Surface/State Correspondence

in AdS/CFT

Reference: M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi, and K. Watanabe, [arXiv:1502.04267]

Noburo Shiba

( Yukawa Institute for Theoretical Physics (YITP), Kyoto Univ.)

GRaB100 @ National Taiwan University 2015/7/9

Collaborators: M. Miyaji, T. Numasawa, T. Takayanagi, and K. Watanabe(YITP)

Introduction

Entanglement Entropy (EE) plays an important role in AdS/CFT correspondence.

Entanglement entropy (EE) is generally defined as the von Neumann entropy

AAAA trS log:

corresponding to the reduced density matrix of a subsystem A .

cA

A

・EE in QFT

In (d+1) dimensional QFT, the entanglement entropy is geometrically defined by separating the spatial manifold into the subsystem and .

cAA

A A

Area law of entanglement entropy

In the vacuum state in local (d+1) dim QFT (d>1)

: constant

:UV cutoff length

subleadinga

AAreaS

dA

1

)(

a

terms

Area law means that the degrees of freedom on the boundary contribute mainly to EE

cA

[d=1: log divergence]

Holographic Entanglement Entropy (HEE)

B

A

z

We omit the time direction

[Ryu,Takayanagi 06; derived by Casini,Huerta,Myers 09, Lewkowycz,Maldacena 13]

1dCFT2dAdS

：

2

1

22

0

2

22

z

dxdxdzRds

d

i i2dAdS

N

AA

G

AreaMinS

4

)(

is the minimal area surface (codim.=2) such that

AA

The HEE suggests the following interpretation:

``A spacetime in gravity = Collections of bits of quantum entanglement’’

This is manifestly realized in the recently found connection between AdS/CFT and tensor networks. Swingle 09, …

B

A

z

1dCFT2dAdS

2

)(

4

)(

PL

A

N

AA

l

Area

G

AreaS

Based on the connection between the AdS/CFT and the tensor network, the Surface/State (SS) correspondence was proposed as a new generalization of holography. Miyaji, Takayanagi 15

Any codimension 2 spacelike convex surface

a quantum state

d

d dualH )(

Gravity

2dM

In this work, we give an explicit formulation of SS correspondence for AdS3/CFT2. Miyaji, Numasawa, Shiba, Takayanagi, Watanabe, 15

Contents

1. Introduction

2. AdS/CFT and tensor network

3. Surface/State correspondence

4. SS correspondence in AdS3/CFT2

5. Conclusion

2. AdS/CFT and tensor network

A tensor network state is a efficient variational ansatz for the ground state wave functions in quantum many-body systems.

[A tensor network diagram = A wave function]

An ansatz should respect the correct quantum entanglement of a ground state.

The geometry of tensor network corresponds to the quantum entanglement.

(2-1)Tensor network

Example: Matrix Product State (MPS)

MPS does not have enough EE to describe 1d quantum critical points (2d CFTs)

CFT

AA SLS ~lnln2~

minint N [#Intersections of ] Aln~ intNSA

In general

・MERA

MERA(Multiscale Entanglement Renormalization Ansatz) Vidal 05

An efficient variational ansatz for CFT ground states.

We add (dis)entanglers to increase entanglement.

AS

min[#Intersections of ] ACFT

ASL ~ln

agrees with the result in 2D CFT

A conjectured relation to AdS/CFT Swingle 09

MERA AdS/CFT

2

22222

2

222 )(

z

xddtdzxddt

eduds

u

uez where

Continuum MERA (cMERA) Haegeman,Osborne,Verschelde,Verstraete 11 Nozaki,Ryu,Takayanagi 12

To remove lattice artifacts, take a continuum limit of MERA:

u

uIR

sKdsiPu )(ˆexp)(

state at scale u IR state IRu

:)(ˆ uK (dis)entangler at length scale ue~

: unentangled state in real space

0AS for any A

The unentangled state is identified with the boundary state. Miyaji, Ryu, Takayanagi, Wen, 14

We apply the idea of quantum quenches.

