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
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