# Critical Behavior and Fractality in Shallow One-Dimensional Quasiperiodic Potentials

## TL;DR

This paper investigates the critical localization and fractal properties of particles in shallow one-dimensional quasiperiodic systems, revealing a universal critical exponent and the nature of the energy spectrum.

## Contribution

It determines the mobility edge, critical potential amplitude, and spectral fractality in shallow quasiperiodic models, connecting these to critical phenomena and spectrum structure.

## Key findings

- Identified the mobility edge separating localized and extended phases.
- Found the universal critical exponent $
u \,\simeq\ 1/3$.
- Showed the spectrum is nowhere dense with nonuniversal Hausdorff dimension.

## Abstract

Quasiperiodic systems offer an appealing intermediate between long-range ordered and genuine disordered systems, with unusual critical properties. One-dimensional models that break the so-called self-dual symmetry usually display a mobility edge, similarly as truly disordered systems in dimension strictly higher than two. Here, we determine the critical localization properties of single particles in shallow, one-dimensional, quasiperiodic models and relate them to the fractal character of the energy spectrum. On the one hand, we determine the mobility edge and show that it separates the localized and extended phases, with no intermediate phase. On the other hand, we determine the critical potential amplitude and find the universal critical exponent $\nu \simeq 1/3$. We also study the spectral Hausdorff dimension and show that it is nonuniversal but always smaller than unity, hence showing that the spectrum is nowhere dense. Finally, applications to ongoing studies of Anderson localization, Bose-glass physics, and many-body localization in ultracold atoms are discussed.