# Characterizing the many-body localization transition through the entanglement spectrum

## TL;DR

This paper investigates the many-body localization transition by analyzing the entanglement spectrum in finite systems, identifying the crossover from criticality to MBL, and establishing bounds on the critical disorder strength.

## Contribution

It provides the first numerical observation of the crossover from criticality to MBL using the entanglement spectrum, offering bounds on the critical disorder strength.

## Key findings

- Identified the crossover from criticality to MBL in finite systems.
- Placed an upper bound on the critical disorder strength for MBL.
- Analyzed the high energy part of the entanglement spectrum to extract universal information.

## Abstract

We numerically explore the many body localization (MBL) transition through the lens of the {\it entanglement spectrum}. While a direct transition from localization to thermalization is believed to obtain in the thermodynamic limit (the exact details of which remain an open problem), in finite system sizes there exists an intermediate `quantum critical' regime. Previous numerical investigations have explored the crossover from thermalization to criticality, and have used this to place a numerical {\it lower} bound on the critical disorder strength for MBL. A careful analysis of the {\it high energy} part of the entanglement spectrum (which contains universal information about the critical point) allows us to make the first ever observation in exact numerics of the crossover from criticality to MBL and hence to place a numerical {\it upper bound} on the critical disorder strength for MBL.