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

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 ${\alpha }_E$ which takes a value in the range [${\alpha }_E^{\min }$,${\alpha }_E^{\max }$]. The fractal dimensions f(${\alpha }_E$) of these singularities in the Cantor set are calculated. This function f(${\alpha }_E$) represents the global scaling properties of the Cantor-set spectrum.