Euclidean Path Integral Optimization in CFT s: holographic complexity and an attempt to a derivation of AdS/CFT

(1)

Euclidean Path Integral Optimization

in CFT’s: holographic complexity and

an attempt to a derivation of AdS/CFT

Nilay Kundu

(2)

Based on the following works ..

1) Phys. Rev. Lett. 119 (2017) no.7, 071602

2) arXiv : 1706.07056 [hep-th] — (Accepted in JHEP)

With … Pawel Caputa, Masamichi Miyaji, Tadashi Takayanagi and Kento Watanabe

Euclidean Path Integral Optimization

in CFT’s: holographic complexity and

an attempt to a derivation of AdS/CFT

(3)

Euclidean Path Integral Optimization

in CFT’s: holographic complexity and

an attempt to a derivation of AdS/CFT

(4)

Euclidean Path Integral Optimization

in CFT’s: holographic complexity and

an attempt to a derivation of AdS/CFT

Towards a BETTER understanding of the basic mechanism behind

(5)

Euclidean Path Integral Optimization

in CFT’s: holographic complexity and

an attempt to a derivation of AdS/CFT

Euclidean path-integral for wave-fn in CFTs

(6)

Euclidean Path Integral Optimization

in CFT’s: holographic complexity and

an attempt to a derivation of AdS/CFT

OPTIMIZATION

Euclidean path-integral for wave-fn in CFTs

(7)

Euclidean Path Integral Optimization

in CFT’s: holographic complexity and

an attempt to a derivation of AdS/CFT

OPTIMIZATION

AdS/CFT Euclidean path-integral for wave-fn in CFTs

(8)

Euclidean Path Integral Optimization

in CFT’s: holographic complexity and

an attempt to a derivation of AdS/CFT

OPTIMIZATION

Holographic Complexity Towards a BETTER

understanding of the basic mechanism behind

AdS/CFT Euclidean path-integral for wave-fn in CFTs

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•

After 20 years of the first proposal by Maldacena … We are still unaware of the basic mechanism behind AdS/CFT conjecture

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•

That means can we aim to demystify this duality beyond known examples …

•

To understand the basic mechanism it may be useful to know what does a suitable measure of information in CFT corresponds to that in the AdS ..

(11)

•

One such quantitative suitable measure happens to be : the entanglement

entropy or von-Neumann entropy …

Holographic Entanglement Entropy

•

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•

Generalization of Hawking -Bekenstein BH entropy ..

•

Idea of Holography ==> unit entanglement per Planck area

•

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•

Interesting interpretation of HEE : “Spacetime in Gravity is built out of collections of bits of quantum entanglement”

•

Generalization of Hawking -Bekenstein BH entropy ..

•

Idea of Holography ==> unit entanglement per Planck area

•

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Tensor Networks are an useful and explicit framework to realize this interpretation

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Tensor Networks are an useful and explicit framework to realize this interpretation

AdS/CFT or more generally holography explicitly realizes

the geometrization of dynamics in CFT’s

Tensor networks uses quantum entanglement to represents

the geometrization of quantum states in many body system

The HEE suggests the following novel interpretation: ``A spacetime in gravity = Collections of bits of quantum entanglement’’

B

A

Planck length ⇒ Manifestly realized in the proposed connection between AdS/CFT and tensor networks ! [Swingle 09] This entanglement/geometry duality may work for any surfaces [Bianchi‐Myers 12]

AdS/CFT ( or more generally holography)

⇒ _{``Geometrization’’ of Dynamics in QFTs}

One important fact for this to work is

Quantum entanglement

⇒ _{``Geometry’’ of Quantum States in many-body system}

① Introduction

Emergent spacetime from Entanglement

Algebraically complicated !

=

Tensor Network

ＱｕａｎｔｕｍＳｔａｔｅｓ

AdS

The HEE suggests that

A spacetime in gravity

= Collections of qubits of quantum entanglement

B

A

γ

A

Planck length

A useful explicit framework for this is the tensor network.

