AdS/CFT Correspondence and

Entanglement Entropy

Tadashi Takayanagi (Kyoto U.)

　 Based on hep-th/0603001 [Phys.Rev.Lett.96(2006)181602] hep-th/0605073 [JHEP 0608(2006)045] by Shinsei Ryu (KITP) and T.T　 hep-th/0611035 to appear in JHEP by Tatsuma Nishioka (Kyoto U.) and T.T.

ISM 06’ Puri, India

① Introduction

In quantum mechanical systems, it is very useful to

measure how entangled ground states are.

An important measure is the entanglement entropy.

[Various Applications]

・ Quantum Information and quantum computer

(entanglement of qubit…….)

・ Condensed matter physics (search for new order parameters..)

・ Black hole physics (what is the origin of entropy?....)

maybe also, String theory? (the aim of this talk)

Definition of 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 .

Now the entanglement entropy is defined by the

von Neumann entropy

A

, Tr totBA

. BAtot HHH

AS

. log Tr AAAAS

Thus the entanglement entropy (E.E.) measures how A and B are entangled quantum mechanically.

(1) E.E. is the entropy for an observer who is only accessible to the subsystem A and not to B.

(2) E.E. is a sort of a `non-local version of correlation functions’. (cf. Wilson loops)

(3) E.E. is proportional to the degrees of freedom. It is non-vanishing even at zero temperature.

Motivation from Quantum Gravity

Q. What are general observables in Quantum Gravity? Holography 　 (AdS/CFT,…..)

(Quantum)Gravity = Lower Dim. Non-gravitational theory (Gauge theory, Matrix Models,…..)

Quantum Mechanics !

We can define the Entanglement Entropy!

We will see 3D E.E.= 4D Wilson loop in AdS/CFT.

In this talk we consider the entanglement entropy in

quantum field theories on (d+1) dim. spacetime

Then, we divide into A and B by specifying the

boundary .

A B

. NRM t

NBA

N

BA

N

An analogy with black hole entropy

As we have seen, the entanglement entropy is definedby smearing out the Hilbert space for the submanifold B. E.E. ~ `Lost Information’ hidden in B

This origin of entropy looks similar to the black hole entropy.

The boundary region ~ the event horizon.

A

BH

? ?Horizon

observerAn

Interestingly, this naïve analogy goes a bit further.

The E.E in d+1 dim. QFTs includes UV divergences.

Its leading term is proportional to the area of the (d-1) dim. boundary

[Bombelli-Koul-Lee-Sorkin 86’, Srednicki 93’]

where is a UV cutoff (i.e. lattice spacing).

Very similar to the Bekenstein-Hawking formula of black hole entropy

terms),subleading(A)Area(

~1

dA aS

A

a

.4

on)Area(horiz

NBH G

S

Of course, this cannot go beyond an analogy since is

finite, but are infinite. Also depends on the matter

content of QFTs, while not.[Instead, one-loop corrections to BH entropy are related to the entanglement

entropy, Susskind-Uglum 94’ and Fiola-Preskill-Strominger-Trevedi 94’ etc.]

In this talk we would like to take a different route to obtain

a direct holographic interpretation of entanglement entropy.

We will apply the AdS/CFT correspondence in order to

interpret the entanglement entropy in CFTs as a geometric quantity in AdS spaces. [cf. Earlier works:

Hawking-Maldacena-Strominger 00’

Maldacena 01’ ]

AS ASBHS

BHS

Contents

①Introduction

②Summary of Our Proposal

③Entanglment Entropy in 2D CFT from AdS3

④ Higher Dimensional Cases

⑤Entanglement Entropy in 4D CFT from AdS5

⑥ 3D Black hole entropy

　　　　 and Entanglement Entropy

⑦AdS Bubbles and Entropy

⑧Conclusions

Setup: AdS/CFT correspondence in Poincare Coordinate

2dAdS

)Coordinate Poincare(AdS 2dNRM

on CFT

t

1d

-1energy)(~z

off)cut (UV az

1z IR UV

.2

211

20

222

z

dxdxdzRds i

di

②　 Summary of Our Proposal

Our Proposal

(1) Divide the space N is into A and B. (2) Extend their boundary to the entire AdS space. This defines a d dimensional surface. (3) Pick up a minimal area surface and call this .

