Multiscaling, Ergodicity and Localization in Quasiperiodic Chains

We report results of numerical simulations of wave-packet dynamics in a class of chains consisting of two types of weakly coupled clusters arranged in a quasiperiodic sequence. Properties of eigenstates are investigated using perturbation theory of degenerate levels in the coupling strength v and by numerical diagonalization. Results show that wave packets anomalously diffuse via a two-step process of rapid and slow expansions, which persist for any v>0. An elementary analysis of the degenerate perturbation expansion reveals that non-localized states may appear only in a sufficiently high order of perturbation theory, which is simply related to the combinatorial properties of the sequences. Numerical diagonalization furthermore shows that eigenstates ergodically spread across the entire chain for v>0, while in the limit as v->0 ergodicity is broken and eigenstates spread only across clusters of the same type, in contradistinction with trivial localization at v=0. An investigation of the effects of a single-site perturbation on wave-packet dynamics shows that, by changing the position or strength of such an impurity, it is possible to control the long-time wave-packet dynamics. By adding a single impurity it is possible to induce wave-packet localization on individual subchains as well as on the whole chain.