and its New DevelopmentsStrings 2011@Uppsala, July 1 Holographic Entanglement Entropy and its New...
Transcript of and its New DevelopmentsStrings 2011@Uppsala, July 1 Holographic Entanglement Entropy and its New...
Strings 2011@Uppsala, July 1
Holographic Entanglement EntropyHolographic Entanglement Entropy and its New Developments
Tadashi Takayanagi y g(IPMU, the University of Tokyo)
Thanks to my collaborators:
Tatsuo Azeyanagi (Riken)Mitsutoshi Fujita (IPMU)j ( )Yasuyuki Hatsuda(YITP)Matt Headrick (Brandeis)
hi iTomoyoshi HirataVeronica Hubeny (Durham)Andreas Karch (Washington U )Andreas Karch (Washington U.)Wei Li (IPMU) Tatsuma Nishioka (Princeton)Noriaki Ogawa (IPMU)Mukund Rangamani (Durham)Shi i R (UC B k l )Shinsei Ryu (UC Berkeley)Ethan Thompson (Washington U.)Erik Tonni (MIT)Erik Tonni (MIT)Tomonori Ugajin (IPMU)
ContentsContents
①① Introduction② Holographic Entanglement Entropy (HEE)③ HEE of Confining Gauge Theories and Higher Derivatives
④ AdS/BCFT and Quantum Entanglement④ AdS/BCFT and Quantum Entanglement
⑤ Towards Holographic Dual of Lattices
⑥ C l i⑥ Conclusions
① Introduction① Introduction
Holography (e.g. AdS/CFT [Maldacena 97] )Holography (e.g. AdS/CFT [Maldacena 97] )
⇒ Non‐perturbative Definition of Quantum Gravity
d+2 dimd+1 dim
M∂ Md+2 dim.d+1 dim.
Boundary Bulk
)()( MZMZ GQM =∂ )()( MZMZ GravityQM ∂
l h h l h l d blTo explore the holography in general setups, we need suitable
physical quantities. [not only AdS, but flat spaces, de Sitter, etc.]
Stationary BH ⇒Mass M, Charge Q, Spin J. (Thermodynamics)
Generic spacetime ⇒ We need much more quantities !(Non‐equilibrium)
We would like to argue that the entanglement entropy (EE)
will be an appropriate quantity.
Definition of Entanglement Entropy
Divide a quantum system into two subsystems A and B.
. BAtot HHH ⊗=A BExample: Spin Chain
W d fi h d d d i i f A bρWe define the reduced density matrix for A byAρ
taking trace over the Hilbert space of Btaking trace over the Hilbert space of B .
N h l i d fi d b hSNow the entanglement entropy is defined by the
von‐Neumann entropyAS
In QFTs, it is defined geometrically:
slice time:N
A BA ∂=∂B A
Various Applications in other subjects
• Quantum Information and Quantum Computing
EE = the amount of quantum information[see e.g. Nielsen‐Chuang’s text book 00]
• Condensed Matter Physics
EE = Efficiency of a computer simulation (DMRG) [Gaite 03,…]
Divergent at phase transition points ![G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, 02,…][ , , , , , ]
i.e. Quantum Critical Points [Sachdev’s talk]
A new quantum order parameter ![Topological entanglement entropy: Kitaev‐Preskill 06, Levin‐Wen 06]
Basic property: Area law p p y
EE in d+1 dim. QFTs (in the ground states) includes UV div. [Bombelli‐Koul‐Lee‐Sorkin 86, Srednicki 93]
A)Area(∂ terms),subleading(A)Area(~ 1 +∂−dAS
ε
where is a UV cutoff (i.e. lattice spacing).ε
Similar to the Bekenstein‐Hawking formula of black hole entropy [ EE = loop corrections to BH entropy[ EE = loop corrections to BH entropy,
Susskind‐Uglum 94,…] .
4on)Area(horiz
NBH G
S =4 NG
② Holographic Entanglement Entropy
(2‐1) Holographic Entanglement Entropy Formula [Ryu‐TT 06][ y ]
)A ( CFT
N
AA 4G
)Area(S γ=
1dCFT +
N
AdS
A
is the minimal area surface (codim.=2) such that
2dAdS +
zBAγ
. ~ and AA AA γγ∂=∂ )direction. timeomit the (We
CommentsCo e ts• In the presence of a black hole horizon, the minimal surfaces
typically wraps the horizon. yp y p
⇒ Reduced to the Bekenstein‐Hawking entropy, consistently.
• We need to replace minimal surfaces with extremal surfaces in the time‐dependent spacetime. [Hubeny‐Rangamani‐TT 07]
• The area formula assumes the supergravity approximation.
