Holographic Entanglement Entropy and Black Holes

Tadashi Takayanagi(IPMU, Tokyo)

based on

arXiv:1008.3439 JHEP 11(2011) with Tomoki Ugajin (IPMU)

arXiv:1010.3700 with Wei Li (IPMU)

Indian Strings Meeting @ Puri, Jan 4-11, 2011

① 　 Introduction

AdS/CFT has been a very powerful tool to understand the

physics of black holes. [Strominger-Vafa 96’ AdS3/CFT2]

In spite of tremendous progresses, there are still several

unsolved or developing important problems on black

holes in quantum gravity.

The one which we would like to discuss here is about

the dynamical aspects where we cannot employ the

supersymmetries.

For example, consider the following problem:

(1) Complete understandings on how the BH information problem is avoided in string theory ?

[For static BH, there have been considerable developments:

e.g. Maldacena 01’, Festuccia-Liu 07’, Hayden-Preskill 07’,

Sekino-Susskind 08’, Iizuka-Polchinski 08’ …]

How to measure creations and annihilations of BHs

in holography ?

In this talk, I would like to argue that the holographic

entanglement entropy is a useful measure for this purpose.

A Quick Sketch of BH Information Problem

A lot of matter⇒Pure state

Black hole formation

Thermal radiations(BH evaporation)⇒Mixed state ??

Contradict with QM !Gravitational collapse

BH formation in AdS/CFT

Thermalization in CFT

BH formation in AdS [e.g. Chesler-Yaffe 08’,

S. Bhattacharyya and S. Minwalla 09’]

In particular, instantaneous

excitations of CFTs are called

quantum quenches.

[e.g. Calabrese-Cardy 05’-10’]

e.g. Mass quench

Time

BH

AdS

t

m(t)m(t)=m

m(t)=0: CFT

Cf. Thermalization in the probe D-brane AdS/CFT [Das-Nishioka-TT 10’]

Thermalization in a probe D-brane

　⇔ Horizon formation in its induced metric on the D-brane

r

Bulk observer

Brane observer

Emergent BH

Probe D-brane

AdS

2

. )...(

)(

brane-Dp Rotating

21

21

2

222

)1(22222

pT

dxdxr

drr

rdrds

H

p

pp

p

Entropy Puzzle in AdS/CFT (a `classical analogue’ of information paradox)

(i) In the CFT side, the von-Neumann entropy remains vanishing under a unitary evolutions of a pure state.

(ii) In the gravity dual, its holographic dual inevitably includes a black hole at late time and thus the entropy looks non-vanishing !

Thus, naively, (i) and (ii) contradict !

We will resolve this issue using entanglement entropy and study

quantum quenches as CFT duals of BH creations and evaporations.

.0)(log)(

),(),(

0

10000

tS(t)ρ(t) Tr ρtS

ttUttU(t)ρ

tottot

tot

Another important question is

(2) Entropy of Schwarzschild BH in flat spacetime ?

Is there any holography in flat spacetime ?

We will present one consistent outline about how the

holography in flat space looks like by employing the

entanglement entropy.

e.g. Volume law (flat space) ⇔ Area law (AdS/CFT)

highly entangled !

Contents

① 　 Introduction

② BH Formations as Pure States in AdS/CFT

③ Entanglement Entropy as Coarse-grained Entropy

④ Holography and Entanglement in Flat Space

⑤ Conclusions

(2-1) Holographic Entanglement Entropy

Divide a given quantum system into two parts A and B.Then the total Hilbert space becomes factorized

We define the reduced density matrix for A by

taking trace over the Hilbert space of B .

A

, Tr totBA

. BAtot HHH A BExample: Spin Chain

② BH formations as Pure States in AdS/CFT

Now the entanglement entropy is defined by the

von-Neumann entropy

In QFTs, it is defined geometrically (called geometric entropy).

. log Tr AAAAS

AS

BA B A

slice time:N

Holographic Entanglement Entropy

The holographic entanglement entropy is given by the area of minimal surface 　　 whose boundary is given by

. [Ryu-TT, 06’]

(`Bekenstein-Hawking formula’ when

is the horizon.)

