  We probe the existence of a many-body localized phase (MBL-phase) in a
spinless fermionic Hubbard chain with algebraically localized single-particle
states, by investigating both static and dynamical properties of the system.
This MBL-phase can be characterized by an extensive number of integrals of
motion which develop algebraically decaying tails, unlike the case of
exponentially localized single-particle states. We focus on the implications
for the quantum information propagation through the system. We provide evidence
that the bipartite entanglement entropy after a quantum quench has an unbounded
algebraic growth in time, while the quantum Fisher information grows
logarithmically.
