Bimodal entanglement entropy distribution in the many-body localization transition

TL;DR

This paper investigates the entanglement entropy distribution in disordered spin chains near the many-body localization transition, revealing bimodal behavior and scaling properties that distinguish localized and ergodic phases.

Contribution

It introduces the cut averaged entanglement entropy and demonstrates its concavity, enabling detailed analysis of entanglement scaling and phase transition characteristics in disordered systems.

Findings

Evidence for bimodal entanglement distribution in the critical region.

Entanglement entropy scales as volume law with a sub-thermal coefficient.

Negative slope of entanglement entropy at intermediate disorder due to rare localized regions.

Abstract

We introduce the cut averaged entanglement entropy in disordered periodic spin chains and prove it to be a concave function of subsystem size for individual eigenstates. This allows us to identify the entanglement scaling as a function of subsystem size for individual states in inhomogeneous systems. Using this quantity, we probe the critical region between the many-body localized (MBL) and ergodic phases in finite systems. In the middle of the spectrum, we show evidence for bimodality of the entanglement distribution in the MBL critical region, finding both volume law and area law eigenstates over disorder realizations as well as within \emph{single disorder realizations}. The disorder averaged entanglement entropy in this region then scales as a volume law with a coefficient below its thermal value. We discover in the critical region, as we approach the thermodynamic limit, that the cut averaged entanglement entropy density falls on a one-parameter family of curves. Finally, we also show that without averaging over cuts the slope of the entanglement entropy \vs subsystem size can be negative at intermediate and strong disorder, caused by rare localized regions in the system.