# Anomalous transport through algebraically localized states in one-dimension

## TL;DR

This paper investigates how algebraic localization in one-dimensional systems with power-law hopping can lead to either conducting or insulating behavior, revealing a phase with super-diffusive transport and a mobility edge separating different conducting states.

## Contribution

It demonstrates that algebraic localization does not necessarily imply insulation and identifies a phase with a mobility edge and super-diffusive transport in a quasiperiodic model.

## Key findings

- Algebraically localized states can be conducting or insulating depending on decay strength.
- Existence of a mobility edge separating delocalized and algebraically localized states.
- Presence of super-diffusive transport signatures near the mobility edge.

## Abstract

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.