# Localization as an entanglement phase transition in boundary-driven Anderson models

## TL;DR

This paper investigates the Anderson localization transition in a boundary-driven non-equilibrium setting, revealing an entanglement phase transition characterized by a change from extensive to area-law scaling of mutual coherence.

## Contribution

It introduces the mutual coherence as a new entanglement witness and analyzes its finite-size scaling across the localization transition in a non-interacting Anderson model.

## Key findings

- Extensive mutual coherence persists at the critical point.
- Mutual coherence transitions to an area-law in the localized phase.
- Analytical and numerical results elucidate the entanglement transition in open systems.

## Abstract

The Anderson localization transition is one of the most well studied examples of a zero temperature quantum phase transition. On the other hand, many open questions remain about the phenomenology of disordered systems driven far out of equilibrium. Here we study the localization transition in the prototypical three-dimensional, noninteracting Anderson model when the system is driven at its boundaries to induce a current carrying non-equilibrium steady state. Recently we showed that the diffusive phase of this model exhibits extensive mutual information of its non-equilibrium steady-state density matrix. We show that that this extensive scaling persists in the entanglement and at the localization critical point, before crossing over to a short-range (area-law) scaling in the localized phase. We introduce an entanglement witness for fermionic states that we name the mutual coherence, which, for fermionic Gaussian states, is also a lower bound on the mutual information. Through a combination of analytical arguments and numerics, we determine the finite-size scaling of the mutual coherence across the transition. These results further develop the notion of entanglement phase transitions in open systems, with direct implications for driven many-body localized systems, as well as experimental studies of driven-disordered systems.