The holographic formula is modified by higher derivatives.[Lovelock: Hung‐Myers‐Smolkin 11, de Boer‐Kulaxizi‐Parnachev 11,
AdS5×S5 in IIB String: Ogawa‐TT to appear]
• In spite of a heuristic argument [Fursaev, 06] , there has been no
l t f H th h b id dcomplete proof. However, there have been many evidences and
no counter examples so far.
[A Partial List of Evidences]
Area law follows straightforwardly [Ryu‐TT 06]
Agreements with analytical 2d CFT results for AdS3 [Ryu‐TT 06]
Holographic proof of strong subadditivity [Headrick‐TT 07]
Consistency of 2d CFT results for disconnected subsystems
[Calabrese‐Cardy‐Tonni 09] with our holographic formula [Headrick 10]
Agreement on the coefficient of log term in 4d CFT (~a+c)[Ryu‐TT 06, Solodukhin 08,10, Lohmayer‐Neuberger‐Schwimmer‐Theisen 09,
Dowker 10, Casini‐Huerta, 10, Myers‐Sinha 10, Casini‐Hueta‐Myers 11]
(2‐2) Holographic Proof of Strong Subadditivity[Headrick‐TT 07]
We can easily derive the strong subadditivity, which is known asWe can easily derive the strong subadditivity, which is known as the most important inequality satisfied by EE. [Lieb‐Ruskai 73]
B ACA A A
BCBACBBA SSSS +≥+⇒ ++++
ABC
=ABC
≥A
BC BCBACBBA ++++
A CACBBA SSSS +≥+⇒ ++
ABC
=ABC
≥AB
C C C
(2‐3) Calculations of HEE
Two analytical examples of the subsystem A:Two analytical examples of the subsystem A:
(a) Infinite strip (b) Circular disk(a) Infinite strip (b) Circular disk
l
B A B B A l1−dL l
xi>1(a) (b) xi>1
z
x1x1
L
i>1
l
(a) (b) i>1
l
zl
Entanglement Entropy for (a) Infinite Strip from AdS
divergence law Area dependnot doesandfinite is termThis cutoff. UVon the
p
d=1 (i.e. AdS3) case: ( )Agrees with 2d CFT results [Holzhey‐Larsen‐Wilczek 94 ; Calabrese Cardy 04]Calabrese‐Cardy 04]
Basic Example of AdS5/CFT4Basic Example of AdS5/CFT4
SYM SU(N) 4SAdS 55 =⇔× N
L( )5
l
The order one deviation is expected since the AdS result corresponds to the strongly coupled Yang‐Mills.
[cf 4/3 in thermal entropy Gubser Klebanov Peet 96][cf. 4/3 in thermal entropy, Gubser‐Klebanov‐Peet 96]
Entanglement Entropy for (b) Circular Disk from AdS[R TT 06][Ryu‐TT 06]
lawAreadivergence
lawArea
f l lConformal Anomaly(~central charge)2d CFT c/3・log(l/ε)
A universal quantity inodd dimensional CFT
f ‘ h ’ 2d CFT c/3 log(l/ε)4d CFT ‐4a・log(l/ε)
⇒ Satisfy ‘C‐theorem’ [Myers‐Sinha 10]
(2‐4) HEE and AdS BH
BγA BCi iAdS
Aγ BH . purenot isCFTtemp.Finite BH AdS
BAtot SS ≠⇔⇔
ρ p BAtotρ
Time tion Thermaliza formation BH ⇔
BHBHofSizebut
,0 .. pure is
∝
=finite
tottot
S
Seiρ
AdS entropy grained-CoarseEE BH.ofSize but
=→∝f
AS
[Arrastia‐Aparicio‐Lopez 10, Ugajin‐TT 10]
③ HEE of Confining Gauge Theories and Higher Derivatives(3‐1) Supergravity Result [Nishioka‐TT 06, Klebanov‐Kutasov‐Murugan 07]
4D N=4 SU(N) SYM on a Scherk‐Schwarz circle
⇔ AdS5 Soliton× S5 [Witten 98]AdS5 Soliton S [Witten 98]
( ) ( ) ./1 2440
222222
22 −+⋅+++−= drrrdrLdydxdtrds θ( ) ( )
.,2~
./1/1
2
02440
22
⎟⎟⎞
⎜⎜⎛
=+
+−
+++
LRR
drrLrrr
dydxdtL
ds
πθθ
θ
We consider the EE when the subsystem A is just a half space:
.2
,2 0⎟⎟⎠
⎜⎜⎝
+r
RRπθθ
yVy
A B
x
Calculation of HEECa cu at o o
2Area)5()5(
rRVS ySUGRA
A == ∫∞π
di )l(
442
)5()5(0
VN
LGG
y
rNN
A ∫
.8
-div.)law(area R
y= ∞=λ
Free Field Calculation
After summing over KK modesAfter summing over KK modes
. 12
-div.)law(area2
FreeYM VNS y
A = 0=λ
⇒ The dependence on λ is non‐trivial.
