Thermal density matrix breaks down the Page curve

TL;DR

This paper investigates how the entanglement entropy and the Page curve behave in a Schwarzschild black hole with finite temperature matter, revealing exponential growth at late times when using the island prescription.

Contribution

It introduces the analysis of entanglement islands and the Page curve with finite temperature conformal matter, extending previous vacuum-based studies.

Findings

Entanglement entropy grows exponentially at late times.

Finite temperature matter alters the traditional Page curve.

The island prescription leads to different entropy behavior in thermal states.

Abstract

In this paper, we study entanglement islands and the Page curve in the eternal four-dimensional Schwarzschild black hole surrounded by finite temperature conformal matter. By finite temperature conformal matter we mean the matter described by the thermal density matrix, rather than the usually considered matter above the Fock vacuum. We take the matter and the black hole at different temperatures and calculate the entanglement entropy for such a setup using the s-wave approximation. As a result, we obtain that at late times the island prescription leads to the exponential growth of the entanglement entropy of conformal matter in thermal vacuum.