(4-1) Tensor Network [See e.g. Cirac-Verstraete 09(review)]

Tensor network states

= 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 ground state. ~Geometry of Tensor Network

④ Tensor Network and Emergent Spacetime

The HEE suggests the following novel interpretation: ``A spacetime in gravity = Collections of bits of quantum entanglement’’

B

A

Planck length ⇒ Manifestly realized in the proposed connection between AdS/CFT and tensor networks ! [Swingle 09] This entanglement/geometry duality may work for any surfaces [Bianchi‐Myers 12]

AdS/CFT ( or more generally holography)

⇒ _{``Geometrization’’ of Dynamics in QFTs} One important fact for this to work is

Quantum entanglement

⇒ _{``Geometry’’ of Quantum States in many-body system}

① Introduction

Emergent spacetime from Entanglement

Algebraically complicated !

=

Tensor Network

ＱｕａｎｔｕｍＳｔａｔｅｓ

AdS

The HEE suggests that

A spacetime in gravity

= Collections of qubits of quantum entanglement

B

A

γ

A

Planck length

A useful explicit framework for this is the tensor network.

(4-1) Tensor Network [See e.g. Cirac-Verstraete 09(review)] Tensor network states

wave functions in quantum many-body systems. [A tensor network diagram = A wave function]

④ Tensor Network and Emergent Spacetime

MERA, Hyperbolic Space and Holographic EE

: size of

steps (a time slice of ) Hyperbolic space

Holographic EE saturates this bound

Entanglement Renormalization

: size of

steps (a time slice of )

Hyperbolic space

: size of

MERA and time slice of AdS and Entanglement entropy

: size of

Hyperbolic space

: size of

Hyperbolic space

: size of

Hyperbolic space

: size of

Hyperbolic space

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Tensor Network : An efficient variational ansatz for the

ground state wave functions in quantum many body systems

The HEE suggests the following novel interpretation:

``

A spacetime in gravity

= Collections of bits of quantum entanglement’’

B

A

Planck length

⇒ Manifestly realized in the proposed connection

between AdS/CFT and tensor networks !

[Swingle 09]

This entanglement/geometry

duality may work

for any surfaces

[Bianchi‐Myers 12]

AdS/CFT ( or more generally holography)

⇒

_{``Geometrization’’ of Dynamics in QFTs}

One important fact for this to work is

Quantum entanglement

⇒

_{``Geometry’’ of Quantum States in many-body system}

① Introduction

Emergent spacetime from Entanglement

Algebraically complicated !

=

Tensor Network

ＱｕａｎｔｕｍＳｔａｔｅｓ

AdS

The HEE suggests that

A spacetime in gravity

= Collections of qubits of quantum entanglement

B

A

γ

A

Planck length

A useful explicit framework for this is the

tensor network

.

(4-1) Tensor Network

[See e.g. Cirac-Verstraete 09(review)]

Tensor network states

=

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 ground state.

~

Geometry of Tensor Network

④ Tensor Network and Emergent Spacetime

A Tensor Network diagram = A many-body wave-fn

Requirement : A good ansatz should reproduce the correct quantum entanglement of the ground state wave-fn

A network of tensors that geometrizes quantum entanglement

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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

Efficiency of Tensor Network Descriptions

Generic States : Exponential … Polynomial !! Efficient !! # of Parameters MPS : Tensors MERA : Tensors Polynomial !! Efficient !! Sites

Tensor Network (TN) [See e.g. the review Cirac-Verstraete 09]

Recent remarkable progresses in numerical algorithms for quantum lattice models:

⇒ Tensor network states

⇒ An ansatz is good if it respects the quantum entanglement of the true ground state.

Quantum wave function ⇔ Geometry of network of entanglement

Tensor Networks …

④ Tensor Network and Emergent Spacetime

Tensor Networks …

④ Tensor Network and Emergent Spacetime

(4-2) MERA

MERA (Multiscale Entanglement Renormalization Ansatz):

⇒ An efficient variational ansatz for CFT ground states.

[Vidal 05]

To increase entanglement in a CFT, we add (dis)entanglers.

1

σ σ₂ σ₃ σ₄ σ₅ _σ₆ _σ₇ _σ₈ σ₁ σ₂ σ₃ σ₄ σ₅ _σ₆ _σ₇ _σ₈

Unitary transf. between 2 spins

Ex. Matrix Product State (MPS) [DMRG: White 92,…, Rommer-Ostlund 95,..] α _β σ Mαβ (σ) 1

σ

₂

σ

_n 1 α α₂ _α₃ α_n Spins n n n n M M M 2 1 , , , 2 1) ( ) ( )] , , , ( Tr[ 2 1 σ σ σ σ σ σ σ σ σ

∑

= Ψ . or , 1,2,..., ↓ =↑ = i i σ χ α Spin chain

Tensor Networks …

wave functions in quantum many-body systems. [A tensor network diagram = A wave function]

④ Tensor Network and Emergent Spacetime

2 type of tensors ==>

Polynomial parameters ==> efficient

Tensor Networks …

(4-1) Tensor Network [See e.g. Cirac-Verstraete 09(review)] Tensor network states

④ Tensor Network and Emergent Spacetime

Polynomial parameters ==> efficient

= Efficient variational ansatz for the ground state wave functions in quantum many-body systems.

