# Duality in power-law localization in disordered one-dimensional systems

## TL;DR

This paper investigates localization properties in one-dimensional disordered systems with power-law hopping, revealing a duality between long-range and short-range hopping regimes and identifying a new universality class.

## Contribution

It introduces a novel universality class of power-law hopping models exhibiting a duality between different decay regimes of wave functions.

## Key findings

- Almost all eigenstates are power-law localized for any $a>0$
- A duality exists between systems with $a<1$ and $a>1$
- Wave function decay follows the same power law in dual regimes

## Abstract

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 $\gamma$ from the localization center.