Gravity Actions from Tensor Networks

Masamichi Miyaji

Yukawa Institute for Theoretical Physics, Kyoto University

[hep-th/1609.04645] with T. Takayanagi and K. Watanabeand ongoing work with

P. Caputa, N. Kundu, T. Takayanagi, K. Watanabe

[Swingle]

Tensor Network Bulk timeslice in AdS/CFT

Equivalent?

• There is a significant similarity between Tensor network and gravity.

• We do not know the precise mechanism of AdS/CFT.

The aim is to find precise relation between Tensor Network and gravity.

Motivation

Tensor network: Network of tensors. By contracting tensors, we can express complicated states.

[White][Vidal]

• RG flow of state.

|↵i|�i

Cab↵�

• Grounds state gets simplified and loses short range entanglement gradually.

• It can simulate ground state of field theory.

a b

Tensor Network

U0acd

Ub0ef

U00ab

|0i |0i

|ci |di |ei |fi

|0i ⌦ |0i

X

a,b

U00ab |ai ⌦ |bi

State without short range quantum entanglement

X

c,d,e,f

Ucd0aUb0

ef

X

ab

U00ab |ci ⌦ |di ⌦ |ei ⌦ |fiGround state

Reduce quantum entanglement

Coarse graining

Tensor Network

• We can identify number of intersecting bonds as area of surface.

(Area of the minimal surface)⇥

• Significant similarity to Ryu-Takayanagi formula in AdS/CFT.

S[⇢A] ⇠

#(Bonds intersecting the minimal bonds surface)

O(1) constant

⇥

S[⇢A] ⇠ O(1) constant

• For certain tensor networks, such as MERA, entanglement entropy of state can be expressed by number of bonds of minimal bond surface.

log (dimensions of bond)⇥

log (dimensions of bond)⇥

Tensor Network and Ryu Takayanagi Formula

[Swingle]

[Swingle]

Tensor network of CFT ground state= Timeslice of bulk spacetime in AdS/CFT

AdSd+1Timeslice of Tensor network

Conjecture

• How should we determine correct tensor network?

Problems

• Relation between gravity and tensor network is still ambiguous.

Tensor Network = Bulk timeslice?

Continuous Tensor Network

• Boundary states have no short range quantum entanglement, so we can use boundary states as .

• Interpolation from vacuum state to unentangled trivial state.

Unentangled trivial state

|⌦1i

[M.M, Ryu, Takayanagi, Wen]

|⌦1i

[Haegeman, Osborne, Verschelde and Verstraete]

Ground state Coarse grained state

|⌦vaci |⌦zi

z: effective lattice spacing

/

sZD�(x, ⌧) e�SE [�]=

ZD�(x, ⌧)

⌧<0e

�R 0�1 d⌧

RdxLE [�(⌧,x)]

�(x, ⌧ = �1) = �B(x)

�(x, 0) = �0(x) �(x,1) = �(x,�1)

= �B(x)

• In Euclidean path integral, wave function is

h�0|⌦vaci

• By deforming this Euclidean path integral, we can simulate tensor network.

• In the deformation, we identify z = �⌧ as effective lattice spacing.

z

z = z0

z = 0

|⌦z0i

In other word, UV cut off is z dependent, and modes with spatial

• By this deformation, we can discard redundant path integral, without changing wavefunction.

Deforming action

momentum larger than 1/z are discarded.

[M.M, Takayanagi, Watanabe]

✓(x) = 0 (x > 1)where

h�0|⌦vaci /Z

D�(x, z)z>0

e

�R 1✏ dz

Rdkd!✓(z

p(k2+!2)/2)LE [�]

�(x, ✏) = �0(x)

We need to eliminate redundant tensors from the path integral, in order to relate tensor network with AdS/CFT and Ryu-Takayanagi formula.

Proposal to fix the deformation

The deformation with minimum redundant tensors corresponds to bulk time slice in AdS/CFT.

• The deformation of the path integral is not unique.

• No criterion relate it to gravity.

Questions

Proposal

Most efficient path integral proposalWork in progress [Pawel, Kundu, M.M, Takayanagi, Watanabe]

• We deform the path integral by Weyl transformation to the CFT, without changing the metric at z = 0.

Most efficient path integral proposal

• Let’s consider path integral representation of the vacuum state.

corresponds to amount of redundancy of the original path integral

compared to deformed one.

A definition of redundancy

• There is a variety of ways to define redundancy of tensor network.

ds

2 = dz

2 + dx

2ds

2 = e

��(x,z)(dz2 + dx

2)

h 0|⌦vac

i =Z

D (x,✏)= 0(x)

e�S[ ] = eSL[�]

ZD

(x,✏)= 0(x)e�S[ ,�]

e�SL[�]

e�SL[�]

We identify the metric as metric on the bulk timeslice.

is determined by minima of

takes maximal value.“The most efficient path integral”

Emergent metric

Most efficient path integral proposal

z z

z = z0

z = 0

ds

2 = e

��(x,z)(dz2 + dx

2)

SL[�]�(x, z)

e�SL[�]

[D�ds

2=e

��(dz2+dx

2)] = eSL [D�ds

2=dz

2+dx

2 ]

where SL is the Liouville action

µ > 0

is minimized when

e�� =4

µz�2

ds

2 =4

µ

dz

2 + dx

2

z

2

S

L

=1

8⇡

Z 1

✏

dz

Z 1

�1dx

h(@

x

�)2 + (@z

�)2 +µ

�

2(e�� � 1)

i

Example: 2d CFTFor Weyl transformation,

In contrast to Einstein-Hilbert action, Liouville action is bounded from below.

So we get AdS solution as true minimum.

c

24⇡

Excited state

For state excited by primary operator with dimension h at

assuming large c, corresponding solution is

Example: 2d CFT

ds2 =4

µ· a2

|w|2(1�a)(1� |w|2a)2dwdw̄

deficit angle geometry with angle 2⇡(1� a)

�a = 1� 12h

c

�

.

In AdS3/CFT2, the relation is a =

r1� 24h

c, consistent when .h << c

w = 0 (|w| < 1)

w

1 2⇡(1� a)

Example: 2d CFTBTZ black holeSimilarly, by considering wave function of Thermo field double state,

| TFDi = 1pZ(�)

X

i

e��Ei |Eii ⌦ |Eii

we get Liouville action.

h ̃1, ̃2| TFDi =

ds

2 =4

µ

· 4⇡2

�

2· dz

2 + dx

2

cos

2( 2⇡z� )

As the minimum of the action, we obtain BTZ solution.

ZD (x, z)

��/4<z<�/4

e

�SCFT

(x,��/4) = ̃1(x)

(x,�/4) = ̃2(x)

• We expressed flow of tensor network states by path integral with deformed action.

Summary

• We fixed one tensor network flow by assuming “most efficient path integral”.

• We identified bulk metric with background metric, which is consistent with several examples.

Entanglement entropy? Quantum corrections? Higher dimensions?

Questions