Tensor Networks …

④ Tensor Network and Emergent Spacetime

Exponential parameters ==> inefficient

Efficiency of Tensor Network Descriptions

Tensor Network Descriptions of Quantum States

Optimization of the Energy

with

Ex: MPS (Matrix Product States)

good for gapped systems

Efficient variational ansatz for ground state wave functions :

Tensor Networks

A “Geometrization” of Quamtum States

( respect the correct Quantum Entanglement )

[DMRG: White 92,…, Rommer-Ostlund 95,… ]

Tensor Networks …

(4-2) MERA

MERA (Multiscale Entanglement Renormalization Ansatz):

⇒ An efficient variational ansatz for CFT ground states.

[Vidal 05]

To increase entanglement in a CFT, we add (dis)entanglers.

1

σ σ₂ σ₃ σ₄ σ5 σ₆ σ₇ σ₈ σ1 σ2 σ3 σ₄ σ5 σ₆ σ₇ σ₈

Ex. Matrix Product State (MPS) [DMRG: White 92,…, Rommer-Ostlund 95,..] α β σ Mαβ(σ) 1 σ σ₂ σ_n 1 α α₂ _α₃ α_n Spins n n n n M M M 2 1 , , , 2 1) ( ) ( )] , , , ( Tr[ 2 1 σ σ σ σ σ σ σ σ σ ∑ = Ψ . or , 1,2,..., ↓ =↑ = i i σ χ α Spin chain

Effective variational ansatz for ground state wave-fn : Tensor Networks that Optimizes the energy

Efficiency of Tensor Network Descriptions

Tensor Network Descriptions of Quantum States

Optimization of the Energy

with

Ex: MPS (Matrix Product States)

good for gapped systems

Efficient variational ansatz for ground state wave functions : Tensor Networks

A “Geometrization” of Quamtum States

( respect the correct Quantum Entanglement )

[DMRG: White 92,…, Rommer-Ostlund 95,… ]

Tensor Networks …

(4-2) MERA

MERA

(Multiscale Entanglement Renormalization Ansatz):

⇒ An efficient variational ansatz for CFT ground states.

[Vidal 05]

To increase entanglement in a CFT, we add

(dis)entanglers

.

1

σ σ ₂ σ₃ _σ ₄ σ₅ _σ₆ _σ₇ _σ₈ σ₁ σ ₂ σ₃ _σ ₄ σ₅ _σ₆ _σ₇ _σ₈

Ex. Matrix Product State (MPS) [DMRG: White 92,…, Rommer-Ostlund 95,..] α β σ Mαβ (σ) 1 σ σ₂ σ_n 1 α α₂ _α₃ α_n Spins n n n n M M M 2 1 , , , 2 1) ( ) ( )] , , , ( Tr[ 2 1 σ σ σ σ σ σ σ σ σ ∑ = Ψ . or , 1,2,..., ↓ =↑ = i i σ χ α Spin chain

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MERA Network

- Isometry MERA (Multi-scale Entanglement Renormalization Ansatz) network

- 2 kinds of Tensors : For CFT ground states, a good TN is ….

- (Dis-) Entangler

Coarse-graining IR (Less Entangled)

UV (More Entangled)

Add (Remove) Entanglement

- Layer by layer RG transformation

Lattice scale changes exponentially

[Vidal 05, 06]

MERA Network

- (Dis-) Entangler

UV (More Entangled)

Entanglement Renormalization [Vidal 05, 06]

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MERA Network

- (Dis-) Entangler

UV (More Entangled)

[Vidal 05, 06]

MERA Network

- (Dis-) Entangler

UV (More Entangled)

Multiscale Entanglement Renormalization Ansatz (MERA)

_{MERA Network}

- (Dis-) Entangler

UV (More Entangled)

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MERA, Hyperbolic Space and Holographic EE

