to study the evolution of these levels when disorder is introduced. We show that this effect also allows us to predict the approximate position of the levels in the disordered system, in spite of its chaotic appearance at first view. We show, against the usual belief, that the nth Stark ladder in general is not formed exclusively from the levels of the nth band. The disorder is introduced in two different ways: by changing the distribution of the blocks or by changing the values of the delta potential intensities. In both cases we start from electrified periodic structures which are gradually perturbed to obtain electrified quasi-periodic structures. We show that the use of Fibonacci sequences as a particular case is not crucial and one can use the electric field to analyze any other type of quasiperiodic systems.

The electronic properties of a Fibonacci chain

Using a tight-binding model and transfer-matrix technique, as well as Lanczos algorithm, we numerically investigate the nature of the electronic states and electron transmission in site, bond and mixing Fibonacci model chains. We rely on the Landauer formalism as the basis for studying the conduction properties of these systems. Calculating the Lyapunov exponent, we also study the localization properties of electronic eigenstates in the Fibonacci chains. The focus is on the significance of the relationship between the transmission spectra and the nature of the electronic states. Our results show that, in contrast to Anderson's localization theorem, in the Fibonacci chains the electronic states are non-localized and the transparent states occurr near the Fermi level.

Quantum intermittency in almost-periodic lattice systems derived from their spectral properties

Physica D: Nonlinear Phenomena, 1997

Hamiltonian tridiagonal matrices characterized by multi-fractal spectral measures in the family of Iterated Function Systems can be constructed by a recursive technique here described. We prove that these Hamiltonians are almost-periodic. They are suited to describe quantum lattice systems with nearest neighbours coupling, as well as chains of linear classical oscillators, and electrical transmission lines. We investigate numerically and theoretically the time dynamics of the systems so constructed. We derive a relation linking the long-time, power-law behaviour of the moments of the position operator, expressed by a scaling function β of the moment order α, and spectral multi-fractal dimensions, D q , via β(α) = D 1−α. We show cases in which this relation is exact, and cases where it is only approximate, unveiling the reasons for the discrepancies.