  One-dimensional quasi-periodic systems with power-law hopping, $1/r^a$,
differ from both the standard Aubry-Azbel-Harper (AAH) model and from power-law
systems with uncorrelated disorder. Whereas in the AAH model all
single-particle states undergo a transition from ergodic to localized at a
critical quasi-disorder strength, short-range power-law hops with $a>1$ can
result in mobility edges. Interestingly, there is no localization for
long-range hops with $a\leq 1$, in contrast to the case of uncorrelated
disorder. Systems with long-range hops are rather characterized by
ergodic-to-multifractal edges and a phase transition from ergodic to
multifractal (extended but non ergodic) states. We show that both mobility and
ergodic-to-multifractal edges may be clearly revealed in experiments on
expansion dynamics.
