© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2512.21995

Entanglement Islands and Thermodynamics of the Black Hole in Asymptotically Safe Quantum Gravity

Sobhan Kazempour, Sichun Sun, Chengye Yu

gr-qchep-thquant-phSubmitted Dec 26, 2025• Announced Dec 29, 2025

Abstract

We study thermodynamic properties and the entanglement island of a black hole in asymptotically safe quantum gravity, analyzing key thermodynamic quantities such as the Hawking temperature, heat capacity, and entropy, as well as the mass-horizon radius relation. Unlike Schwarzschild black holes, the temperature decreases with mass near the evaporation endpoint, signaling a phase transition and possible stable remnant. The entanglement entropy of Hawking radiation is obtained both with and without island contributions. Without islands, the radiation entropy grows linearly indefinitely, leading to the information paradox. By including island contributions and extremizing the generalized entropy functional, we resolve this paradox. At late times, the radiation entropy saturates at the Bekenstein-Hawking entropy, confirming unitary evolution. From this, we derive the Page time and scrambling time by equating early- and late-time entanglement entropies. The result of this study establishes the finiteness of the radiation entropy and consistency with quantum mechanics.

Related papers

Where the Quantum World Meets Gravity: Entanglement Entropy and the Thermodynamics of Black Holes

L. G. Zazunov, Michael Shatnev

Black holes are windows into the deepest structures of reality, where quantum mechanics, thermodynamics, and geometry converge. In this pedagogical preprint we explore how quantum entanglement entropy provides a microscopic foundation for black hole thermodynamics. Starting from rst principles of quantum information theory, we introduce entanglement entropy, the area law, and Hawking radiation, then proceed to modern results: the Ryu-Takayanagi formula, quantum extremal surfaces, the Page curve, replica wormholes, and the island formula. Throughout, the emphasis is on physical intuition. No prior knowledge of string theory or loop quantum gravity is assumed.

•Mar 28, 2026•0 citations

Notes on the islands in the CGHS black holes

Takanori Anegawa, Norihiro Iizuka

We study the islands and the Page curve in the 1+1-dimensional eternal CGHS black hole models. Without islands, the entanglement entropy of the radiation grows linearly at late time. However with an island, its growth stops at the value of almost twice of the black hole entropy. Therefore an island emerges at the late time, and the entanglement entropy of the radiation shows the Page curve.

•Apr 3, 2020•0 citations

Evaporation of black holes in flat space entangled with an auxiliary universe

Akihiro Miyata, Tomonori Ugajin

Abstract We study a thermofield double type entangled state on two disjoint universes A and B, where one of the universes is asymptotically flat containing a black hole. As we increase the entanglement temperature, this black hole receives back-reaction from the stress–energy tensor of the state. This results in lengthening of the wormhole region in the black hole interior, and decreasing of its horizon area, both of which are key features of an evaporating black hole. We then compute the entanglement entropy in universe A through the island formula, and argue that it naturally follows the Page curve of an evaporating black hole in flat space. We also study the effects of local operations in the gravitating universe with the black hole. We find that they accelerate the evaporation of the black hole, thereby disrupting the entanglement between the two universes. Furthermore, we observe that, depending on whether the operation can be regarded as a local operation and classical communication or not, the behavior of the entanglement entropy changes. In particular, when the operation is made neither in the entanglement wedge of the radiation system nor that of the black hole, the transition between the island phase and the no-island phase can happen multiple times.

arXiv:2104.00183•Progress of Theoretical and Experimental Physics•Dec 8, 2021•28 citations

Islands and entanglement entropy in $d$ -dimensional curved backgrounds

Filip Landgren, Arvind Shekar

A large part of the discussion on entanglement islands has explored the specific setup of $2d$ JT gravity with a flat heatbath coupled to a $2d$ CFT. In this paper, we consider a more general setup and treatment of islands in a $d-$dimensional AdS black hole background. The quantum fields modeling the Hawking radiation have a scale and are consistently inherited from a conformal parent theory; their symmetries are thus compatible with those of curved backgrounds. We demonstrate explicitly that the existence of islands is sensitive to the choice of CFT used to model the Hawking radiation. We compute the renormalised entanglement entropy of conformal fields on a negatively curved background in $d$ dimensions at zero temperature as well as the thermal regulated entropy of an entangling region near the UV boundary. Using the latter quantity as the entropy of the Hawking radiation, we find that islands never emerge for $d>2$.

arXiv:2401.01653•Jan 3, 2024•0 citations

$q$-Deformed Quantum Mechanics and the Thermodynamics of Black Hole/White Hole Spectral pair

S. Jalalzadeh, R. Jalalzadeh, H. Moradpour

In this work, we investigate the thermodynamics of Schwarzschild black and white holes within a $q$-deformed Wheeler--DeWitt framework. By introducing a $q$-deformed Heisenberg--Weyl algebra at a root of unity, we derive a finite-dimensional Hilbert space, a bounded mass spectrum, and an adiabatic invariant leading to a bounded entropy-mass relation. The deformation results in a universal logarithmic correction, as well as a minimum temperature and a maximum entropy that matches the de Sitter bound. Also, we examine the interpretation of a cold remnant, which is dynamically stable because its radiation rate approaches zero, even though its heat capacity remains negative. We also explore the holographic implications of this limited entropy. Our results thus provide a consistent semiclassical picture, where quantum deformation naturally introduces an entropy bound, avoids divergences at the final evaporation stage, and suggests a smooth transition from quantum gravity to cosmology.

gr-qcCross-listedarXiv:2603.27621•Mar 31, 2026

Probing Hawking radiation through capacity of entanglement

Kohki Kawabata

We consider the capacity of entanglement in models related with the gravitational phase transitions. The capacity is labeled by the replica parameter which plays a similar role to the inverse temperature in thermodynamics. In the end of the world brane model of a radiating black hole the capacity has a peak around the Page time indicating the phase transition between replica wormhole geometries of different types of topology. Similarly, in a moving mirror model describing Hawking radiation the capacity typically shows a discontinuity when the dominant saddle switches between two phases, which can be seen as a formation of island regions. In either case we find the capacity can be an invaluable diagnostic for a black hole evaporation process.

