Universal slow growth of entanglement in interacting strongly disordered systems

TL;DR

This paper explains the universal logarithmic growth of entanglement entropy in many-body localized systems as a dephasing effect caused by exponentially small interactions, supported by numerical simulations.

Contribution

It introduces a universal mechanism for slow entanglement growth in many-body localized phases, linking it to dephasing from interaction-induced energy corrections.

Findings

Entanglement grows as  imes ext{ln}(Vt/\u211d) for weak interactions.

Long-time entanglement saturation depends on initial state participation ratios.

Logarithmic growth is a universal feature across spatial dimensions in MBL phases.

Abstract

Recent numerical work by Bardarson et. al. [Phys. Rev. Lett. 109, 017202 (2012)] revealed a slow, logarithmic in time, growth of entanglement entropy for initial product states in a putative many-body localized phase. We show that this surprising phenomenon results from the dephasing due to exponentially small interaction-induced corrections to the eigenenergies of different states. For weak interactions, we find that the entanglement entropy grows as \xi ln (Vt/\hbar), where V is the interaction strength, and \xi is the single-particle localization length. The saturated value of the entanglement entropy at long times is determined by the participation ratios of the initial state over the eigenstates of the subsystem. The proposed mechanism is illustrated with numerical simulations of small systems. Our work shows that the logarithmic entanglement growth is a universal phenomenon characteristic of the many-body localized phase in any number of spatial dimensions, and reveals a broad hierarchy of dephasing time scales present in such a phase.