Algebraic localization in disordered one-dimensional systems with long-range hopping

The transport of excitations between particles in many physical systems may be mapped to single-particle models with power-law hopping, $1/r^a$. For randomly placed particles, these models present an effective peculiar disorder that leads to surprising localization properties. We prove that in one-dimensional systems all eigenstates remain algebraically localized for any value of $a>0$. Moreover, we show that there is an effective duality between models with $a1$, which present similar localization properties.