Entanglement dynamics and eigenstate correlations in strongly disordered quantum many-body systems

TL;DR

This paper develops a microscopic theory explaining the ultraslow, logarithmic growth of entanglement in many-body localized quantum systems through eigenstate correlations involving quartets of eigenstates, revealing a hierarchy of timescales.

Contribution

It introduces a novel microscopic framework based on eigenstate quartets to understand entanglement dynamics in the MBL phase, surpassing the traditional l-bit phenomenology.

Findings

Entanglement growth is dominated by spectral correlations within special eigenstate quartets.

The spatial structure of quartets explains the hierarchy of timescales in entanglement dynamics.

The theory accounts for the logarithmic time dependence of entanglement in MBL systems.

Abstract

The many-body localised phase of quantum systems is an unusual dynamical phase wherein the system fails to thermalise and yet, entanglement grows unboundedly albeit very slowly in time. We present a microscopic theory of this ultraslow growth of entanglement in terms of dynamical eigenstate correlations of strongly disordered, interacting quantum systems in the many-body localised regime. These correlations involve sets of four or more eigenstates and hence, go beyond correlations involving pairs of eigenstates which are usually studied in the context of eigenstate thermalisation or lack thereof. We consider the minimal case, namely the second R\'enyi entropy of entanglement, of an initial product state as well as that of the time-evolution operator, wherein the correlations involve quartets of four eigenstates. We identify that the dynamics of the entanglement entropy is dominated by the spectral correlations within certain special quartets of eigenstates. We uncover the spatial structure of these special quartets and the ensuing statistics of the spectral correlations amongst the eigenstates therein, which reveals a hierarchy of timescales or equivalently, energyscales. We show that the hierarchy of these timescales along with their non-trivial distributions conspire to produce the logarithmic in time growth of entanglement, characteristic of the many-body localised regime. The underlying spatial structures in the set of special quartets also provides a microscopic understanding of the spacetime picture of the entanglement growth. The theory therefore provides a much richer perspective on entanglement growth in strongly disordered systems compared to the commonly employed phenomenological approach based on the $\ell$-bit picture.