# Energy level dynamics across the many-body localization transition

## TL;DR

This paper investigates how energy levels evolve across the many-body localization transition in a disordered spin model, revealing non-universal behavior in the ergodic phase and distinct dynamics in the localized phase.

## Contribution

It provides a detailed comparison of level dynamics measures across the transition, highlighting deviations from universal predictions and phase-dependent behaviors.

## Key findings

- Level curvatures and fidelity susceptibilities match GOE distributions after rescaling.
- Level dynamics do not show universal behavior in the ergodic phase.
- Distinct dynamics are observed in the localized phase, explained by local integrals of motion.

## Abstract

The level dynamics across the many body localization transition is examined for XXZ-spin model with a random magnetic field. We compare different scenaria of parameter dependent motion in the system and consider measures such as level velocities, curvatures as well as their fidelity susceptibilities. Studying the ergodic phase of the model we find that the level dynamics does not reveal the commonly believed universal behavior after rescaling the curvatures by the level velocity variance. At the same time, distributions of level curvatures and fidelity susceptibilities coincide with properly rescaled distributions for Gaussian Orthogonal Ensemble of random matrices. Profound differences exists depending on way the level dynamics is imposed in the many-body localized phase of the model in which the level dynamics can be understood with the help of local integrals of motion.