surface/state correspondence in ads/cftweb.phys.ntu.edu.tw/stringmeeting/grab100/lectures...1....
TRANSCRIPT
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