AAS

N

AA 4G

)Area(S

2dAdS

z

A

)direction. timeomit the (We

B

A

1dCFT

A

A

Comments

• We need to employ extremal surfaces in the time-dependent spacetime. [Hubeny-Rangamani-TT, 0705.0016]

• In the presence of a black hole horizon, the minimal surfaces typically

wraps the horizon.

⇒ 　 Reduced to the Bekenstein-Hawking entropy, consistently.

• Many evidences and no counter examples for 5 years, in spite of

the absence of complete proof.

EE from AdS BH

(i) Small A (ii) Large A

A A B

A AB B

HorizonEvent

pure.not is tot ifSS BA

(2-2) Resolution of the Puzzle via Entanglement Entropy

Our Claim: The non-vanishing entropy appears only after coase-

graining. The von-Neumann entropy itself is vanishing

even in the presence of black holes in AdS.

First, notice that the (thermal) entropy for the total system can be found

from the entanglement entropy via the formula

This is indeed vanishing if we assume the pure state relation SA=SB.

).(lim0||

BAB

tot SSS

Indeed, we can holographically show this as follows:

[Hubeny-Rangamani-TT 07’]

A

2dAdS globalin

formation holeBlack

Time

. )( such that

tohomotopic surfaces extremal

, 4

Areamin)(

tA(t)γ

A(t)(t)γ

G

(t)γtS

A

A

N

AA

(t)γA

AB

A

B

BA A

BContinuous deformation leads to SA=SB

BH

Therefore, if the initial state does not include BHs, then always

we have SA=SB and thus Stot=0. 　 In such a pure state system,

the total entropy is not useful to detect the BH formation.

Instead, the entanglement entropy SA can be used to probe

the BH formation as a coarse-grained entropy.

[CFT calculation: Calabrese-Cardy 05’,..., (Finite Size effect our work⇒ )

Time evolutions of Hol EE: Hubeny-Rangamani-TT 07’

Arrastia-Aparicio-Lopez 10’, Albash-Johnson 10’

Balasubramanian-Bernamonti-de Boer-Copland-Craps-

Keski-Vakkuri-Müller-Schäfer-Shigemori-Staessens 10’]

t

Quantum quench

BH entropy

Scoase-grained ＝ SA(t)-Sdiv

③ 　 Entanglement Entropy as Coarse-grained Entropy

[Ugajin-TT 10’]

(3-1) Evolution of Entanglement Entropy and BH formation

As an explicit example , consider the 2D free Dirac fermion on a circle.

AdS/CFT: free CFT quantum gravity

with a lot of quantum corrections !

Assuming that the initial wave function 　　　 flows into a boundary

fixed point as argued in [Calabrese-Cardy 05’], we can approximate by

where is the boundary state. The constant is a regularization

parameter and measures the strength of the quantum quench:

.

0

, 0 Be H

B

1~ m

Calculations of EE (Replica method)

. ),(),(

])Tr[(

:expression the toleads This

.(y) :onbosonizati theemployed weAlso

dim.lowest ith w

),( Introduce

. ])Tr[(

2

1

2

12

22)(

11)(2

)(

),()(

1

N

Na

H

aaHN

A

yiX

yyXN

ai

a

N

NANA

BeB

ByyyyeB

e

r ex operatoted verta-th twis

eyy

S

The final result of entanglement entropy is given by

This satisfies

. 20 and offcut UV where

,

2422

2422

2

2224log

6

12log

3

1),(

11

6

2

1

2

1

a

iiitiiiti

iitii

atSA

σB

A

)recurrence Poincare an theshorter th(much

theoryfield free the tospecial Recurrence ),(),(

State Pure ).,()2,(),(

tStS

tStStS

AA

BAA

cf. Finite Size System at Finite Temperature (2D free fermion c=1 on torus) [07’ Azeyanagi-Nishioka-TT]

.

sinh

1cot)1(

2

)1(

)1)(1(log

3

1

sinhlog

3

1

1

12/2

/2/2/2/2

m

mm

m

e

eeee

aS

m

m

im

mm

A

σ

SA Thermal Entropy

thermal

1

0

sinh2coth2)1(

3

)]()1([lim

:show alsocan We

S

lll

l

SS

l

l

AA

Time evolution of entanglement entropy

divA StS 2.0),(

t

Time

BH

BHHolography

BH formation and evaporation in extremely quantum gravity

No information paradox at all in either side !

