# Holographic Entanglement Entropy of Local Quenches in AdS$_4$/CFT$_3$: A Finite-Element Approach

## TL;DR

This paper develops a finite-element numerical method to compute holographic entanglement entropy in higher-dimensional CFTs, revealing early-time agreement with the first law and complex late-time behavior for local quenches in AdS$_4$/CFT$_3$.

## Contribution

It introduces a finite-element approach for calculating holographic entanglement entropy in higher dimensions and explores its dynamics after local quenches, highlighting deviations from the first law.

## Key findings

- Early-time entanglement growth matches the first law for small conformal dimension.
- Deviations from the first law occur at later times, showing sub-linear growth.
- Large conformal dimension leads to qualitatively different time dependence than in 2D cases.

## Abstract

Understanding quantum entanglement in interacting higher-dimensional conformal field theories is a challenging task, as direct analytical calculations are often impossible to perform. With holographic entanglement entropy, calculations of entanglement entropy turn into a problem of finding extremal surfaces in a curved spacetime, which we tackle with a numerical finite-element approach. In this paper, we compute the entanglement entropy between two half-spaces resulting from a local quench, triggered by a local operator insertion in a CFT$_3$. We find that the growth of entanglement entropy at early time agrees with the prediction from the first law, as long as the conformal dimension $\Delta$ of the local operator is small. Within the limited time region that we can probe numerically, we observe deviations from the first law and a transition to sub-linear growth at later time. In particular, the time dependence at large $\Delta$ shows qualitative differences to the simple logarithmic time dependence familiar from the CFT$_2$ case. We hope that our work will motivate further studies, both numerical and analytical, on entanglement entropy in higher dimensions.