# Sub-ballistic growth of R\'enyi entropies due to diffusion

## TL;DR

This paper reveals that higher Renyi entropies grow sub-ballistically, specifically as abft, in diffusive quantum systems, contrasting with the linear growth of von Neumann entropy, due to local quantum fluctuations governed by diffusion.

## Contribution

It demonstrates that higher Renyi entropies grow sub-ballistically in diffusive systems and provides a theoretical framework and evidence from models and spin chains.

## Key findings

- Higher Renyi entropies grow as abft in diffusive systems.
- The second Renyi entropy exhibits hydrodynamic tails with three distinct power laws.
- The results connect quantum fluctuations in conserved densities to entanglement growth.

## Abstract

We investigate the dynamics of quantum entanglement after a global quench and uncover a qualitative difference between the behavior of the von Neumann entropy and higher R\'enyi entropies. We argue that the latter generically grow \emph{sub-ballistically}, as $\propto\sqrt{t}$, in systems with diffusive transport. We provide strong evidence for this in both a U$(1)$ symmetric random circuit model and in a paradigmatic non-integrable spin chain, where energy is the sole conserved quantity. We interpret our results as a consequence of local quantum fluctuations in conserved densities, whose behavior is controlled by diffusion, and use the random circuit model to derive an effective description. We also discuss the late-time behavior of the second R\'enyi entropy and show that it exhibits hydrodynamic tails with \emph{three distinct power laws} occurring for different classes of initial states.