# Evolutions of entanglement and complexity after a thermal quench in massive gravity theory

## TL;DR

This paper investigates how graviton mass, charge, and boundary region size influence the evolution of holographic entanglement entropy and complexity after a thermal quench in massive gravity, revealing new discontinuity and peak behaviors.

## Contribution

It introduces the effects of graviton mass and charge on HEE and HC evolution, highlighting universal discontinuity emergence and multiple peaks in complexity growth.

## Key findings

- Discontinuity in HEE and HC evolution appears for larger charge or boundary size.
- Graviton mass accelerates the system reaching stability and alters final entanglement and complexity values.
- Multiple peaks in HC evolution can occur due to graviton mass, reverting to single peak at high charge.

## Abstract

We study the evolution of the holographic entanglement entropy (HEE) and the holographic complexity (HC) after a thermal quench in $1+1$ dimensional boundary CFTs dual to massive BTZ black holes. The study indicates how the graviton mass $m_g$, the charge $q$, and also the size of the boundary region $l$ determine the evolution of the HEE and HC. We find that for small $q$ and $l$, the evolutions of the HEE and the HC is a continuous function. When $q$ or $l$ is tuned larger, the discontinuity emerges, which could not observed in the neutral AdS$_3$ backgrounds. We show that, the emergence of this discontinuity is a universal behavior in the charged massive BTZ theory. With the increase of graviton mass, on the other hand, no emergence of the discontinuity behavior for any small $q$ and $l$ could be observed. We also show that the evolution of the HEE and HC both become stable at later times, and $m_g$ speeds up reaching to the stability during the evolution of the system. Moreover, we show that $m_g$ decreases the final stable value of HEE but raises the stable value of HC. Additionally, contrary to the usual picture in the literature that the evolution of HC has only one peak, for big enough widths, we show that graviton mass could introduce two peaks during the evolution. However, for large enough charges the one peak behavior will be recovered again. We also examine the evolutions of HEE and HC growths at the early stage, which an almost linear behavior has been detected.