Duality in Power-Law Localization in Disordered One-Dimensional Systems

The transport of excitations between pinned particles in many physical systems may be mapped to single-particle models with power-law hopping, 1/r^a. For randomly spaced particles, these models present an effective peculiar disorder that leads to surprising localization properties. We show that in one-dimensional systems almost all eigenstates (except for a few states close to the ground state) are power-law localized for any value of a > 0. Moreover, we show that our model is an example of a new universality class of models with power-law hopping, characterized by a duality between systems with long-range hops (a < 1) and short-range hops (a > 1), in which the wave function amplitude falls off algebraically with the same power γ from the localization center.