12)(
RA
p
(3‐2) Higher Derivative Corrections [Ogawa‐TT to appear]
We take into account the R4 correction in IIB string theory:
. ...8)3(
161 4
3'10
)10( ⎥⎦
⎤⎢⎣
⎡++−= ∫ WRgdx
GS
NIIB
αζπ
The correction to HEE can be calculated by using the replica trick:
816 ⎦⎣GNπ
[Cf. thermal entropy: Gubser‐Klebanov‐Tseytlin 98]
)3(1 32 ⎞⎛ ζVN.
264)3(
81div.) law (area 2
32
⎟⎟⎠
⎞⎜⎜⎝
⎛+−⋅+=
−λζ
RVN
S yA
Indeed, HEE increases as the coupling gets smaller !
N=4 SYM on A Twisted Circle
Twist
Supersymmetric PointMaximall SUSY Breaking(⇔ AdS Soliton)
0.2 0.4 0.6 0.8 1
-0.05
Twist parameter
0 15
-0.1 Free Yang‐Mills
-0.2
-0.15
AdS side (Strongly coupled YM)Higher derivative-0.25
EE
AdS side (Strongly coupled YM)Higher derivative corrections
EE
(3‐3) Confinement/deconfinement phase transition
EE can be an order parameter.p
HEE of pure SU(N) YM Lattice Result for pure YM
l
HEE of pure SU(N) YM Lattice Result for pure YM
Sfinite10
]
123× 24, β=5.70
163× 32, β=5.70
0.05
0.150
0.1 0.350.30.250.2l 5
A| )
∂S
/ ∂l
[fm
-3]
16 × 32, β=5.70
163× 32, β=5.75
163× 32, β=5.80
163× 32, β=5.85
-0.05
IR:Confinement(disconnected)
0
( 1
/ |∂A
|
-0.1
-0.15
(disconnected) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
l [fm]
-5
[4D SU(3), Nakagawa‐Nakamura‐
[Nishioka‐TT 06, Klebanov‐Kutasov‐Murugan 07]
Motoki‐Zakharov 09]
④AdS/BCFT and Quantum Entanglement
(4‐1) AdS/BCFT
h i h l hi d l f if ld i hWhat is a holographic dual of CFT on a manifold with
Boundary (BCFT) ?
CFTd: SO(d,2) ⇔ AdSd+1
BCFTd: SO(d‐1 2) ⇔ AdSdBCFTd: SO(d 1,2) ⇔ AdSd
B
?
Bounda
BCFT?
[cf Defect CFT Karch‐Randall 00 DeWolfe‐Freedman‐Ooguri 01
ary
[cf. Defect CFT Karch‐Randall 00, DeWolfe‐Freedman‐Ooguri 01,
Janus CFT Bak‐Gutperle‐Hirano 03, Clark‐Freedman‐Karch‐Schnabl 04]
AdS/BCFT Proposal [TT 11 + work in progress with Fujita and Tonni]
In addition to the standard AdS boundary M, we include an extra boundary Q, such that ∂Q=∂M.
11∫∫ .)(
81)2(
161
∫∫ −−−Λ−−=Q
Qmatter
NN matter
NE LKh
GLRg
GI
ππ
EOM at boundary leads to the Neumann b c on Q :
Qthe Neumann b.c. on Q :
. 8 QabNabab TGKhK π=−
NM
Conformal inv ⇒
abNabab
bQb ThT −=Conformal inv. ⇒ .abab ThT
(4‐2) Simplest Example
Consider the AdS slice metric:
. )/(cosh 2)(
222)1( dAdSdAdS dsRdds ρρ +=+
Restricting the values of ρ to solves the
boundary condition with*ρρ <<∞−
boundary condition with. tanh1 *
RRdT ρ−
=
*ρρ =N (AdS) Q
−∞=ρ ∞=ρMρ ρAdS bdy
(4‐3) Holographic Boundary Entropy
The entanglement entropy in 2D CFT
( ) g p y py
with a boundary looks like [Holzhey‐Larsen‐Wilczek 94 ; Calabrese‐Cardy 04,
R t i C l b C d 09 C i i H t 09]
l
B ARecent review: Calabrese‐Cardy 09, Casini‐Huerta 09]
ll lc⎟⎞
⎜⎛
B A
, log log6
glcSA +⎟⎠⎞
⎜⎝⎛=ε
where log g is the boundary entropy [Affleck‐Ludwig 91].[Earlier holographic calculations: Yamaguchi 02 (Defect CFT),[Earlier holographic calculations: Yamaguchi 02 (Defect CFT),
Azeyanagi‐Karch‐Thompson‐TT 07 (Non‐SUSY Janus),
Chiodaroli‐Gutperle‐Hung, 10 (SUSY Janus) ]
In our setup, HEE can be found as follows
. 4
log64
14
Length **
NNNA G
lcdGG
S ρε
ρρ
+=== ∫ ∞−NNN
Boundary Entropy
Comment 1. can be confirmed from the disk
cylinder partition function. Nbdy GS 4/*ρ=
.21log)/sinh(
24**
2
2
⎟⎟⎠
⎞⎜⎜⎝
⎛−−++=
RrRrr
GRI
NDisk
ρεερ
ε ⎠⎝
.23
*
NBHCylinder G
TlcI ρπ+⋅⋅=
Comment 2. The null energy condition for TQ leads to a
3 NG
holographic g‐theorem.
