Page Curve and Phase Transition in deformed Jackiw-Teitelboim Gravity Cheng-Yuan Lu1Ming-Hui Yu1Xian-Hui Ge1Li-Jun Tian1

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Page Curve and Phase Transition in deformed

Jackiw-Teitelboim Gravity

Cheng-Yuan Lu1,Ming-Hui Yu1,Xian-Hui Ge1∗,Li-Jun Tian1

1Department of Physics, Shanghai University, Shanghai 200444, China

Abstract

We consider the entanglement island in a deformed Jackiw-Teitelboim black hole in

the presence of the phase transition. This black hole has the van der Waals-Maxwell-like

phase structure as it is coupled with a Maxwell ﬁeld. We study the behavior of the Page

curve of this black hole by using the island paradigm. In the ﬁxed charge ensemble,

we discuss diﬀerent situations with diﬀerent charges that inﬂuence the system’s phase

structure. There is only a Hawking-Page phase transition in the absence of charges,

which leads to an unstable small black hole. Hence, the related Page curve does not

exist. However, a van der Waals-Maxwell-like phase transition occurs in the presence

of charges. This yields three black hole solutions. The Page curve of the middle size

black hole does not exist. For the extremal black hole, the Page time approaches zero

in the phase transition situation but becomes divergent without the phase transition.

In a word, we study the Page curve and the island paradigm for diﬀerent black hole

phases and in diﬀerent phase transition situations.

* Corresponding author. gexh@shu.edu.cn

arXiv:2210.14750v2 [hep-th] 19 Mar 2023

Contents

1 Introduction 3

2 Set up 4

3 Entanglement Entropy 7

3.1 Cutoﬀ surface far away horizon . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2 Cutoﬀ surface near horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Page curve and Page time 11

5 Phase transition of the black hole 13

5.1 Q=0 ....................................... 13

5.2 Q6=0 ....................................... 15

6 Discussion and Conclusion 21

1 Introduction

It is widely believed that studying the information loss paradox is signiﬁcant for revealing

quantum gravity. In 1975, Stephen Hawking discovered that the black hole would emit black-

body radiation which does not carry any information about the interior of the black hole

[1, 2]. If a black hole is formed by the pure state, then the whole process of black hole

evaporation corresponds to the evolution from the pure state to the mixed state, which

breaks the unitary principle of quantum mechanics. If quantum mechanics is respected, the

ﬁne-grained entropy, also called entanglement entropy or von-Neuman entropy, of Hawking

radiation should drop to zero at the end of the whole process. However, Don Page pointed

out that if black hole evaporation is a unitary process, then the entanglement entropy of the

radiation should linearly increase at the early time, and decrease after a special time – Page

time, this curve is called the Page curve [3, 4]. Recently, there is a great breakthrough in

this issue. By using the island formula [5], the Page curve emerges naturally. It leads to a

decreasing ﬁne-grained entropy for the Hawking radiation due to the emergence of the island

in the black hole interior. The ﬁne-grained entropy of Hawking radiation can be expressed

as [6, 7]

Srad = MinIExtIArea(∂I)

4GN

+SvN (R∪I),(1.1)

where Irepresents the island, GNis the Newton constant, Ris the radiation region and ∂I

is the boundary of the island. Crucially, the calculations include the contribution of islands

inside the black hole to the radiation entropy. In order to obtain the right answer, we should

ﬁrst extremize the generalized entropy and then pick up the minimum value to match this

formula. Before the Page time, there are no islands, so the generalized entropy without the

island is the smaller one. However, the generalized entropy with the island has more weight

in the process of minimization after the Page time. Therefore, the entanglement wedge

not only includes the outgoing Hawking radiation, but also the black hole interior purifying

partner.

The island formula was ﬁrst proposed in a solvable two dimensional evaporating black hole

model which is constructed by coupling a bath to a nearly-AdS2black hole [8]. Diﬀerent from

the evaporating black hole, the island of the eternal black hole is well established and outside

the event horizon [9]. Although the island formula is derived from Jackiw-Teitelboim(JT)

black holes, its application is beyond the context of Anti-de Sitter(AdS) spacetime. It can

be applied to other spacetime backgrounds, such as the other AdS2black hole [10–17], the

two-dimensional(2D) dilaton gravity [18–24], the higher-dimensional spacetime [25–34], the

cosmology [35–42], the boundary conformal ﬁeld theory(BCFT) [43–49]. Interestingly, there

are also some related works for the Sachdev-Ye-Kitaev(SYK) model [50–55].

