AdS/CFT Correspondence and

Entanglement Entropy

Tadashi Takayanagi (Kyoto U.)

弦理論と場の理論 --- 量子と時空の最前線 @ 近畿大 07’

Thanks to my collaborators:

Shinsei Ryu (Santa Barbara) Matt Headrick (Stanford) Mukund Rangamani , Veronica Hubeny (Durham), Tomoyoshi Hirata, Tatsuma Nishioka (Kyoto) .

① 　Introduction

Ten years have already been passed since the AdS/CFT correspondence was discovered by J. Maldacena.

Many affirmative evidences have been accumulated for this conjectured duality, especially in the celebrated example of

Therefore, nowadays, many people believe the AdS/CFTis true at least in this particular example.

theory.Mills-YangSuper SU(N) (planar) 4N 4D

todual is which ,SAdSon string IIB Type 55

SYM SU(N) 4N 4D SAdSon string IIB 55

4

122 )8(

'

1

SYM 4N ofsymmetry -R SO(6)

symmetry conformal 4D SO(2,4)

YM

s

NgR

Ng

branes-D3 NNear horizon limit 0'

UV-IR relation

The coordinate z plays the role of length scale in the sense

of the RG scale in QFT.

The boundary z=0 corresponds to the UV limit.

.

:AdS of Coordinate Poincare

21

22222

)2(

2d

z

dxdtdzRds

d

i idAdS

.scale)(Energy scaleLength -1z

IR

z z

UV

z 0

2dAdS

1/z The energy of extended F-string∝Boundary

1 Energy z

Also many other successful examples have been found.

(i) The deformation of (i.e. R-sym)

This leads to less supersymmetric CFTs.

remains.try supersymme 1Nthen

manifold,Einstein -SasakiX If

. manifoldEinstein X S requires EOM

5

55

5S

(ii) The deformation of (UV-IR relation)

horizon

AdS BH UV IR

capped off

AdS

5AdS

entropy) zero-non (i.e. 0znear horizon Event

IR) in the freedom of degrees extra (i.e. re temperatuFinite (a)

　

0.znear geometry theof ncedisappeara The

t)confinemen (e.g. IR in the gap Mass (b)

holeBlack AdS

bubble) (AdSSoliton AdS

In spite of these remarkable development, we still do not

clearly understand the reason why the AdS/CFT is true.

Remember that the AdS/CFT is an explicit realization of

holography. A systematic proof of AdS/CFT will be related

to the proof of the holography itself.

The idea of holography

(d+2) dimensional (d+1) dimensional Quantum gravity Non-gravitational local theory (e.g. QM, QFT, CFT, etc.)Equivalent

Often, lives in the boundary of (d+2) dim. spacetime

Origin of Holography: Entropy Bound

If we remember the history of holography (’t Hooft, Susskind),

it is speculated from the idea of entropy bound in gravity.

When we stuff a certain region A with a lot of matter,

eventually it collapses into a black hole.

BH

A horizon Event

A

GMrA 2

Ex. 4D Schwarzschild Black hole

In Einstein gravity theory with or without , the entropy

of a black hole is given by the Bekenstein-Hawking formula,

proportional to the area of the horizon (= )

. 2

12

1 2221

22

drdrr

GMdt

r

GMds

A

. 4G

)Area(

N

AS

This consideration leads to the entropy bound in a region A

This bound tells us that the maximum amount of degrees

of freedom in a region is proportional not to the volume

but to the area of its boundary.

This suggests the gravity theory is equally described

by a non-gravitational theory in one dimension lower.

Holography

. 4G

)Area(

N

ASA

In this way, the correspondence of degrees of freedom (or information) between the gravity and its dual theory plays a crucial role in the understanding of holography.

In the AdS/CFT set up, this raises the following question:

Which region in the AdS does encode the information included in a certain region in the dual CFT? We would like to argue that this is answered by looking at the quantity called entanglement entropy.

This is closely related to the `inverse problem’:

Some information in QFT Holographic Dual metric

(Wilson loops, (e.g. AdS, AdS BH,….)

Correlation functions

Entanglement entropy)Spin chain Lattice QCDMatrix QM : :

Contents

① 　 Introduction 　 (+ Brief Review of AdS/CFT)

② 　 Entanglement Entropy in QFT (Review of EE)

③ 　 Holographic Entanglement Entropy

④ 　 BH Entropy as Entanglement Entropy

⑤ Conclusions and Discussions

② 　 Entanglement Entropy in QFT

(2-1) 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 .

. BAtot HHH

, Tr totBA A

B.in n informatio on the dependnot does if

],[Tr ]Tr[ :Note

A

AAAtotAA

O

OOO

Now the entanglement entropy is defined by the

von Neumann entropy w.r.t the reduced density matrixAS

. log Tr AAAAS

The simplest example

Consider a system with two ½ spins (two qubit)

2

1 (i)

BBAA

2 / (ii)B

ABA

? ?

