Spectral properties of two coupled Fibonacci chains

Abstract

The Fibonacci chain, i.e., a tight-binding model where couplings and/or
on-site potentials can take only two different values distributed according to the
Fibonacci word, is a classical example of a one-dimensional quasicrystal. With its
many intriguing properties, such as a fractal eigenvalue spectrum, the Fibonacci chain
offers a rich platform to investigate many of the effects that occur in three-dimensional
quasicrystals. In this work, we study the eigenvalues and eigenstates of two identical
Fibonacci chains coupled to each other in different ways. We find that this setup allows
for a rich variety of effects. Depending on the coupling scheme used, the resulting
system (i) possesses an eigenvalue spectrum featuring a richer hierarchical structure
compared to the spectrum of a single Fibonacci chain, (ii) shows a coexistence of Bloch
and critical eigenstates, or (iii) possesses a large number of degenerate eigenstates,
each of which is perfectly localized on only four sites of the system. If additionally, the
system is infinitely extended, the macroscopic number of perfectly localized eigenstates
induces a perfectly flat quasi band. Especially the second case is interesting from an
application perspective, since eigenstates that are of Bloch or of critical character
feature largely different transport properties. At the same time, the proposed setup
allows for an experimental realization, e.g., with evanescently coupled waveguides,
electric circuits, or by patterning an anti-lattice with adatoms on a metallic substrate.

Super band gaps and periodic approximants of generalised Fibonacci tilings

“…That is, F n+1 = F n ∪ F n−1 , which is an equivalent definition for the golden mean Fibonacci tiling. Generalised Fibonacci tilings have been studied extensively in the literature for various elastic, mechanical and Hamiltonian systems [18][19][20][21][22][23][24]. Complex patterns of stop and pass bands have been observed, whose features include large stop bands across multiple frequency scales and self-similar properties.…”