Equivalence classes of Fibonacci lattices and their similarity properties

Abstract

We investigate, theoretically and experimentally, the properties of Fibonacci lattices with arbitrary spacings. Different from periodic structures, the reciprocal lattice and the dynamical properties of Fibonacci lattices depend strongly on the lengths of their lattice parameters, even if the sequence of long and short segment, the Fibonacci string, is the same. In this work we show that by exploiting a self-similarity property of Fibonacci strings under a suitable composition rule, it is possible to define equivalence classes of Fibonacci lattices. We show that the diffraction patterns generated by Fibonacci lattices belonging to the same equivalence class can be rescaled to a common pattern of strong diffraction peaks thus giving to this classification a precise meaning. Furthermore we show that, through the gap labeling theorem, gaps in the energy spectra of Fibonacci crystals belonging to the same class can be labeled by the same momenta (up to a proper rescaling) and that the larger gaps correspond to the strong peaks of the diffraction spectra. This observation makes the definition of equivalence classes meaningful also for the spectral and therefore dynamical and thermodynamical properties of quasicrystals. Our results apply to the more general class of quasiperiodic lattices for which similarity under a suitable deflation rule is in order

Emergence of anomalous dynamics from the underlying singular continuous spectrum in interacting many-body systems

“…It is interesting to compare these features with those of the on-site Fibonacci model (OFM), showing a purely SC energy spectrum [36] induced by its quasiperiodic geometry [37,38] and displaying no phase transition. The OFM is obtained by setting n = λ( (n + 1)/τ − n/τ ) in Eq.…”

Dynamics and energy spectra of aperiodic discrete-time quantum walks

“…3 that the effect of different coin operations on different sites is to open new gaps inside the two above mentioned main energy bands, thus creating sub-bands. The position of these new gaps can be determined in first order perturbation theory (i.e., considering the case |θ 2 − θ 1 | ≈ 0) by finding that they open where the intensity of the Fourier transform of w(x) is higher [41][42][43][44]. Within the two main bands the structure is therefore determined by the spatial distribution of the two coins and consequently by the geometry of the chosen arrangement.…”

A study of the brightest peaks in the diffraction pattern of Fibonacci gratings

“…The diffracted intensity is therefore the product of a structure term that contains the details of the interaction between the scattered wavefield and the potential, and a second term with the information on the lattice geometry, which in this context plays a role analogous to the Laue sums in the kinematical theory of x-ray diffraction. In our analysis we will drop for the moment the term (3) and focus our attention on the 'geometrical' term, equation (4).…”

“…Moreover for aperiodic structures, the intensity of the peaks is influenced not only by the structural factor, but also by the geometry of the lattice. This means that two aperiodic structures, even sharing the same topology, but with slightly different lattice parameters, are characterized by completely different diffraction patterns: not only the positions, but the intensity of the diffraction peaks is strongly changed in an unpredictable way [4]. The diffraction pattern of aperiodic structures therefore consists of a complicated sequence of strong maxima emerging from a background of noisy smaller peaks.…”