? Entangled

Not Entangled

For the quantum state

Then we find the reduced density matrix

Finally we obtain the entanglement entropy as follows

This takes the maximal value when .

. sin cosTrA

22 AAABA

.sin log sin 2cos log cos 2|Tr 221

nn

AA nS

2logAS2

1cos2

. sin cosB

ABA

Here, we consider the entanglement entropy

(or geometrical entropy) in (d+1) dim. QFT

Then, we divide into A and B by specifying the

boundary .

A B

. NRM t

NBA

N

BA

N

. BAtot HHH

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.

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

Area Law of E.E.

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 a

SA

a

.4

on)Area(horiz

NBH G

S

(2-2) Entanglement Entropy in 2D CFT

Let us see the lowest dimensional example i.e. 2D CFTs.

First we review how to compute the entanglement entropy in 2D CFT. [ Holzhey-Larsen-Wilczek 94’,…, Calabrese-Cardy 04’]

A basic strategy is to first calculate as a certain

partition function and then to take the derivative of n

. | Tr log | Tr 1A1A

nn

Ann

AA nnS

nATr

In the path-integral formalism, the ground state wave

function can be expressed as follows

in the path-integral 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

In this way, we obtain the following representation

where is the partition function on the n-sheeted Riemann surface .

To evaluate , let us first consider the case where the CFT is defined by a complex free scalar field .

It is useful to introduce n replica fields on a complex plane .

, )(

Tr 1

nnn

A Z

Z

nZ

nZ

C1 n

n ,, 21

n

Then we can obtain a CFT equivalent to the one on

by imposing the boundary condition

By defining the conditions are diagonalized

n

),())(( ),())(( 12

12 vzvzeuzuze k

ikk

ik

vu

plane-z

k 1k

k1k

),(~

))((~

),(~

))((~ /22/22 vzevzeuzeuze k

nikikk

nikik

knikn

kk en

/210

1~

Using the orbifold theoretic argument, these twisted

boundary conditions are equivalent to the insertion of

(ground state) twisted vertex operators at z=u and z=v.

This leads to the following answer

For general CFTs, we can extend this analysis in a bit more

abstract way. In the end, we obtain

.)()()(Tr 1

0

)/1(3

1

//

n

k

nn

nknkn

A vuvu

.)(Tr )/1(

6 nnc

nA vu

Now the entanglement entropy is obtained as follows

(l is the length of A and the total system is infinitely long. )

).( log3

uvla

lcSA

If we consider the total system is a circle with the total

length L, then we instead find (l is again the length of A)

At finite temperature of a infinitely long system we find

. sinlog3

L

l

a

LcSA

. sinhlog3

l

a

cSA

1

(2-3) Higher Dimensional Case

In principle, we can compute the entanglement entropy

following the formula

However, its explicit evaluations are extremely complicated

and the analytical results have been restricted to some

special case of free field theories.

A motivation of the holographic method

. )(

Tr 1

nnn

A Z

Z

)Coordinate Poincare(AdS 2d

N

z

B

A

A Surface Minimal

)direction. timeomit the (We

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

③ 　 Holographic Entanglement Entropy(3-1) Holographic Formula

Holographic Calculation [Ryu-T]

(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

Comments:(i) We assumed a static asymptotically AdS space and considered the minimal surface on a time-slice.

e.g. pure AdS, AdS-Schwarzschild black hole

(ii) In the case of non-static background, we require that the surface is an extremal surface in the Lorentzian spacetime. [Hubeny-Rangamani-T]　　 e.g. Kerr-AdS black hole, Black hole formation proce

ss

Killing Horizon

Apparent Horizon(Dynamical Horizon)

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

The information in Bis encoded here.

Leading divergence and Area law

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 a

S

(3-2) A proof of the holographic formula [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 bulk-boundary relation 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

(3-3) 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

UV-IR duality

In this holographic calculation, the UV-IR duality is manifest

UV IR

Z Z

IRUV

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

(3-4) Higher Dimensional Cases

A B A BA

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

(3-5) 　 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 first 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.

[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

(3-6) Holographic Strong Subadditivity

It is known that the entanglement entropy satisfies

an interesting relation called strong subadditivity.

　　　　　　　　　　　　　 [Lieb-Ruskai 73’ ; See also Nielsen-Chuang’ text book 00’]

This is the most strong property of the entropy known at present.

. CBBABCBA SSSS

B AC

強劣加法性

This is an analogue of the second law of thermodynamics

and represents the concavity of the von-Neumann entropy.

A Very Quick Holographic Proof [Headrick-T, Hirata-T]

Note: This proof can be applied to not only to asymptotically AdS

spaces but also to any spacetimes with holographic duals.

