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 ⇡