Download A Holographic Interpretation of Entanglement Entropy

Indian Strings Meeting @ Puri, Jan 4-11, 2011
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)
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
Gravitational collapse
Black hole formation
Thermal radiations
(BH evaporation)
⇒Mixed state ??
Contradict
with QM !
BH formation in AdS/CFT
Thermalization in CFT
BH formation in AdS
Time
[e.g. Chesler-Yaffe 08’,
S. Bhattacharyya and S. Minwalla 09’]
BH
AdS
In particular, instantaneous
excitations of CFTs are called
quantum quenches.
[e.g. Calabrese-Cardy 05’-10’]
e.g. Mass quench
m(t)=m
m(t)
m(t)=0: CFT
t
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
AdS
Bulk observer
Probe D-brane
Rotating Dp - brane
r 2( p 1)
2
ds  (r   )d  2 p
dr
r 2p
 r 2 (dx12  ...  dx 2p 1 ) .
2
2
2
 p
TH 
2
Brane observer
Emergent BH
r
2
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.
ρtot (t)  U (t , t 0 ) 0 0 U (t , t 0 ) 1
 S (t )  Tr ρtot (t) log ρtot (t)  S (t 0 )  0.
(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.
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
②
③
④
⑤
Introduction
BH Formations as Pure States in AdS/CFT
Entanglement Entropy as Coarse-grained Entropy
Holography and Entanglement in Flat Space
Conclusions
② BH formations as Pure States in AdS/CFT
(2-1) Holographic Entanglement Entropy
Divide a given quantum system into two parts A and B.
Then the total Hilbert space becomes factorized
H tot  H A  H B .
Example: Spin Chain
A
B
We define the reduced density matrix  A for A by
 A  TrB tot ,
taking trace over the Hilbert space of B .
Now the entanglement entropy
von-Neumann entropy
SA
is defined by the
S A  TrA  A log A
.
In QFTs, it is defined geometrically (called geometric entropy).
N : time slice
B
A
A  B
Holographic Entanglement Entropy
The holographic entanglement entropy S A is given by the
area of minimal surface  A whose boundary is given by A .
[Ryu-TT, 06’]
CFTd 1
A
Area( A )
SA 
4G N
A
B
AdSd  2
z
(We omit the time direction.)
(`Bekenstein-Hawking formula’ when
 A is the horizon.)
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
Event Horizon
A
A
B
A
S A  SB
A
if  tot is not pure.
B
B
(2-2) Resolution of the Puzzle via Entanglement Entropy
Our Claim: The non-vanishing entropy appears only after coasegraining. 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
Stot  lim ( S A  S B ).
| B|  0
This is indeed vanishing if we assume the pure state relation SA=SB.
Indeed, we can holographically show this as follows:
 Areaγ A(t) 
S A (t )  min 
 ,
4GN


γ A(t)  extremal surfaces homotopic to A(t)
Time
such that γ A(t)  A(t ) .
γ A(t)
[Hubeny-Rangamani-TT 07’]
A
BH
A
A
B
A
A B
B
Black hole formation
in global AdSd  2
B
Continuous deformation leads to SA=SB
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.
Scoase-grained ＝SA(t)-Sdiv
Quantum
quench
BH entropy
t
[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-CrapsKeski-Vakkuri-Müller-Schäfer-Shigemori-Staessens 10’]
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 0 flows into a boundary
fixed point as argued in [Calabrese-Cardy 05’], we can approximate by
0  e H B ,
where B is the boundary state. The constant  is a regularization
parameter and measures the strength of the quantum quench:
m ~  1
.
Calculations of EE (Replica method)
S A   N Tr[(  A ) N ]
N 1
.
Introduce  ( a ) ( y , y )  e
i
a
X ( y, y )
N
 a-th twisted vertex operator
with lowest dim.
Also we employed the bosonization :  (y)  e iX ( y ) .
This leads to the expression :
Tr[(  A ) N ] 
N 1
2

a 
N 1
2
B e  2H   ( a ) ( y1 , y1 )   (  a ) ( y 2 , y 2 ) B
Be
 2H
B
.
The final result of entanglement entropy is given by
2
2
 i i 
   it i 
1 
  1 