A

B

C

=

A

B

C

A

B

C

④ 　 BH Entropy as Entanglement Entropy(4-1) Extremal BH entropy as Entanglement Entropy

[Work in progress with Azeyanagi and Nishioka]

Historically, the quantum entanglement has been

considered as an interesting candidate of the microscopic

origin of the Bekenstein-Hawking black hole entropy.

Nowadays, the microscopic explanation of (BPS) extremal

BHs has been established (essentially using AdS3/CFT2). [Strominger-Vafa 96’, ….]

However, it has not been understood well how the BH

entropy from the viewpoint of AdS2/CFT1.

Consider the metric of 4D extremal charged black hole.

The AdS2/CFT1 suggests that the micro state counting in the dual conformal QM (CFT1) will account for the BH entropy.

We would like to claim that in this setup the BH entropy is equal to the entanglement entropy of two CQMs.

2

2

2222

22

2

22

2222

22

2

22

)(

)(

SAdS

drdr

dtr

ds

drdrrr

rdt

r

rrds

The crucial point is that the space in the global

coordinate has two boundaries while have a single

one. 3dAdS

2AdS

AdS2CQM(A) CQM(B)

. )(cos 2

222

dd

ds

time

])Tr[( An

n-sheets

Cut

! 4 BH 4D)4(

2

41

)2( SG

rS

NG

entA

N

AdS2/CFT1 This is true even if we take the higher derivatives into account.

(4-2) Brane-World Black hole Entropy [Emparan hep-th/0603081]

Entropy of 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 ~

⑤ Conclusions and Discussions

We have reviewed the holographic computation of

entanglement entropy via the AdS/CFT. This clarifies

Region A in CFTd Region γA in AdSd+1

Dual

B

A γA

4

)Area()1(A

d

N

entA G

S

Many checks have been done for this holographic relation.

1) Complete agreement for AdS3/CFT2

[Ryu-T]

2) A proof from the bulk-boundary relation

[Frusaev]

3) Semi-quantitative test for AdS5*S5 / N=4 SYM [Ryu-T]

4) Exact agreement between the log term of EE

from AdS5 and the central charge in 4D CFT

[Ryu-T]

5) Holographic proof of strong subadditivity

[Hirata-T, Headrick-T]

6) Agreement between the EE in compactified YM and

the holographic one from AdS solition

[Nishioka-T]

Future Problems

(i) Relation to Covariant Entropy Bound (Bousso Bound)?

The holographic computation suggests [Hubeny-Rangamani-T]

(ii) Is there any second law as in the thermal entropy? An interesting observation is that the EE is increasing in the process of black hole formation process.

(iii) Applications to condensed matter physics ?

(iv) Metric extraction from EE ?

]Min[ BoussoentA A

SS

References

(i) Our original papers:

　 Ryu-T, hep-th/0603001, PRL96(2006)181602.

　　　　　 Ryu-T, hep-th/0605073, JHEP0608:045,2006.

Hirata-T, hep-th/0608213, JHEP 0702:042,2007.

Nishioka-T, hep-th/0611035, JHEP 0701:090,2007.

Headrick-T, arXiv:0704.3719 

Hubeny-Rangamani-T, arXiv:0705.0016

　　 笠 - 高柳 , 日本物理学会誌 62(2007)421

(ii) Related papers:[BH and EE]Bombelli-Koul-Lee-Sorkin, PRD34(1986) 373Srednicki, hep-th/9303048, PRL71(1993) 666 [EE in CFT]Holzhey-Larsen-Wilczek, hep-th/9403108, NPB424(1994)443Calabrese-Cardy, hep-th/0405152, JSM0406(2004)002Casini-Huerta, cond-mat/0511014, JSM0512(2005)012

[Quantum Phase Transition and EE] Vidal-Latorre-Rico-Kitaev, quant-ph/0211074, PRL90(2003)227902

[Topological Entanglement Entropy]Kitaev-Preskill, hep-th/0510092, PRL110404(2006)96Levin-Wen, cond-mat/0510613, PRL96(2006)110405 Fendley-Fisher-Nayak, cond-mat/0609072, JSM126(2007)1111

[Earlier Discussions on AdS/CFT and EE]

Hawking-Maldacena-Strominger, hep-th/0002145,JHEP0105(2001)001

Maldacena, hep-th/0106112, JHEP0304(2003) 021

[Recent Discussions on AdS/CFT and EE]

Emparan, hep-th/0603081, JHEP0606(2006)012

Iwashita-Kobayashi-Shiromizu-Yoshino, hep-th/0606027, PRD74(2006).

Fursaev, hep-th/0606184, JHEP0609(2006)018

Solodukhin, hep-th/0606205, PRL97(2006)201601

Hammersley, arXiv:0705.0159

Minton-Sahakian, arXiv:0707.3786 

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