Wavefunctions, quantum diffusion, and scaling exponents in golden-mean quasiperiodic tilings

Thiem, Stefanie; Schreiber, Michael

doi:10.1088/0953-8984/25/7/075503

Cited by 23 publications

(30 citation statements)

References 74 publications

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“…It is well known that the spectrum of the Fibonacci Hamiltonian is fractal [37], as illustrated in the top panel of fig. (2) which shows the integrated density of states, idos(E) defined by the fraction of states of energy smaller than E. The fractal dimensions of the spectrum can be computed in the limits ρ ∼ 1 [31] and ρ 1 [27]. The structure of the eigenstates is also well understood in these two limits [19,21,32,39]. Away from these limits, however, the structure of the eigenstates is not known, with the notable exception of the E = 0 state at the center of the spectrum shown in fig.…”

Section: A the Fibonacci Chain And Hopping Hamiltonianmentioning

confidence: 99%

Critical eigenstates and their properties in one- and two-dimensional quasicrystals

et al. 2017

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We present exact solutions for some eigenstates of hopping models on one and two dimensional quasiperiodic tilings and show that they are "critical" states, by explicitly computing their multifractal spectra. These eigenstates are shown to be generically present in 1D quasiperiodic chains, of which the Fibonacci chain is a special case. We then describe properties of the ground states for a class of tight-binding Hamiltonians on the 2D Penrose and Ammann-Beenker tilings. Exact and numerical solutions are seen to be in good agreement.In quasicrystals, the arrangement of atoms is nonperiodic yet long-range ordered. This type of quasiperiodic order is best illustrated by considering tilings. These are structures obtained by packing a small number of basic elements, or tiles, according to certain specified rules. Quasiperiodic tilings can exhibit non-crystallographic symmetries: in this article we will focus on the AmmannBeenker tiling ( fig. (1)) that has a eight-fold orientational symmetry, and on the celebrated Penrose tiling which has a five-fold symmetry.Single-electron properties of 1D quasicrystals are rather well understood: many spectral properties of quasiperiodic chains are known, and the eigenstates are also well characterized. In contrast, even the simplest models of 2 and 3 dimensional quasicrystals have resisted theoretical investigations. Consider for example the Ammann-Beenker tiling ( fig. (1)), and a tightbinding model for non-interacting electrons hopping between nearest neighbor vertices on this tiling:Although this Hamiltonian is simple, no solutions were known for any of its eigenstates, apart from trivial confined eigenstates at the middle of the spectrum [17]. Taking their cue from periodic crystals, where Bloch states are given by ψ k (r) = u k (r)e ik.r with u a periodic function, Kalugin and Katz [15] proposed that the ground state of the model (1) be given by a product of two factors, namelyIn this expression, κ is a real constant (note that in [15] the notation λ = exp(2κ) is used). The pre-exponential factor C(i) of the ansatz (2) is a quasiperiodic function, a site-dependent amplitude which depends only on the arrangement of the atoms around the site i. In other words, C(i) C(j) if the arrangement of the atoms around site i matches the arrangement of the atoms around site j out to a large distance. The non-local height field h(i) in the exponential depends on the geometrical properties of the tiling. We will refer to eigenstates of this form henceforth as Sutherland-Kalugin-Katz -or SKK eigenstates.In this article, we consider generalizations of the results of Kalugin and Katz to other tight-binding models. We consider first the relatively simple case of models on 1D quasiperiodic chains, and show that they admit eigenstates of a form similar to that given in Eq. (2). We next consider the 2D case, for the Ammann-Beenker and Penrose tilings. and we show that the ground states continue to have the SKK structure even when the Hamiltonian in Eq.(1) is generalized to include onsite potenti...

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Section: A the Fibonacci Chain And Hopping Hamiltonianmentioning

confidence: 99%

Critical eigenstates and their properties in one- and two-dimensional quasicrystals

et al. 2017

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“…Thus the wavefunctions are not multifractal at this order in ρ and multifractality appears only at the nextto-leading order, as discussed in the next section. This first-order description of the wavefunctions has been compared to numerical results in [20], where the agreement was found not to be very good. We argue that this is because the wavefunctions becomes rapidly multifractal as ρ is increased.…”

Section: B Fractal Dimensions Of the Wavefunctionsmentioning

confidence: 99%

Fractal dimensions of wave functions and local spectral measures on the Fibonacci chain

2016

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We present a theoretical framework for understanding the wavefunctions and spectrum of an extensively studied paradigm for quasiperiodic systems, namely the Fibonacci chain. Our analytical results, which are obtained in the limit of strong modulation of the hopping amplitudes, are in good agreement with published numerical data. In the perturbative limit, we show a new symmetry of wavefunctions under permutation of site and energy indices. We compute the wavefunction renormalization factors and from them deduce analytical expressions for the fractal exponents corresponding to individual wavefunctions, as well as their global averages. The multifractality of wavefunctions is seen to appear at next-to-leading order in ρ. Exponents for the local spectral density are given, in extremely good accord with numerical calculations. Interestingly, our analytical results for exponents are observed to describe the system rather well even for values of ρ well outside the domain of applicability of perturbation theory.Introduction As distinct from periodic crystals on the one hand, where electronic states are typically extended, and disordered systems on the other hand, where states are typically localized for low dimension and/or strong enough disorder, electronic states in quasicrystals are believed to have an intermediate "critical" character. The study of tight binding models on the Fibonacci chain, a one dimensional paradigm for quasicrystalline structures, is particularly important, as a first step towards understanding the physics of these systems. These models have been extensively investigated theoretically, as in [1][2][3][4][5][6]. There have also been many experimental studies of electronic properties of this model. To cite some recent works, in [7] investigated transport due to topologically protected edge states in a Fibonacci photonic waveguide. In [8], the density of states of a Fibonacci tight-binding model was studied by direct observation of polariton modes in a one-dimensional cavity, and shown to have the fractal structure, log-periodic oscillations and gaps labeled as predicted by theory [4].

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“…By keeping in mind the main results reviewed in this work concerning the spatial structure of the wave functions belonging to fractal and QPS, it could be concluded that the recent trend of abandoning the former use of "critical wave function" to generically refer to all eigenstates belonging to singular continuous spectra should be encouraged [92]. In fact, whereas the electronic states found in the Aubry-André self-dual model can be properly regarded as being strictly critical ones (in the sense that they undergo a metal-insulator critical transition) this criterion no longer holds for the states found in both fractal and QPS, albeit all of them display multifractal properties.…”

Section: Outlook and Perspectivesmentioning

confidence: 99%

On the Nature of Electronic Wave Functions in One-Dimensional Self-Similar and Quasiperiodic Systems

Maciá

2014

ISRN Condensed Matter Physics

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The interest in the precise nature of critical states and their role in the physics of aperiodic systems has witnessed a renewed interest in the last few years. In this work we present a review on the notion of critical wave functions and, in the light of the obtained results, we suggest the convenience of some conceptual revisions in order to properly describe the relationship between the transport properties and the wave functions distribution amplitudes for eigen functions belonging to singular continuous spectra related to both fractal and quasiperiodic distribution of atoms through the space.

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Wavefunctions, quantum diffusion, and scaling exponents in golden-mean quasiperiodic tilings

Cited by 23 publications

References 74 publications

Critical eigenstates and their properties in one- and two-dimensional quasicrystals

Critical eigenstates and their properties in one- and two-dimensional quasicrystals

Fractal dimensions of wave functions and local spectral measures on the Fibonacci chain

On the Nature of Electronic Wave Functions in One-Dimensional Self-Similar and Quasiperiodic Systems

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