(4) The E.E. is given by naively applying the Bekenstein-Hawking formula

as if were an event horizon.

A

A

.4

)Area()2(

A

dN

A GS

A

)Coordinate Poincare(AdS 2d

N

z

B

A

A Surface Minimal

)direction. timeomit the (We

]98' Maldacena Yee,-[Reyn computatio loop Wilson cf.

Motivation of this proposal

Here we employ the global coordinate of AdS space and

take its time slice at t=t0.

t

t=t0

Coordinate globalin

AdS 2d

AB A? ?

.saturated) is bound (Bousso boundentropy

strict most thegives surface area Minimal

observerAn

Leading divergence

For a generic choice of , a basic property of AdS gives

where R is the AdS radius.

Because , we find

This agrees with the known area law relation in QFTs.

A

terms),subleading()Area(

~)Area(1

AA

d

d

aR

AA

terms).subleading(A)Area(

~1

dA aS

A proof of the holographic formula via GKP-Witten

[Fursaev hep-th/0606184]

In the CFT side, the (negative) deficit angle is localized on .

Naturally, it can be extended inside the bulk AdS by solving Einstein equation. We call this extended surface.

Let us apply the GKP-Witten formula in this background with the deficit angle .

A

A

)( Z iGravitySCFT e

)1(2 n

][Tr nAA

)1(2 n

sheets n

(cut)A

The curvature is delta functionally localized on the deficit

angle surface:

rms.regular te)()1(4 AnR

).1(4

)Area( ...

16

1 A2 n

GRgdx

GS

N

d

Ngravity

.4

)Area(

)(Z

Zlog trlog A

1

nA

Nn

nA Gnn

S

! surface minimal 0 AgravityS

Consider AdS3 in the global coordinate

In this case, the minimal surface is a geodesic line which

starts at and ends at

( ) .

Also time t is always fixed

e.g. t=0.

). sinh cosh( 2222222 dddtRds

0 ,0 0 ,/2 Ll

AB AL

l2

offcut UV:0

③ Entanglement Entropy in 2D CFT from AdS3

The length of , which is denoted by , is found as

Thus we obtain the prediction of the entanglement entropy

where we have employed the celebrated relation

[Brown-Henneaux 86’]

A || A

.sinsinh21||

cosh 20

2

L

l

RA

,sinlog34

||0

)3(

L

le

c

GS

N

AA

.2

3)3(

NG

Rc

Furthermore, the UV cutoff a is related to via

In this way we reproduced the known formula [Cardy 04’]

0

.~0

a

Le

.sinlog3

L

l

a

LcSA

Finite temperature caseWe assume the length of the total system is infinite.Then the system is in high temperature phase .

In this case, the dual gravity background is the BTZ black Hole and the geodesic distance is given by

This again reproduces the known formula at finite T.

1L

.sinhcosh21||

cosh 20

2

l

RA

. sinhlog3

l

a

cSA

Geometric Interpretation

(i) Small A (ii) Large A

A A B

A AB B

HorizonEvent

entropy.nt entangleme the to )3/(

on contributi thermal the toleads This horizon. ofpart a

wraps rature),high tempe (i.e. large isA When A

lTcSA

Now we compute the holographic E.E. in the Poincare metric

dual to a CFT on R1,d. To obtain analytical results,

we concentrate on the two examples of the subsystem A

(a) Straight Belt (b) Circular disk

ll

1dL

④ Higher Dimensional Cases

Entanglement Entropy for (a) Straight Belt from AdS

.21

21

2 where

,)1(2

2/1

11

)2(

d

dd

dd

dN

d

A

ddd

C

l

LC

a

L

Gd

RS

divergence law Area

vely.quantitatiresult CFT the

it with compare toginterestin isIt

cutoff. UVon the

dependnot does and finite is termThis

Entanglement Entropy for (b) Circular Disk from AdS

. !)!1/(!)!2()1( .....

