Entanglement Entropy from AdS/CFT

Tadashi Takayanagi (Kyoto Univ.)

　 Based on　　　　　 hep-th/0603001, 0605073, 0608213, 0611035,　　　　　 arXiv:0704.3719, 0705.0016, 0710.2956, 0712.1850

with S. Ryu (KITP), V. Hubeny, M. Rangamani (Durham) M. Headrick (Stanford) T. Azeyanagi, T. Hirata, T. Nishioka (Kyoto) A. Karch and E. Thompson (Washington)

Miami 2007 Celebrating ten years of AdS/CFT

AdS/CFT has been successfully studied for ten years as the best example of holography.

Holography is expected to relate quantum gravity to various quantum mechanical systems.

To understand holography from general viewpoints, we need to find a universal quantity as a physical observable.

In this talk, we would like to point out that the quantity called entanglement entropy is such a candidate.

① Introduction

What is the entanglement entropy (EE) ?

A measure how much a given quantum state is quantum mechanically entangled (or complicated).

(A) It is universal in that it is well defined in any quantum mechanical systems.

(B) In QFT, it has the properties of both entropy and correlation functions.

21 (i)

BBAA

2 / (ii)B

ABA

? ?

? Entangled

Not Entangled

0 SA

2 log SA

The simplest example: two electrons (two qubits)

Definition of entanglement entropy

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

. BAtot HHH

A BExample: Spin Chain

Now the entanglement entropy is defined by the

von-Neumann entropy

. log Tr AAAAS

AS

We define the reduced density matrix for A by

taking trace over the Hilbert space of B .

A

, Tr totBA

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

. BAtot HHH

An analogy with black hole entropyAs we have seen, the entanglement entropy is definedby smearing out the Hilbert space for the submanifold B. EE ~ `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 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

, A)Area(~ 1

dA a

SA

a

.4

on)Area(horiz

NBH GS

2dAdS

)Coordinate Poincare(AdS 2dNRMon CFT

t

1d

-1energy)(~z

off)cut (UV az

1z IR UV

② Holographic Entanglement Entropy

We assume the AdS/CFT in the supergravity limit.But we can generalize this to other setups of holography.

Example. Poincare coordinate

Holographic Computation [06’ Ryu-TT]

(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 d

NA GS

A

)Coordinate Poincare(AdS 2d

N

z

B

A

A Surface Minimal

)direction. timeomit the (We

Comments

(i) We can compare AdS and CFT results exactly in AdS3/CFT2 case and find perfect agreements.

(ii) The holographic formula can be derived from the bulk-boundary relation of GKPW. [Fursaev 06’]

(iii) It is straightforward to derive the area law divergence of EE, induced from the warp factor of AdS spaces.

(iv) In the presence of the horizon, the EE for the total system coincides with the BH entropy. [This is consistent with earlier pioneering works: Maldacena 01’ and Hawking-Maldacena-Strominger 00’ ; See also Emparan 06’]

(v) The EE generally obeys the inequality called strong subadditivity. Our holographic formula clearly satisfies this. [Headrick-TT 07’]

(vi) In the time-dependent b.g., we need a covariant description. It turns that the minimal surface is replaced with the extremal surface in any Lorentzian spacetime. [Hubeny-Rangamani-TT 07’]

(3-1) EE from AdS3/CFT2 Consider AdS3 in the global coordinate

In this case, the minimal surface is a geodesic line which starts at and ends at ( ) .

). sinh cosh( 2222222 dddtRds

0 ,0 0 ,/2 Ll

AB ALl2

offcut UV:0

③ Various Examples

Boundary

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

Ll

RA

,sinlog34

||0

)3(

Llec

GS

N

AA

.23

)3(NGRc

Furthermore, the UV cutoff a is related to via

In this way we reproduced the known formula [94’ Holzhey-Larsen-Wilczek, 04’ Calabrese-Cardy ]

(In 2D CFT, we can analytically compute EE in various setups owing to the conformal map technique. But this is not so in higher dimensions.) [04’ Calabrese-Cardy ]

0

.~0

aLe

.sinlog3

Ll

aLcSA

Finite temperature caseWe assume the length of the total system is infinite.

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.

.sinhcosh21||cosh 20

2

l

RA

. sinhlog3

l

acSA

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

Let us compute the holographic EE dual to a CFT on R1,d.We concentrate on the following two examples.

(a) Infinite Strip (b) Circular disk

ll

1dL

(3-2) Higher Dimensional Case

A A

Entanglement Entropy for (a) Infinite Strip from AdS

.21

212 where

,)1(2

2/1

11

)2(

ddd

dd

dN

d

A

dddC

lLC

aL

GdRS

divergence law Area

vely.quantitatiresult CFT theit 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

dalq

alp

dpalp

alp

alp

dGRS

d

d

dd

dd

dN

dd

A

divergence law Area

Conformal Anomaly(~central charge)

A universal quantity which characterizes odd dimensional CFT

(3-3)　 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.

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

lLN

aLNS AdSA

L

l

Comparison with 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’]

.078.0 2

22

2

22

lLN

aLNKS freeCFTA

.051.0 2

22

2

22

lLN

aLNKS AdSA

(3-4) EE in confining gauge theories In many confining examples (AdS soliton, KS solution), we find a phase transition when we change the width l of A. [Nishioka-TT 06’, Klebanov-Kutasov-Murugan 07’]

SA-Sdiv

lConfinement/ deconfinement transition

zl

Minimal Surface

Disconnected Surfacesl

Twist parameter

Entropy (finite part)

Free Yang-Mills (sum of KK modes)

AdS side (Strongly coupled YM)

Supersymmetric Point

Comparison with Free Yang-Mills on Twisted Circle

AB

⑥ Conclusions• We have found the holographic area formula of entangle

ment entropy via AdS/CFT duality.

[Other our recent topics]• If we apply our argument to the AdS2/CFT1, we find EE of two CQMs = Wald entropy formula. [Azeyanagi-Nishioka-TT, 07’]

• In the defect or interface CFTs, we can extract the boundary entropy (g-function) from EE. It is holographically computable in Janus b.g.

[Azeyanagi-Karch-Thompson-TT, 07’]

Can we reconstruct the spacetime metric from the information of EE in the dual CFT ??

Boundary entropy obtained from 2D Interface CFT and Janus

Free interface CFT

3D Janus

Deformation

Boundary entropy　　　　（ =log g)