1
2 1
 4 2 
 2 2 
S A (t ,  )  log  log
6
3
a 6
 2  2it  i i   2  2it  i i 
 i 
    1 
  1 

4
2  
4
2 
 2 

where a  UV cut off and 0    2 .
B
This satisfies
σ
A
,
cf. Finite Size System at Finite Temperature
(2D free fermion c=1 on torus) [07’ Azeyanagi-Nishioka-TT]
 
1 
S A  log sinh 
3  a
 
 (1  e 2 /  e 2m /  )(1  e 2 /  e 2m /  ) 
 1 
    log 

 2m /  2
3
(
1

e
)
i 1



 m 
m

  1
cot
m


 
(1)

SA Thermal Entropy
 2

.
m
 m 
m 1

sinh 
  
σ
Time evolution of entanglement entropy
S A (t ,  ) 0.2  S div
Time
Holography
BH
BH
t
Quantum
quench
Quantum quench in free CFT
BH formation and evaporation
in extremely quantum gravity
No information paradox at all in either side !
(3-2) More comments
First of all, we can confirm that the scalar field X
(= bosonization of the Dirac fermion) is thermally excited:
Oscillator s : X( ,  )  (αn , α~n ).
En    n n 
n
2 (e
 left  Trright e  2H
4n

 1)
Teff 
1
4
.
n
n
(

)
(

)
B B   e  4mn  m 0 0 m
n!
n!
m 0 n 0


.
Correlation functions Size of BH
Seff (t )
e
e
ikX (t , 0 )
ikX ( t , 0 )  ikX ( 0 , 0 )
e
eikX ( t , ) e  ikX ( 0,0 )
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
Quantum
quench
Poincare recurrence
BH entropy
t
Universal (for any CFT)
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 R d 1 :
ds 2  d 2   2 d 2d .
UV cut off     
Note: We may regard this as a logarithmic version of Lifshitz backgrounds:
2
ds Flat
dr 2
 2  (log r ) 2 d 2d
r
2
 ds Lif

dr 2
 2  r 2 z dt 2  r 2 dx 2
r
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:
1
S
4G N


d 1

dx
g
f
(

)



.


Then we find that all n-point functions scale simply:
(   ) d 1
O1 ( y1 )O2 ( y 2 ) On ( y n ) 
g ( y1 , y 2 ,, y n ).
GN
⇒ Only divergent terms appear !
Adding the (non-local) boundary counter term,
all correlation functions become trivial !
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 !
d
( )
SA ~
d 1


  Sin 
2

4G N
B
d 1
Area ( A )  Vol( A)
in the limit   0 !
.
A
A
S
θ
R
d 1
This unusual volume law implies that the subsystem A gets
maximally entangled with B when A is infinitesimally small.
A 
 M 1, M 2,..., Mn
M 1, M 2,..., Mn   i  i
M 1, M 2 ,.., Mn
Therefore, we have the trivial correlation functions:
O1O2 On  Tr[  AO1O2 On ]  0.
At the same time, the volume law argues that the holographic
dual of flat space is given by a highly non-local theory.
Calculations of EE in Non-local Field Theory
S   (d) d
For the field theory action,
g [  f ()   ] ,
we can calculate EE for θ=π as follows:
ZN

SA 
log
N
( Z 1 )1 / N
,
1  ds
 
Tr (Sd /Z ) e  sf (   ) .
N
2  s
log Z N
N 1

p
In the end we find,  f ( x)  x



xq
 f ( x)  e



L
SA   
a
L
SA   
a
d 2
.
d 2 2 q
.
Thus the volume law corresponds to the non-local action q=1/2 i.e.
S   dx
d
g [  e
 2
] .
[Note: a bit similar to open string field theory]
Possible Relation to Schwarzschild BH Entropy
1
Unruh Temp. at the boundary : TU 
2
.
(   ) d 1
1
1
SA ~
~

.
d 1
GN
G N (TU )
This can be comparable to the Schwarzschild BH entropy:
S BH
1
1
~

.
d 1
G N (TBH )
This suggests that the BH entropy may be interpreted as
the entanglement entropy…
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