First Law of Entanglement Entropy vs Black Hole Entropy in an Anomalous Theory

TL;DR

This paper investigates the validity of the first law of entanglement entropy within topologically massive gravity, showing it holds under certain conditions despite bulk instabilities and discrepancies with black hole entropy.

Contribution

It demonstrates that the entanglement first law remains valid in TMG when bulk equations of motion are satisfied, even with bulk instabilities and differences from black hole entropy.

Findings

The entanglement entropy variation satisfies the first law under bulk equations of motion.

Black hole entropy differs from holographic entanglement entropy in boosted frames.

The entanglement first law is unaffected by discrepancies between entanglement and black hole entropy.

Abstract

In this note we explore the validity of the first law of entanglement entropy in the context of the topologically massive gravity (TMG). We found that the variation of the holographic entanglement entropy under perturbation from the pure AdS background satisfies the first law upon imposing the bulk equations of motion in a given time slice, despite the appearance of instabilities in the bulk for generic gravitationl Chern-Simons coupling $\mu$. The black hole entropy is different from the holographic entanglement entropy in a general boosted frame. This discrepancy however do not affect the entanglement first law.