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AdS/CFT correspondence and tensor networks
Masamichi Miyaji (YITP)
M. M, T. Takayanagi, S. Ryu, X. Wen : JHEP 1505 (2015) 152 M. M, Takayanagi : PTEP 2015 (2015) 7, 073B03
Overview of today’s talk
AdSd+1
AdSd+1/CFTdU00�1�2
U0�1
�1�2 U�20�3�4
U0�1↵1↵2
MERA
• General formulation of continuous MERA
• Generalization of the correspondence to other spacetimes (Surface/State correspondence)
Equivalent?
To do
Plan of today’s talk
1. Quantum entanglement and AdS/CFT
2. Tensor networks
3. Surface/state correspondence
1. Quantum entanglement and AdS/CFT
Quantum entanglement
| "i| #i± | #i| "ip2
Example: two spin system
Measurement on either spin affects the measurement on the other spin.
Measurement on either spin doesn’t affect the measurement on
the other spin.
(| #i+ | "i)(| #i+ | "i)2
Different parts of a quantum state are correlated because it is a superposition of states.
Entanglement entropy
H = HA ⌦HAc
A Ac
⇢A := trHAc⇢
SA := �trHA⇢Alog⇢A
State ⇢Reduced density matrix
Entanglement entropyA Ac
| "ih" |+ | #ih# |2
(| #i+ | "i)(| #i+ | "i)2
Sleft = log 2
Sleft = 0
Example
Application• Order parameter for phase transitions.
• Used to classify class of states.
S(0,L) =c
3log
L
✏
2d CFT
2d gappedS(0,L) = Constant
AdS/ CFT correspondenceConjecture on equivalence between gravity(string) theory on AdS and conformal field theory on the boundary.
When either side is weakly coupled, the other side is strongly coupled.
Though there are numerous evidences, there is no direct derivation and still a conjecture.
ZCFT [�(x)] = Zgravity[�(x) = limz!0
z
�O�d�(x, z)]
Note
AdSd+1
Gravity on
CFTd on boundary
[Maldacena][Gubser, Klebanov,Polyakov][Witten],………
Non-perturbative gravity may be studied by using dual field theory.
| CFT i $ |�gravityi
Ryu Takayanagi formulaEntanglement entropy of region A of dual Euclidean CFT of Einstein gravity is given by
• Proved using AdS/CFT and replica trick for static case.[Lewkowycz, Maldacena]
• 1 entanglement per Planck area.
SA =Area(�)
4GN
where is the minimal area surface whose edge coincides with that of A.
��A
Note
• Generalization of Bekenstein-Hawking formula for BH entropy.
AdSd+1
[Ryu, Takayanagi]
Connectedness and entanglement
A pair of disconnected spacetimes is dual to a tensor product of dual states.
AdS AdS
CFT1 CFT2
|E0i ⌦ |E0i
Entanglement of CFT plays essential role for the connectedness of dual spacetime.
Example
No entanglement
Singularity
Singularity
Horizon
HorizonCFT1 CFT2| i =
X
i
e��Ei2 |Eii ⌦ |Eii
SCFT1 = Thermal entropy of CFT1
Eternal black hole is dual to thermofield double state, whose E.E is proportional to temperature.
=⇡2
GN�
Thermofield double state
Eternal black hole
In 2d
Connectedness = entanglement
CFT1CFT2
In general,
Small entanglement in CFT Large distance in bulk
I(A,B) � (hOAOBi � hOAihOBi)22|OA|2|OB |2
where
I(A,B) = SA + SB � SA[B
is called mutual information which measures entanglement between A and B.
Since for operators with large conformal dimensions,hOAOBi ⇠ e�md(A,B)
I(A,B) ⇠ (hOAOBi � hOAihOBi)22|OA|2|OB |2 / e�md(A,B)
Einstein equation and entanglement
Assuming Ryu-Takayanagi formula for E.E, the first law of E.E for sphere is equivalent to linearized Einstein equation in the bulk.
[Nozaki, Numasawa, Prudenziati, Takayanagi][Lashkari, MacDermott, Raamsdonk]
@2zH
ii +
d+ 1
z@zH
ii + @j@
jHii � @i@jHij = 0
Linearized Einstein equation
⇢A = e�HA
�SA = �hHAi
�SA = �hHAi
First law
So entanglement of CFT states determines structure of dual spacetime.
First law of E.E is an analogue of first law of thermodynamics.
where
AdS/CFT and RG flowRadial direction of AdS corresponds to energy scale of RG flow.