Quantum quench

Quantum quench in free CFT

(3-2) More comments

First of all, we can confirm that the scalar field X

(= bosonization of the Dirac fermion) is thermally excited:

. !

)(00

!

)( Tr

. 4

1

)1(2

).~ , ( ),X( :sOscillator

0 0

42rightleft

4

nneBBe

Te

nE

αα

nm

m n

nmmnH

effnnnn

nn

Correlation functions

)(tSeff

)0,(tikXe

)0,0()0,( ikXtikX ee

)0,0(),( ikXtikX ee

Size of BH

Radiations

Exponential decay⇒Thermal

Semi Classical AdS BHs ?

For semi classical BHs, which are much more interesting, the dual CFT

gets strongly coupled as usual and thus the analysis of the time

evolution is difficult.

However, we can still guess what will happen especially in the AdS3

BH case.

Scoase-grained ＝ SA(t)-Sdiv

t

Quantum quench

BH entropy

Universal (for any CFT)

Poincare recurrence

④ Holography and Entanglement in Flat Space [Li-TT 10’]

The entanglement entropy is a suitable observable in

general setup of holography as in our BH example.

Motivated by this, finally we would like to discuss what a

holography for the flat spacetime looks like.

So, simply consider

Note: We may regard this as a logarithmic version of Lifshitz backgrounds:

offcut UV

. : 22221d

d dddsR

22222

22Lif

222

22Flat )(log xdrdtr

r

drdsdr

r

drds z

d

Correlation functions

We can compute the holographic n-point functions following

the bulk-boundary relation just like in AdS/CFT.

Assuming a scalar in Rd+1 like the dilaton:

Then we find that all n-point functions scale simply:

⇒ 　 Only divergent terms appear !

Adding the (non-local) boundary counter term,

all correlation functions become trivial !

.)(4

1 1

fgdx

GS d

N

).,,,()(

)()()( 21

1

2211 nN

d

nn yyygG

yOyOyO

Holographic Entanglement Entropy (HEE)

Though this result seems at first confusing, actually it is

consistent with the holographic entanglement entropy.

It is easy to confirm that the HEE follows the volume law rather

than the standard area law !

.4

2Sin)(

~

11

N

dd

A GS

dS

1dRAA

B

θ

! 0limit in the

)(Vol)(Area

AA

This unusual volume law implies that the subsystem A gets

maximally entangled with B when A is infinitesimally small.

Therefore, we have the trivial correlation functions:

At the same time, the volume law argues that the holographic

dual of flat space is given by a highly non-local theory.

.0][T 2121 nAn OOOrOOO

iiMnMM

A MnMMMnMM ,...,2,1,...,2,1,..,2,1

Calculations of EE in Non-local Field Theory

we can calculate EE for θ=π 　 as follows:

In the end we find,

Thus the volume law corresponds to the non-local action q=1/2 i.e.

[Note: a bit similar to open string field theory]

, ])([ )( action, theory field For the fgdS d

.Tr2

1log ,

)(log )(

)/Z(S1

/11

dsf

N

N

NN

A es

dsZ

Z

Z

NS

N

. )(

. )(

22

2

qd

Ax

d

Ap

a

LSexf

a

LSxxf

q

. ][ 2

egdxS d

Possible Relation to Schwarzschild BH Entropy

This can be comparable to the Schwarzschild BH entropy:

This suggests that the BH entropy may be interpreted as

the entanglement entropy…

. )(

11~

)(~

1

1

dUNN

d

A TGGS

. 2

1T : boundary at the Temp.Unruh U

.)(

11~

1d

BHNBH TG

S

⑤ Conclusions

• We raised a puzzle on the entropy in the thermalization in AdS/CFT and resolved by showing the total von-Neumann entropy is always vanishing in spite that the AdS includes BHs at late time.

• We present a toy model of holographic dual of BH formations and

evaporations using quantum quenches.

• We discussed a consistent picture of holography for flat space

and argued that the dual theory is should be non-local and is highly

entangled.

Future problems: possible holography for de Sitter spaces and

cosmological backgrounds