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.