O(x) O(y) O(x) O(y)
UV of QFT IR of QFT Near boundary Deep interior
This fact motivates us to interpret AdS/CFT in terms of some version of RG.
z ds2AdS =dz2 + dx2
z2
µ ⇠ 1
zRG energy scale:
z zSmall Largez0
z0
AdSd+1Boundary
IRUV
[Maldacena][Gubser, Klebanov,Polyakov][Witten],[Susskind, Witten],…
AdS/CFT and RG flowOne approach: Holographic Wilsonian RG flow
The way to integrate out d.o.f in dual field theory is unknown and needs to be investigated.
Question
Z =
ZD�̃(l) IR(�̃(l)) UV (�̃(l))
�̃(l) IR(�̃(l))
UV (�̃(l))
AdSd+1
Divide gravity path integral into 2 parts
Natural assumption
gives Wilsonian action of dual field theory
with e�S(l) =
ZD�̃(l)e
R�̃(l)O UV (�̃(l))
Boundary
[Heemskerk, Polchinski]
IR(�̃(l)) =
ZDMl e
R�̃(l)O
/Z
DMl e�S(l)
Fields with unknown cut off
2. Tensor networks
Tensor networks
Multi-scale Entanglement Renormalization Anzatz(MERA)
Matrix Product State(MPS), Density, Matrix Renormalization Group(DMRG)
Efficient method to produce ground state of given Hamiltonian.
• Real space, variational method: Using Hamiltonian.
• No sign problem: No Monte-Carlo method unlike lattice gauge theory.
• Efficiency: Computable on usual computers
• Gapped systems
• Critical or gapped systems
Examples
Wilsonian numerical renormalization group
RG perspectiveTake reduced density matrix of two contiguous spins, and project Hilbert space to the direction with non zero spectrum.
We obtain effective coarse grained lattice.
⇢(2) =X
a
�a|aiha|
P (2) =X
a:�a 6=0
|aiha|Projection by
Simulation perspective�2�1
�1 �2
�2�1
X
�1�2
U00(1)�1�2
|�1i|�2i
X
�1...
U0�1(2)�1�2
U�20(2)�3�4
X
�1�2
U↵1↵2(1)�1�2
|�1i|�2i|�3i|�4i
|0i|0i0 0
w
w ⌦ w
w† ⌦ w† · · ·
The rank of reduced density matrix of half lattice is upper bounded by .
=c
3Log L
= Constant
(critical)
(gapped)
So NWRG can’t reproduce correct ground states of critical theory, unless N increases polynomially with system size.
Shalf log�
�
= �dimension of each bond
Problem of WNRG
SA � ⇠ Lc3
� ⇠ Constant
0
In 2d,
In RG perspective, dimension of effective lattice decrease slowly, so the method is not so useful.
Inefficient
Efficient
Entanglement renormalization
↵1 ↵2 ↵3 ↵4
�1 �2 �3
�3�2�1
U0↵1�1�2
U�0�1�1�2
�0
|↵i
�1 �2
X
�1�2
U0↵�1�2
|�1i|�2i
Isometry
Disntangler
�2�1
�1 �2
X
�1�2
U�1�2�1�2
|�1i|�2i
|�1i|�2i
IR
Fine graining
In order to increase the amount of entanglement, in addition to fine graining transformation (isometry), we use local unitary transformation called disentangler which add short range spatial entanglement.
UV
Isometry Disntangler
Entanglement renormalization
+
=↵
[Vidal][Evenbly, Vidal]
w
u
Choose u and w in order to minimize energy of the state.
Entanglement renormalization preserves locality: length of support of local operator doesn’t change a lot.
RG perspective
�̂
�̂0
Reduce short range entanglement by disentangler
Take reduced density matrix of two contiguous spins, and project Hilbert space to the direction with non zero spectrum.
We obtain effective coarse grained lattice.
Locality
⇢(2) =X
a
�a|aiha|
P (2) =X
a:�a 6=0
|aiha|Projection by
Disentangler
u† ⌦ u† · · ·
w† ⌦ w† · · ·
Multi-scale Entanglement Renormalization Anzatz
| MERAi =X
�1,�2
U00�1,�2
X
�1,...,�4
U0,�1
�1,�2U�2,0�3,�4
X
↵1,...,↵6
U0�1↵1,↵2
U�2,�3↵3,↵4
U�4,0↵5,↵6
|↵1, ...,↵6i
Layer of entanglement renormalization. Tensor network for gapped or gapless system
MERA:
U00�1�2
U0�1
�1�2 U�20�3�4
U0�1↵1↵2
For critical systems, conformal symmetry can be incorporated naturally by taking identical tensors at each layer.
