Dynamics of many-body localized systems: logarithmic lightcones and $\log \, t$-law of $\alpha$-R\'enyi entropies

TL;DR

This paper rigorously demonstrates that many-body localized systems exhibit a logarithmic lightcone and a slow, logarithmic-in-time growth of entanglement, contrasting with Anderson localization, using Lieb-Robinson bounds without assuming LIOMs.

Contribution

It provides a rigorous analysis of entanglement dynamics in MBL systems based on Lieb-Robinson bounds, with or without assuming local integrals of motion.

Findings

Logarithmic lightcone in MBL systems.

Logarithmic growth of $ ext{α}$-Rényi entropies over time.

Contrast with Anderson localized phases showing no entanglement growth.

Abstract

In the context of the Many-Body-Localization phenomenology we consider arbitrarily large one-dimensional local spin systems, the XXZ model with random magnetic field is a prototypical example. Without assuming the existence of exponentially localized integrals of motion (LIOM), but assuming instead that the system's dynamics gives rise to a Lieb-Robinson bound (L-R) with a logarithmic lightcone, we rigorously evaluate the dynamical generation, starting from a generic product state, of $ \alpha$-R\'enyi entropies, with $ \alpha $ close to one, obtaining a $\log \, t$-law, that denotes a slow spread of entanglement. This is in sharp contrast with Anderson localized phases that show no dynamically generated entanglement. To prove this result we apply a general theory recently developed by us in arXiv:2408.00743 that quantitatively relates the L-R bounds of a local Hamiltonian with the dynamical generation of entanglement. Assuming instead the existence of LIOM we provide new independent proofs of the known facts that the L-R bound of the system's dynamics has a logarithmic light cone and show that the dynamical generation of the von Neumann entropy has for large times a $ \log \, t$-shape. L-R bounds, that quantify the dynamical spreading of local operators, may be easier to measure in experiments in comparison to global quantities such as entanglement.