# Entanglement growth after inhomogenous quenches

## TL;DR

This paper investigates how initial inhomogeneities in conserved quantities influence entanglement growth in quantum systems, revealing a diffusive $
oot t$ scaling of entanglement features through an effective model and numerical simulations.

## Contribution

It introduces a generalized minimal cut model incorporating local entropy density and connects entanglement dynamics with KPZ and diffusion equations, supported by numerical evidence.

## Key findings

- Entanglement features grow as $
oot t$ due to initial inhomogeneities.
- Diffusive charge transport constrains entanglement growth to $
oot t$.
- Features persist even after charge density equilibrates.

## Abstract

We study the growth of entanglement in quantum systems with a conserved quantity exhibiting diffusive transport, focusing on how initial inhomogeneities are imprinted on the entropy. We propose a simple effective model, which generalizes the minimal cut picture of \textit{Jonay et al.} in such a way that the `line tension' of the cut depends on the local entropy density. In the case of noisy dynamics, this is described by a Kardar-Parisi-Zhang (KPZ) equation coupled to a diffusing field. We investigate the resulting dynamics and find that initial inhomogeneities of the conserved charge give rise to features in the entanglement profile, whose width and height both grow in time as $\propto\sqrt{t}$. In particular, for a domain wall quench, diffusion restricts entanglement growth to be $S_\text{vN} \lesssim \sqrt{t}$. We find that for charge density wave initial states, these features in the entanglement profile are present even after the charge density has equilibrated. Our conclusions are supported by numerical results on random circuits and deterministic spin chains.