# Self-dual quasiperiodic systems with power-law hopping

## TL;DR

This paper introduces a family of self-dual quasiperiodic models with power-law hopping, revealing complex localization transitions, singular continuous spectra, and unique conductivity properties that extend the understanding of quasiperiodic systems beyond the Aubry-Andre model.

## Contribution

The paper generalizes the Aubry-Andre model to include power-law hopping, uncovering new localization phenomena and spectral features in self-dual quasiperiodic systems.

## Key findings

- Localization transition splits into three for p ≤ 2.1
- Presence of singular continuous spectra and self-similar states
- Frequency-dependent conductivity with sample-to-sample fluctuations

## Abstract

We introduce and explore a family of self-dual models of single-particle motion in quasiperiodic potentials, with hopping amplitudes that fall off as a power law with exponent $p$. These models are generalizations of the familiar Aubry-Andre model. For large enough $p$, their static properties are similar to those of the Aubry-Andre model, although the low-frequency conductivity in the localized phase is sensitive to $p$. For $p \leq 2.1$ the Aubry-Andre localization transition splits into three transitions; two distinct intermediate regimes with both localized and delocalized states appear near the self-dual point of the Aubry-Andre model. In the intermediate regimes, the density of states is singular continuous in much of the spectrum, and is approximately self-similar: states form narrow energy bands, which are divided into yet narrower sub-bands; we find no clear sign of a mobility edge. When $p < 1$, localized states are not stable in random potentials; in the present model, however, tightly localized states are present for relatively large systems. We discuss the frequency-dependence and strong sample-to-sample fluctuations of the low-frequency optical conductivity, although a suitably generalized version of Mott's law is recovered when the power-law is slowly decaying. We present evidence that many of these features persist in models that are away from self-duality.