For t<0, we assume a state is the ground state of the massive Hamiltonian . Then at t=0, we suddenly change the Hamiltonian into as in [Calabrese,Cardy 05].

mH

CFTH

the ground state of mH )( m

In this setup, the state at t=0 is identified with the boundary state:

Bt )0(

mH

CFTHt

3. Surface/State correspondence

The surface/state correspondence is motivated by the tensor network description of holography. Miyaji, Takayanagi 15

Based on this connection between the AdS/CFT and the tensor network, the Surface/State (SS) correspondence was proposed as a new generalization of holography. Miyaji, Takayanagi 15

Any codimension 2 spacelike convex surface

a quantum state

d

d dualH )(

Gravity

2dM

4. SS correspondence in AdS3/CFT2 Miyaji, Numasawa, Shiba, Takayanagi, Watanabe, 15

We give an explicit formulation of SS correspondence for AdS3/CFT2. It is useful to start with the symmetry of global AdS3 space:

)sinhcosh( 2222222 dddtRds

whose isometry is generated by RL RSLRSL ),2(),2(

[Maldacena, Strominger 98]

In particular, we are interested in the SL(2,R) subgroup which preserves the time slice t=0 (i.e. H2) of the AdS3.

They are generated by which annihilates the boundary states.

,~

000 LLl ,~

111 LLl 111

~LLl

The SL(2,R) action which maps to the point is given by

0),(

)(2

110),(

ll

lieeg

0t

0]~

[ BLL nn

cMERA for the ground state of CFT2 is formulated as:

0

0

)(ˆexp0 BsKdsiP

boundary (Ishibashi) state for the identity sector

If we act the SL(2,R) transformation we find ),( g

0

0

),( )(ˆexp0 BsKdsiP

where 1

),( ),()(ˆ),()(ˆ guKguK

More generally, we can describe the diffeomorphism by taking into account

nnn LLl

~,...)3,2|(| n

0

0

)(ˆexp0 BsKdsiP h

11 )()()()(ˆ)()(ˆ uhuhiuhuKuhuK uh

where with n nn luhuh )(exp)( 0)0( nh

We can define a dual state for any surface as

)( uu

0)(ˆexp)( BsKdsiP

u

hu

0B

u

This transformation is interpreted as the deformation of the intermediate surface , which allows us to choose any possible foliation of the time slice.

u

How to describe the bulk local excitation

We argue the following identification:

bulk0),(

BsKdsiP

0

),( )(ˆexp),(

Bulk local operator

Ishibashi state for primary

This is because the local operator insertion does not change the bulk metric (= entanglement).

B

),(

We argue this state is evaluated as

Jeeg HLLi

)

~(

200

),(),(

)',','(),,( tt Our inner product in the 2D CFT perfectly matches with the known expression of bulk to bulk propagator of a free massive scalar in AdS3.

We can compute the information metric:

ba

ab dxdxG 1|),(),(|

)sinh(1 222

2inf

2

ddds

2c By choosing 1 c (as in AdS/CFT)

Ishibashi state for primary ),2( RSL

Time slice of AdS3

some UV cutoff

5. Conclusion ・We give an explicit formulation of SS correspondence for AdS3/CFT2.

・The boundary states in CFT is identified as an IR state and the bulk diffeomorphism is naturally taken into account.

・We give an identification of bulk local operators which reproduces bulk scalar propagators on AdS3.

・We also calculate the information metric for a locally excited state and show that it is given by that of 2d hyperbolic manifold, which is argued to describe the time slice of AdS3.

Future problems: ・BTZ black hole ・Finding explicit expression of disentangler

)(ˆ uK

・Boundary states in 2D CFTs

A boundary states (Ishibashi states) is a state which satisfies a conformally invariant boundary condition:

0]~

[ BLL nn

)(

10

;,;,Nd

jRL

N

jNjNB

Ishibashi state is explicitly given by:

SL(2,R) Ishibashi state is explicitly defined by:

0k

RLkkJ k

LLk )( 1 k

RLk )~

( 1 where