Electronic Energy Spectra of Square and Cubic Fibonacci Quasicrystals

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. 1 Background and Motivation As we celebrate the Silver Jubilee of the 1982 discovery of quasicrystals [1], and highlight the achievements of the past two and a half decades of research on quasicrystals, we are reminded that there still remains a disturbing gap in our understanding of their electronic properties. Among the open questions is a lack of understanding of the dependence of electronic properties—such as the nature of electronic wave functions, their energy spectra, and the nature of electronic transport—on the dimension of the quasicrystal. In an attempt to bridge some of this gap, we [2,3] have been studying the spectrum and electronic wave functions of an off-diagonal tight-binding hamiltonian on the separable n-dimensional Fibonacci quasicrystals1 [4]. The advantage of using such separable models, despite the fact that they do not occur in nature, is the ability to obtain exact results in one, two, and three dimensions, and compare them directly to each other. Here we focus on the energy spectra of the 2-dimensional (2d) and 3-dimensional (3d) Fibonacci quasicrystals to obtain a quantitative understanding of the nature of the transitions between different spectral behaviors in these crystals, as their dimension increases from 1 up to 3. In particular, we consider the transitions between different regimes in the spectrum, taking into account the existence of a regime in which the spectrum contains both singular continuous The reader is referred to Refs. [5] and [6] for precise definitions of the terms ‘crystal’ and ‘quasicrystal’.