Entanglement entropy of MERA
⇢A
⇢(1)A
⇢(2log2L)A
= �dimension of each bond
|0ih0| =
weak subadditivity of E.E SA[B SA + SB
S[⇢(2Log2L)A
] + Log2L⇥ 2Log�
= Log2L⇥ 2Log�
SA S[⇢(1)A ] + 2Log�
Entanglement entropy of reduced density matrix is bounded by number of bonds of “minimal bond surface”.
number of bonds of surface γSA ⇡ ⇥ log�
To keep the dimension of Hilbert space steady during coarse graining, we add residual d.o.f at each step.
|0i |0i|0i|0i
| (1)MERAi =
X
�1,�2
U00�1,�2
X
�1,...,�4
U0,�1
�1,�2U�2,0�3,�4
|0,�1, ...,�4, 0i
| (2)MERAi =
X
�1,�2
U00�1,�2
|0, 0, �1, �2, 0, 0i
|0i|0i
�1 �2�3
�4
�1 �2
|↵i
�1 �2
X
�1�2
U0↵�1�2
|�1i|�2i
Isometry↵
�1 �2
↵|0i
X
�1�2
U0↵�1�2
|�1i|�2i
|0i|↵i
Note is not entangled with other spins.|0i
w w
We show conformal boundary states have no real space entanglement. So they are candidates of IR states of cMERA.
Continuous MERAMERA
|0i|0i · · · |0i|vacuumi
cMERA|vacuumi |⌦i
Unitary
IR state of MERA is given by a state with no short range quantum entanglement.
IR state
|vacuumi = Pexp(�i
Z 0
�1du K̂(u))|⌦i
disentangler
Entanglement of conformal boundary states
2✏
hB|O(x)O(y)...|Bi ' hB|O(x)|BihB|O(y)|Bi...
e�✏H |Bi : Conformally invariant boundary state
Correlation function
Correlation functions on conformal boundary state factorize.
Mass deformation
Ground state of such mass deformed theory have no short range entanglement, and is boundary state.
S(�) ! S(�) +M
D�2��
Zd
Dx (�(x)� �(x))2
�(x)� �(x) = 0
SA =1
3log
4✏
⇡a
2d free fermion
: UV cut off
Entanglement entropy of interval is given by
a
M ⇡ ⇤ : Cut off scale
[M.M, Takayanagi, Ryu, Wen]
Image charges
O(x) O(y)
3. Surface/state correspondence
AdS/CFT and tensor network
⇥ log�
number of bonds of surface γ
Significant similarity between geometric expression of entanglement entropy in AdS/CFT and MERA.
SA =Area(�)
4GN
Ryu-Takayanagi formula:
Entanglement entropy is given by area of minimal surface γ devided by Newton constant.
MERA:
Entanglement entropy is bounded by number of bond of minimal surface γ. This bound is often approximately saturated.
�
A A
�
SA
[Evenbly, Vidal(2014)]
ProposalMERA tensor network for CFT ground state corresponds to spacetime in AdS/CFT in such a way that density of bonds is proportional to area of intersecting minimal surface.
number of bonds of surface γSA ⇡ ⇥ log�
Area(�)
4GN=
Note
Can be applied to gapped or critical system.
Degrees of freedom which are integrated out in path integral is manifest geometrically.
[Swingle]
• Can we formulate this proposal in background independent manner?
• Can we see emergence of classical bulk at large N?
Can we determine MERA at large N?
No special direction?
Area of arbitrary surface in the bulk?
No boundary?
• More gravity observables from tensor networks?
Questions
Can we obtain Einstein equation from MERA?
|�(�⇤)i = |Bi |�(0)i = |vacuumi|�(u)i
Tensor network state are obtained via contraction of tensors.
Usually, they start with a tensor at the center, and contract tensors homogeneous and isotropically.
So surface in the tensor network defines state in CFT Hilbert space.
One can do same thing in different ways.
|�0(u)i|�0(0)i = |Bi |�0(0)i = |vacuumi
With identical tensor network as usual method, one can starts contraction from different point, and proceed in various ways.
E.E of subsystems of such states are given by number of bonds on “minimal bond surface” γ.
number of bonds of surface γ
⇥ log�
SA ⇡ A
A
�
Surface/state correspondence
Codimension 2, topologically trivial, convex closed surfaces in spacetime correspond to pure states that describe gravity interior of these surfaces. • Entanglement entropy of these states is given by area of minimal surface.
• Area of surface gives log of dimension of effective Hilbert space.
Note• Background independent. No special direction nor boundary of spacetime is necessary.
