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

Abstract

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].

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

“…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.…”

“…We recall that this state can be built using the so-called trace-map method, used for instance in [8] to obtain a description of this state, and in particular to compute its fractal dimensions. The fractal dimensions of this state can also be computed exactly using a renormalizationgroup approach [19]. We wish to show now that the E = 0 state is an SKK like state (2).…”

Diffusive transport in a quasiperiodic Fibonacci chain: Absence of many-body localization at weak interactions

“…However, also within quasiperiodic systems the Fibonacci model offers several important advantages going beyond the much more commonly studied AAH model. The noninteracting model is critical, showing eigensystem (multi)fractality [8][9][10][11]60 , for any potential amplitude and therefore serves as one of the simplest deterministic systems with anomalous transport 61 in closed and open setting 62 . We note that (multi) fractality is typical at Anderson transitions 63 .…”

Gap structure of 1D cut and project Hamiltonians

“…Using the recursive spectrum construction of figure (2) we see that G l can be partitioned into 3 subsets containing the gap labels of the left, central and right clusters of energy bands. We have the following recursive relations [9]:…”