Up to now, most studies have not yet considered the inﬂuence of the black hole phase

transition on the Page curve. Considering the possibility that the phase transition may

change the structure of the Page curve, we would like to study the relationship among the

island formula, the Page curve, and the phase transition. Based on this motivation, this

paper studies a special 2D black hole solution that yields van der Waals-Maxwell-like phase

structure in deformed JT gravity [62–64]. Following [9], we consider a two-side eternal black

hole coupled to ﬂat thermal baths. Then we use these strategies to obtain the Page curve

and discuss the eﬀect of the phase transition on the Page curve.

This paper is organized as follows. In section 2, we discuss the black hole solution which is

derived from JT-Maxwell gravity and introduce our toy model – the two-sided deformed JT

black hole coupled thermal baths. In section 3, we calculate the evolution of entanglement

entropy without island and display the paradox. Then, we consider an island that would

avoid the divergence of the entanglement entropy at late times. In section 4, based on the

previous result, we plot the Page curve and derive the scrambling time. In section 5, we

considered these results in combination with the phase transition. We discuss the change

of the Page time with diﬀerent charges in the canonical ensemble. We consider both the

Hawking-Page phase transition without charges and the van der Waals-Maxwell-like phase

transition with charges. Finally, we give a summary and some remarks in the last section.

Hereafter,we use the unit ~=kB=c= 1.

2 Set up

In this section, we brieﬂy review the deformed JT gravity with a Maxwell ﬁeld [58–62].

We can write down the action for the deformed JT gravity with metric gand dilaton coupled

to conformal matter, the action can be written as follows [56, 57],

S[g, φ, Φ] = Stop[g] + SJT[g, φ] + Sm[g, Φ]

=φ0

16πGNZM

√−gR + 2 Z∂M

√−γK

16πGNZM

√−gφ(R+ 2) + 2 Z∂M

√−γφb(K−1)

+Sm[g, Φ].

(2.1)

The ﬁrst term Stop is a purely topological term and only contributes a constant φ0χ,

where χis the Euler characteristic of the corresponding manifold M. The second term

SJT is the consequence of dimensional reduction. The dilaton φ, the Gibbons-Hawking-York

boundary term, a counter-term with boundary metric γand curvature K, and boundary

dilaton value φbare included in SJT. The last term Smis simply a 2D CFT action for a

matter ﬁeld, which does not couple to the dilaton directly.

Hereafter, we consider a deformed JT gravity plus a Maxwell ﬁeld. The action for this

theory reads as [62]

Scharg

JT =φ0

2ZM

d2x√−gR + 2 Z∂M

√−γKdτ

2ZM

d2x√−gφR +V(φ)

l2−1

2Z(φ)F2

2Z∂M

dτ√−γKµZ(φ)Fµν Aν+Z∂M

√−γφbKdτ

+Z∂M

√−γLctdτ,

(2.2)

where lis the Ads radius, Fis the Maxwell ﬁeld and Lct is the counterterms. The ﬁrst

bracket is the topological term for JT gravity. Through the variation of this action, we can

obtain the black hole solution of the above theory in the Schwarzschild coordinates,

ds2=−f(r)dt2+dr2

f(r),(2.3a)

V(φ)=2φ+a0

φη, φ(r) = r, (2.3b)

f(r) = r2−r2

l2+a0

(1 −η)l2r1−η−r1−η

H−Q2

(1 −ζ)l2r1−ζ−r1−ζ

H,(2.3c)

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PageCurveandPhaseTransitionindeformedJackiw-TeitelboimGravityCheng-YuanLu1,Ming-HuiYu1,Xian-HuiGe1,Li-JunTian11DepartmentofPhysics,ShanghaiUniversity,Shanghai200444,ChinaAbstractWeconsidertheentanglementislandinadeformedJackiw-Teitelboimblackholeinthepresenceofthephasetransition.Thisblackholehasthe...

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Page Curve and Phase Transition in deformed Jackiw-Teitelboim Gravity Cheng-Yuan Lu1Ming-Hui Yu1Xian-Hui Ge1Li-Jun Tian1

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