)],....3(2/[)2(,)1( where

,

odd) (if log

even) (if

)2/(2

2/)1(

31

1

2

2

1

3

3

1

1)2(

2/

ddq

ddpdp

da

lq

a

lp

dpa

lp

a

lp

a

lp

dG

RS

d

d

dd

dd

dN

dd

A

divergence

law Area

universal. is thusandanomaly conformal theto

related is Actually theories.field with compare toquantities

ginterestin are and cutoff on the dependnot do termsThese

q

⑤　 Entanglement Entropy in 4D CFT from AdS5

Consider the basic example of type IIB string on ,

which is dual to 4D N=4 SU(N) super Yang-Mills theory.

We study the straight belt case.

In this case, we obtain the prediction from supergravity

(dual to the strongly coupled Yang-Mills)

We would like to compare this with free Yang-Mills result.

55 SAdS

.)6/1(

)3/2(2

2 2

223

2

22

l

LN

a

LNS AdS

A

Free field theory result

On the other hand, the AdS results numerically reads

The order one deviation is expected since the AdS

result corresponds to the strongly coupled Yang-Mills.

The AdS result is smaller as expected from the

interaction .

[cf. 4/3 problem in thermal entropy, Gubser-Klebanov-Peet 96’]

.078.02

22

2

22

l

LN

a

LNKS freeCFT

A

.051.02

22

2

22

l

LN

a

LNKS AdS

A

2],[ ji

⑥ 3D Black hole entropy and Entanglement Entropy [Emparan hep-th/0603081]

Entropy of 3D quantum black hole = Entanglement Entropy

)Coordinate Poincare(AdS4z

B

A

A :HorizonEvent

3)(dsolution blackhole

Myers-Horowitz-Emparan

)2(3

1)1( )1(

~ :world-Brane 3D

dN

dd

N GR

adG

Ra

Raz ~

⑦ 　 AdS bubbles and Entropy

Compactify a space coordinate xi in AdS space and impose

the anti-periodic boundary condition for fermions.

Closed string tachyons in IR region

)Coordinate Poincare(AdS 2d

-1energy)(~z IR UV

Anti-periodic

The end point of closed string tachyon condensation is conjectured to be the AdS bubbles (AdS solitons).

(Horowitz-Silverstein 06’)

Closed string

tachyon

Cap off !

We expect that the entropy will decrease under the

closed tachyon condensation and this is indeed true!

2dAdS

z IR UV

Anti-Periodic

2dAdS

z IR UVAnti-Periodic

Here we consider the twisted AdS bubbles

dual to the N=4 4D Yang-Mills with twisted boundary

conditions. In general, supersymmetries are broken.

Closed string tachyon condensation on a twisted circle [David-Gutperle-Headrick-Minwalla]

The metric can be obtained from the double Wick rotation

of the rotating 3-brane solution.

The metric of the twisted AdS bubble

The entanglement entropies computed in the free Yang-

Mills and the AdS gravity agree nicely!

This is another evidence for our holographic formula !

Twist parameter

Entropy

Free Yang-Mills

AdS side (Strongly coupled YM)

Supersymmetric Point

Ｃｆ . Casimir Energy=ADM mass

Energy

Twist parameter

3

4

AdS

freeYM

E

E

8

9

AdS

freeYM

E

E

SUSY

Free Yang-Mills

AdS gravity

⑧ Conclusions

• We have proposed a holographic interpretation of entanglement entropy via AdS/CFT duality.

• Our proposal works completely in the AdS3/CFT2 case. We presented several evidences in the AdS5/CFT4 case.

• The log term of E.E. in 4D CFTs is determined by the central charge a.

• The E.E. in the twisted AdS bubble offers us a quantitative test of AdS/CFT in a non-SUSY background.

E.E.= Area of minimal surface Spacetime metricReproduce

The wave function can be expressed as the path-

integral from t=-∞to t=0 in the Euclidean formalism

x

t

,0t

nintegratioPath

-t

t

Next we express in terms of a path-integral. BA Tr

abA ][

B A

b

a0t

x

t

B

Finally, we obtain a path integral expression of the trace

as follows. kaAbcAabAn

A ][][][Tr

nATr

a

ab

b

ly.successive boundarieseach Glue

n surfaceRiemann sheeted-

over integralpath a

n

sheets n

cut