•Jul 10, 2023•68 citations

We study thermodynamic properties and the entanglement island of a black hole in asymptotically safe quantum gravity, analyzing key thermodynamic quantities such as the Hawking temperature, heat capacity, and entropy, as well as the mass-horizon radius relation. Unlike Schwarzschild black holes, the temperature decreases with mass near the evaporation endpoint, signaling a phase transition and possible stable remnant. The entanglement entropy of Hawking radiation is obtained both with and without island contributions. Without islands, the radiation entropy grows linearly indefinitely, leading to the information paradox. By including island contributions and extremizing the generalized entropy functional, we resolve this paradox. At late times, the radiation entropy saturates at the Bekenstein-Hawking entropy, confirming unitary evolution. From this, we derive the Page time and scrambling time by equating early- and late-time entanglement entropies. The result of this study establishes the finiteness of the radiation entropy and consistency with quantum mechanics.

Black holes are windows into the deepest structures of reality, where quantum mechanics, thermodynamics, and geometry converge. In this pedagogical preprint we explore how quantum entanglement entropy provides a microscopic foundation for black hole thermodynamics. Starting from rst principles of quantum information theory, we introduce entanglement entropy, the area law, and Hawking radiation, then proceed to modern results: the Ryu-Takayanagi formula, quantum extremal surfaces, the Page curve, replica wormholes, and the island formula. Throughout, the emphasis is on physical intuition. No prior knowledge of string theory or loop quantum gravity is assumed.

Abstract We study a thermofield double type entangled state on two disjoint universes A and B, where one of the universes is asymptotically flat containing a black hole. As we increase the entanglement temperature, this black hole receives back-reaction from the stress–energy tensor of the state. This results in lengthening of the wormhole region in the black hole interior, and decreasing of its horizon area, both of which are key features of an evaporating black hole. We then compute the entanglement entropy in universe A through the island formula, and argue that it naturally follows the Page curve of an evaporating black hole in flat space. We also study the effects of local operations in the gravitating universe with the black hole. We find that they accelerate the evaporation of the black hole, thereby disrupting the entanglement between the two universes. Furthermore, we observe that, depending on whether the operation can be regarded as a local operation and classical communication or not, the behavior of the entanglement entropy changes. In particular, when the operation is made neither in the entanglement wedge of the radiation system nor that of the black hole, the transition between the island phase and the no-island phase can happen multiple times.

A large part of the discussion on entanglement islands has explored the specific setup of

2d

JT gravity with a flat heatbath coupled to a

2d

CFT. In this paper, we consider a more general setup and treatment of islands in a

d-

dimensional AdS black hole background. The quantum fields modeling the Hawking radiation have a scale and are consistently inherited from a conformal parent theory; their symmetries are thus compatible with those of curved backgrounds. We demonstrate explicitly that the existence of islands is sensitive to the choice of CFT used to model the Hawking radiation. We compute the renormalised entanglement entropy of conformal fields on a negatively curved background in

d

dimensions at zero temperature as well as the thermal regulated entropy of an entangling region near the UV boundary. Using the latter quantity as the entropy of the Hawking radiation, we find that islands never emerge for

d>2

.

q

-Deformed Quantum Mechanics and the Thermodynamics of Black Hole/White Hole Spectral pair

In this work, we investigate the thermodynamics of Schwarzschild black and white holes within a

q

-deformed Wheeler--DeWitt framework. By introducing a

q

-deformed Heisenberg--Weyl algebra at a root of unity, we derive a finite-dimensional Hilbert space, a bounded mass spectrum, and an adiabatic invariant leading to a bounded entropy-mass relation. The deformation results in a universal logarithmic correction, as well as a minimum temperature and a maximum entropy that matches the de Sitter bound. Also, we examine the interpretation of a cold remnant, which is dynamically stable because its radiation rate approaches zero, even though its heat capacity remains negative. We also explore the holographic implications of this limited entropy. Our results thus provide a consistent semiclassical picture, where quantum deformation naturally introduces an entropy bound, avoids divergences at the final evaporation stage, and suggests a smooth transition from quantum gravity to cosmology.

We consider the capacity of entanglement in models related with the gravitational phase transitions. The capacity is labeled by the replica parameter which plays a similar role to the inverse temperature in thermodynamics. In the end of the world brane model of a radiating black hole the capacity has a peak around the Page time indicating the phase transition between replica wormhole geometries of different types of topology. Similarly, in a moving mirror model describing Hawking radiation the capacity typically shows a discontinuity when the dominant saddle switches between two phases, which can be seen as a formation of island regions. In either case we find the capacity can be an invaluable diagnostic for a black hole evaporation process.