# Out of Time Order Correlations in the Quasi-Periodic Aubry-Andr\'e model

## TL;DR

This paper investigates out of time ordered correlators (OTOC) in the Aubry-Andre9 model, revealing universal behaviors, wavefront velocities, and bounds on late-time equilibration across different phases.

## Contribution

It provides a comprehensive analysis of OTOC dynamics in the Aubry-Andre9 model, including a new finite-time equilibration bound applicable to quadratic models.

## Key findings

- Power-law behavior in early time regime.
- Universal form of wavefront regime and characteristic velocity extraction.
- Finite-time equilibration bounds for OTOC in the extended phase.

## Abstract

We study out of time ordered correlators (OTOC) in a free fermionic model with a quasi-periodic potential. This model is equivalent to the Aubry-Andr\'e model and features a phase transition from an extended phase to a localized phase at a non-zero value of the strength of the quasi-periodic potential. We investigate five different time-regimes of interest for out of time ordered correlators; early, wavefront, $x=v_Bt$, late time equilibration and infinite time. For the early time regime we observe a power law for all potential strengths. For the time regime preceding the wavefront we confirm a recently proposed universal form and use it to extract the characteristic velocity of the wavefront for the present model. A Gaussian waveform is observed to work well in the time regime surrounding $x=v_Bt$. Our main result is for the late time equilibration regime where we derive a finite time equilibration bound for the OTOC, bounding the correlator's distance from its late time value. The bound impose strict limits on equilibration of the OTOC in the extended regime ans is valid not only for the Aubry-Andr\'e model but for any quadratic model model. Finally, momentum out of time ordered correlators for the Aubry-Andr\'e model are studied where large values of the OTOC are observed at late times at the critical point.