Many-body localisation implies that eigenvectors are matrix-product states

TL;DR

This paper establishes a rigorous connection between many-body localisation and the entanglement structure of eigenvectors, showing that localisation implies eigenvectors can be approximated by matrix-product states, especially in one dimension.

Contribution

It proves that strong dynamical localisation in many-body systems leads to eigenvectors with clustering correlations and an entanglement area law, linking dynamical and entanglement properties.

Findings

Eigenvectors exhibit clustering correlations under localisation.

In 1D, eigenvectors satisfy an entanglement area law.

Part of the spectrum allows for mobility edges with transport.

Abstract

The phenomenon of many-body localisation received a lot of attention recently, both for its implications in condensed-matter physics of allowing systems to be an insulator even at non-zero temperature as well as in the context of the foundations of quantum statistical mechanics, providing examples of systems showing the absence of thermalisation following out-of-equilibrium dynamics. In this work, we establish a novel link between dynamical properties - the absence of a group velocity and transport - with entanglement properties of individual eigenvectors. Using Lieb-Robinson bounds and filter functions, we prove rigorously under simple assumptions on the spectrum that if a system shows strong dynamical localisation, all of its many-body eigenvectors have clustering correlations. In one dimension this implies directly an entanglement area law, hence the eigenvectors can be approximated by matrix-product states. We also show this statement for parts of the spectrum, allowing for the existence of a mobility edge above which transport is possible.