Holographic Dual of Disk Holographic Dual of Cylinder
Qo og ap c ua o s o og ap c ua o Cy de
QQ Q
M N M N Black HoleM N M N Horizon
z z
⑤ Towards Gravity Dual of Lattices [work in progress with Ryu]y(5‐1) Holographic Dual of Two Points (or 2 qubits)
Thermal AdS32
2222 )( dxzhdzdtRds ⎟⎟
⎞⎜⎜⎛
++−=
A B .020
2
222
2~ , /1)(
,)(
zxxzzzh
dxzzhzz
Rds
π+−=
⎟⎟⎠
⎜⎜⎝
++
N . 00
⎥⎥⎤
⎢⎢⎡
⋅≤ 0 arctan|)(| : RTzzzxN
HEE is calculated as follows:⎥⎥⎦⎢
⎢⎣ − 22
0
0)(
|)(|TRzhz
A and B aremaximally entangled
bdyN
z
zN
A SGzhz
dzGRS === ∫ 4
2)(4
2 *0
*
ρ
maximally entangled NN GzhzG 4)(4
(5‐2) How does Holographic Dual of Lattices look like ?
Holographic dual of many pointsg p y p
A Minimal Surface for ALL+1
AγAB
.log3
~4
|| LcG
S AA
γ=
L A
BN
34GN
The separation does not have xΔ
1
2xΔ
pany direct physical meaning in CFT. But, it does in the AdS gravity.
Emergent AdS space in the IR of a lattice theory
N
Emergent AdS space in the IR of a lattice theory(continuum limit)
This argument looks a bit similar to the calculation framework g
called MERA (multiscale entanglement renormalization ansatz).
[MERA: Vidal 06; Relation to AdS/CFT: Swingle 09’]; / g ]
�Space directionA
Curve Aγ
ing
A
B d ][#MiS
e‐grain
||
Bonds][#Min
⎤⎡
≈ASA
γγ
Coarse .
4||
Min~ A⎥⎦
⎤⎢⎣
⎡
NGA
γγ
���� �� ���� ���� ������� ������� ������� �� ��� ����� ����������� ��� ��������Disentangler Coarse‐graining [O h h h l hi l i
���� �� ������ ���
�� �� ��� �� ������ �� ����� �� ���� �� �� ������� �������� ���� �� �� ���� ��
������� ������ ��� �� �� ������ �� �� ���� ���� ��� ������ ���� ��� ���� ���� ����
�� ����� �� ���� �� �� �������� �� ������ � �� �� �� ����� ������������ �� ��� ����� �
Coarse graining
Fig taken from B. Swingle 0905.1317
[Other approaches to holographic lattices: e.g. Kachru‐Karch‐Yaida 09, Lee 10]
⑥ Conclusions
• The entanglement entropy (EE) is a useful bridge between gravity and cond mat physics [ f S hd ’ t lk]between gravity and cond‐mat physics. [cf. Sachdev’s talk]
Gravity Entanglement Cond‐mat.
systemsμνg Area≈AS Ψy
• EE can be a universal quantity for holography in
μ A
• EE can be a universal quantity for holography in general spacetimes.
[ h l h i fl Li TT 10[e.g. holography in flat space: Li‐TT 10
⇒highly entangled and non‐local gravity dualAdS Lorentzian wormholes: Fujita‐Hatsuda‐TT 11AdS Lorentzian wormholes: Fujita Hatsuda TT 11
⇒The EE between two boundaries are actually vanishing. ]
• EE is non‐zero even for pure states (cf. thermal entropy).
⇒ A quantum order parameter at zero temp⇒ A quantum order parameter at zero temp.e.g. Useful for BH formation = Thermalization (quantum quench)
• We proposed a holographic dual of BCFT.
⇒ Here again HEE played an important role.
⇒ This holography can also be useful in AdS/CMT⇒ This holography can also be useful in AdS/CMT.e.g. Edge states of QHE, Topological Insulators ?
A h l hi SC l li d b d i ?Any holographic SC localized on boundaries ?
Non‐equilibrium systems with boundaries ?
: