# Signatures of information scrambling in the dynamics of the entanglement spectrum

## TL;DR

This paper investigates how the entanglement spectrum's level repulsion signals information scrambling and thermalization in non-integrable spin chains, revealing different timescales for various spectral features.

## Contribution

It uncovers the relationship between entanglement spectrum dynamics and operator spreading, providing analytical and numerical insights into the timescales of thermalization.

## Key findings

- Level repulsion onset varies with entanglement energy.
- Entanglement spectrum reflects operator front shape.
- Level repulsion spreads faster than full thermalization.

## Abstract

We examine the time evolution of the entanglement spectrum of a small subsystem of a non-integrable spin chain following a quench from a product state. We identify signatures in this entanglement spectrum of the distinct dynamical velocities (related to entanglement and operator spreading) that control thermalization. We show that the onset of level repulsion in the entanglement spectrum occurs on different timescales depending on the "entanglement energy", and that this dependence reflects the shape of the operator front. Level repulsion spreads across the entire entanglement spectrum on timescales close to when the mutual information between individual spins at the ends of the subsystem reaches its maximum; this timescale is much shorter than that for full thermalization of the subsystem. We provide an analytical understanding of this phenomenon and show supporting numerical data for both random unitary circuits and a microscopic Hamiltonian.