Electronic Energy Spectra and Wave Functions on the Square Fibonacci Tiling

We study the electronic energy spectra and wave functions on the square Fibonacci tiling, using an off-diagonal tight-binding model, in order to determine the exact nature of the transitions between different spectral behaviors, as well as the scaling of the total bandwidth as it becomes finite. The macroscopic degeneracy of certain energy values in the spectrum is invoked as a possible mechanism for the emergence of extended electronic Bloch wave functions as the dimension changes from one to two. 1 Background and Motivation We continue our initial studies [1] of the off-diagonal tight-binding hamiltonian on the square Fibonacci tiling [2], in order to gain a better quantitative understanding of the nature of the transitions between different spectral behaviors in this 2-dimensional (2d) quasicrystal. We also consider more carefully the transition of the spectrum from singular-continuous to absolutely continuous, in going from one to two dimensions, and the implications of this transition on the possible emergence of extended Bloch wave functions. The square Fibonacci tiling is constructed by taking two identical grids—each consisting