Critical wave functions and a Cantor-set spectrum of a one-dimensional quasicrystal model

TLDR: The electronic properties of a tight-binding model which possesses two types of hopping matrix element arranged in a Fibonacci sequence are studied and the fractal dimensions f(ae) represents the global scaling properties of the Cantor-set spectrum.

Abstract: The electronic properties of a tight-binding model which possesses two types of hopping matrix element (or on-site energy) arranged in a Fibonacci sequence are studied. The wave functions are either self-similar (fractal) or chaotic and show ``critical'' (or ``exotic'') behavior. Scaling analysis for the self-similar wave functions at the center of the band and also at the edge of the band is performed. The energy spectrum is a Cantor set with zero Lebesque measure. The density of states is singularly concentrated with an index ${\ensuremath{\alpha}}_{E}$ which takes a value in the range [${\ensuremath{\alpha}}_{E}^{\mathrm{min}}$,${\ensuremath{\alpha}}_{E}^{\mathrm{max}}$]. The fractal dimensions f(${\ensuremath{\alpha}}_{E}$) of these singularities in the Cantor set are calculated. This function f(${\ensuremath{\alpha}}_{E}$) represents the global scaling properties of the Cantor-set spectrum.