• Structure of entanglement of surface state is explicitly given.
Proposal
[M.M, Takayanagi]
Convex surface and minimal surface
Convex surface: minimal surface of any area on convex surface is located inside.
If outside, E.E. is determined by tensors which is independent of the surface state.
For any partition of minimal surface, entanglement entropy of corresponding surface is additive.
NoConvex
This means that degrees of freedom on minimal surface are not entangled each other. They are entangled with outside of A.
A = tAi SA =X
SAi
Ai
Minimal surface
Convexity
Area of surface = Effective entropy
Dividing surface into infinitesimally small pieces and consider minimal surfaces of each piece.
Using
We obtain
�i⌃i
⌃
⌃�i
⌃i
SurfaceArea(minimal)
4GN⇡ ⇥ log�Number of bonds
Area(⌃)
4GN⇡ ⇥ log�Number of bonds on ⌃
=log dimH⌃
Area(⌃)
4GN=
X
i
S⌃i =X
i
Area(�i)
4GN
Flat spaceds
2 = �dt
2 + dx
2 + dy
2
Minimal surface is given by straight line.
| ⌃iL
SA =L
4GN
De Sitter space
Entanglement entropy for straight line state obeys volume law.
ds
2 = R
2(�dt
2 + cosh
2t d⌦2
S2)
| ⌃i
t = 0
t = 0SA =
L
4GN This means that any point is only entangled with antipodal point.
Entanglement entropy for great circle state also obeys volume law.
L
In both cases, states have highly nonlocal entanglement.
Pathological behavior in other time slices: negative E.E.
This means that any point is only entangled with points at infinity.
Summary
• We identified tensor networks as bulk spacetime, and generalized to AdS/CFT by attaching states to each cxdimension 2 surfaces in the bulk.
• Entanglement of boundary field theory plays important role in emergence of classical dual spacetime.
• MERA, one of tensor networks, is conjectured to describe bulk spacetime in AdS/CFT.
• Such new conjecture can be used in more general spacetimes, other than AdS.
Appendix
Correlation function
Entanglement renormalization
Scaling local operator
U†U = 1U : V 0 ! V
U�0aU
† = �a�a
�0a
�aV
V 0
�00a
=1
L2Log2�ah0(Log2L)|�(Log2L)
a
(0)2|0(Log2L)i
h0|�a(L)�a(0)|0i
=1
�2a
h00|�0a(L
2)�0
a(0)|00i
Log2L steps
�a =1
2Log2�a
Matrix product states
Generalization to higher dimensions is known: Projected Entangled Pairs
| MPSi =X
↵1,↵2...,↵N
Tr(A↵11 A↵2
2 ... A↵NN )|↵1,↵2...,↵N i
A↵b,c
b c
|↵i↵1 ↵2
(X
↵
A↵A↵† = 1 )
Number of degrees of freedom is NMD2 which is exponentially smaller than MN, assuming D is kept finite.
Consider a spin system of length N, where each spin has M degrees of freedom. Prepare M D×D matrices for each spin.
Typical states in Hilbert space have far larger entanglement entropy: MPS is designed to aim at a tiny corner of the full Hilbert space.
↵1 ↵2
full Hilbert space
SA(L) ⇡ constSchunch Wolf Verstraete Cirac(2008)
MPS with finite D is expected to have constant entanglement entropy, same as that of ground state of gapped 1+1 system.
Density Matrix Renormalization GroupDMRG begins with coarse grained lattice and coarse grained wave function, and fine grain them iteratively like renormalization group in opposite direction.
|1lefti, |2lefti, ..., |Dlefti
| i
trright| ih | =X
1a
�a|aleftihaleft|
| 0i =
X
1aD,1bD
Ms1s2ab |alefti|s1i|s2i|brighti
|a0
lefti =X
s
X
1aD
Aaa0 (s)|alefti|si
�1 � �2 � ...�D � �D+1 � ...
MPS
Minimization of energy
Choose
Fine graining
In order to simulate ground state correctly, resulting states should be able to reproduce correct entanglement entropy. The rank of reduced density matrix can be estimated as
In DMRG, the rank of reduced density matrix of half lattice is upper bounded by D.
�1 � �2 � ...�D � �D+1 � ...
trright| ih | =X
1a
�a|aleftihaleft|
=c
3Log L
= Constant Constant’
!X
1aD
�a|aleftihaleft|
(critical)
(gapped)
So DMRG can’t reproduce correct ground states of critical theory, unless N=D increases polynomially with system size.
⇢A =X
1aN
1
N|aiha|
SA = Log N
N ⇡ Lc3
N ⇡