# Entanglement Complexity in Quantum Many-Body Dynamics, Thermalization and Localization

## TL;DR

This paper investigates the spectral properties of entanglement in quantum many-body systems, revealing distinct patterns for thermalization, Anderson localization, and many-body localization, and linking these to entanglement complexity.

## Contribution

It introduces a spectral analysis of entanglement to distinguish dynamical phases and connects entanglement complexity with level spacing statistics.

## Key findings

- Poisson statistics for Anderson localization
- Wigner-Dyson statistics for thermalization and MBL
- Entanglement complexity correlates with level spacing distribution

## Abstract

Entanglement is usually quantified by von Neumann entropy, but its properties are much more complex than what can be expressed with a single number. We show that the three distinct dynamical phases known as thermalization, Anderson localization, and many-body localization are marked by different patterns of the spectrum of the reduced density matrix for a state evolved after a quantum quench. While the entanglement spectrum displays Poisson statistics for the case of Anderson localization, it displays universal Wigner-Dyson statistics for both the cases of many-body localization and thermalization, albeit the universal distribution is asymptotically reached within very different time scales in these two cases. We further show that the complexity of entanglement, revealed by the possibility of disentangling the state through a Metropolis-like algorithm, is signaled by whether the entanglement spectrum level spacing is Poisson or Wigner-Dyson distributed.