  Localization in one-dimensional disordered or quasiperiodic non-interacting
systems in presence of power-law hopping is very different from localization in
short-ranged systems. Power-law hopping leads to algebraic localization as
opposed to exponential localization in short-ranged systems. Exponential
localization is synonymous with insulating behavior in the thermodynamic limit.
Here we show that the same is not true for algebraic localization. We show, on
general grounds, that depending on the strength of the algebraic decay, the
algebraically localized states can be actually either conducting or insulating
in thermodynamic limit. We exemplify this statement with explicit calculations
on the Aubry-Andr\'e-Harper model in presence of power-law hopping, with the
power-law exponent $\alpha>1$, so that the thermodynamic limit is well-defined.
We find a phase of this system where there is a mobility edge separating
completely delocalized and algebraically localized states, with the
algebraically localized states showing signatures of super-diffusive transport.
Thus, in this phase, the mobility edge separates two kinds of conducting
states, ballistic and super-diffusive. We trace the occurrence of this behavior
to near-resonance conditions of the on-site energies that occur due to the
quasi-periodic nature of the potential.
