Fractal dimensions of wave functions and local spectral measures on the Fibonacci chain

We present a theoretical framework for understanding the wave functions and spectrum of an extensively studied paradigm for quasiperiodic systems, namely the Fibonacci chain. Our analytical results, which are obtained in the limit of strong modulation of the hopping amplitudes, are in good agreement with published numerical data. In the perturbative limit, we show a symmetry of wave functions under permutation of site and energy indices. We compute the wave-function renormalization factors and from them deduce analytical expressions for the fractal exponents corresponding to individual wave functions, as well as their global averages. The multifractality of wave functions is seen to appear at next-to-leading order in $\ensuremath{\rho}$. Exponents for the local spectral density are given, in extremely good accord with numerical calculations. Interestingly, our analytical results for exponents are observed to describe the system rather well even for values of $\ensuremath{\rho}$ well outside the domain of applicability of perturbation theory.