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
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