Lyapunov exponent for pure and random Fibonacci chains

We study the Lyapunov exponent for electron and phonon excitations, in pure and random Fibonacci quasicrystal chains, using an exact real-space renormalization-group method, which allows the calculation of the Lyapunov exponent as a function of the energy. It is shown that the Lyapunov exponent on a pure Fibonacci chain has a self-similar structure, characterized by a scaling index that is independent of the energy for the electron excitations, ''diagonal'' or "off-diagonal" quasiperiodic, but is a function of the energy for the phonon excitations. This scaling behavior implies the vanishing of the Lyapunov exponent for the states on the spectrum, and hence the absence of localization on the Fibonacci chain, for the various excitations considered. It is also shown that disordered Fibonacci chains, with random tiling that introduces phason flips at certain sites on the chain, exhibit the same Lyapunov exponent as the pure Fibonacci chain, and hence this type of disorder is irrelevant, either in the case of electron or phonon excitations.