# Fractal x-ray edge problem at the critical point of the Aubry-Andr\'e model

## TL;DR

This paper investigates the x-ray edge problem at the critical point of the Aubry-André model, revealing complex, fractal-induced dynamics and broad distributions in response to local quenches, with implications for cold atom experiments.

## Contribution

It provides a numerical analysis of the Anderson orthogonality catastrophe at a fractal critical point, highlighting non-monotonic behavior and site-dependent responses due to fractal wavefunctions.

## Key findings

- Overlap and Green function show non-monotonic evolution with system size and time.
- Fractal density of states causes alternating regimes in response.
- Broad distributions of observables emerge from site-dependent responses.

## Abstract

We study the Anderson orthogonality catastrophe, and the corresponding x-ray edge problem, in systems that are at a localization transition driven by a deterministic quasiperiodic potential. Specifically, we address how the ground state of the Aubry-Andre model, at its critical point, responds to an instantaneous local quench. At this critical point, both the single-particle wavefunctions and the density of states are fractal. We find, numerically, that the overlap between post-quench and pre-quench wavefunctions, as well as the "core-hole" Green function, evolve in a complex, non-monotonic way with system size and time respectively. We interpret our results in terms of the fractal density of states at this critical point. In a given sample, as the post-quench time increases, the system resolves increasingly finely spaced minibands, leading to a series of alternating temporal regimes in which the response is flat or algebraically decaying. In addition, the fractal critical wavefunctions give rise to a quench response that varies strongly from site to site across the sample, which produces broad distributions of many-body observables. Upon averaging this broad distribution over samples, we recover coarse-grained power laws and dynamical exponents characterizing the x-ray edge singularity. We discuss how these features can be probed in ultra-cold atomic gases using radio-frequency spectroscopy and Ramsey interference.