: size of

Hyperbolic space

MERA, Hyperbolic Space and Holographic EE

: size of

Hyperbolic space

MERA, Hyperbolic Space and Holographic EE

: size of

Hyperbolic space

MERA, Hyperbolic Space and Holographic EE

: size of

Hyperbolic space

MERA, Hyperbolic Space and Holographic EE

: size of

Hyperbolic space

MERA, Hyperbolic Space and Holographic EE

: size of

Hyperbolic space

MERA, Hyperbolic Space and Holographic EE

: size of

Hyperbolic space

MERA, Hyperbolic Space and Holographic EE

: size of

Hyperbolic space

MERA, Hyperbolic Space and Holographic EE

: size of

Hyperbolic space

MERA, Hyperbolic Space and Holographic EE

: size of

Hyperbolic space

MERA, Hyperbolic Space and Holographic EE

: size of

Hyperbolic space

MERA, Hyperbolic Space and Holographic EE

: size of

Hyperbolic space

MERA, Hyperbolic Space and Holographic EE

: size of

Hyperbolic space

MERA and time slice of AdS and Entanglement entropy

MERA, Hyperbolic Space and Holographic EE

: size of

Hyperbolic space

MERA, Hyperbolic Space and Holographic EE

: size of

Hyperbolic space

MERA, Hyperbolic Space and Holographic EE

: size of

Hyperbolic space

MERA, Hyperbolic Space and Holographic EE

: size of

Hyperbolic space

MERA, Hyperbolic Space and Holographic EE

: size of

Hyperbolic space

MERA, Hyperbolic Space and Holographic EE

: size of

Hyperbolic space

MERA, Hyperbolic Space and Holographic EE

: size of

Hyperbolic space

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 

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What we learn from tensor networks :

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MERA geometry is equal to time slice of AdS

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It sort of qualitatively works … but several doubts remains … Evidences :

1) A real-space RG structure of MERA :: The extra radial direction and evolution in bulk AdS

2) If the entanglement bound in MERA saturates :: it coincides with the HEE

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Doubts :

1) Is it possible to explore locality beyond the AdS scale ..

2) Why do we need to have saturation of entanglement entropy ..

3) Is the full conformal symmetry structure realizable ?

4) Ultimately we want to go beyond the lattice formulation and see the AdS in true continuum sense ..

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Doubts :

1) Is it possible to explore locality beyond the AdS scale ..

2) Why do we need to have saturation of entanglement entropy ..

3) Is the full conformal symmetry structure realizable ?

4) Ultimately we want to go beyond the lattice formulation and see the AdS in true continuum sense ..

We propose an alternative method using OPTIMIZATION of

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An “OPTIMIZATION’’ of the Euclidean Path Integral ( ˜'(x)) = Z Y x Y ✏<z<₁ D'(x, z) ! e SCF T(') Y x ('(✏, x) '(x))˜

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An “OPTIMIZATION’’ of the Euclidean Path Integral ( ˜'(x)) = Z Y x Y ✏<z<₁ D'(x, z) ! e SCF T(') Y x ('(✏, x) '(x))˜

The ground state

wave-fn as a functional of the UV boundary

value of the fields in the CFT

The measure of the path-integration over the fields in the CFT

The CFT action

The delta-fn ensuring the boundary value for the field at UV

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An “OPTIMIZATION’’ of the Euclidean Path Integral x z= 0 0 0 0 0 0 0 0 0 0 Hyperbolic Space z Flat Space

Euclidean path Integral can be rewritten as MERA

network but we would like to get rid of the possible lattice artifact

Another motivation: Tensor network renormalization

[Evenbly-Vidal 14, 15]

We can rewrite the Euclidean path-integral into a MERA

tensor network.

Picture taken from

Evenbly-Vidal

arXiv:1502.05385

In this talk, we would like to have a general argument

which does not assume any particular tensor networks to

avoid any lattice artifacts.

( ˜'(x)) = Z Y x Y ✏<z<₁ D'(x, z) ! e SCF T(') Y x ('(✏, x) '(x))˜

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Optimization of Path-Integral

Tensor Network

Renormalization

(TNR) = Optimization of TN

Optimization of

Path-integral

Euclidean

Time (-z)

Space (x)

Hyperbolic Space = Time slice of AdS3

ε

Lattice Constant

[Evenbly-Vidal 14, 15] [Miyaji-Watanabe-TT 16]

⑤ Optimization of Path-Integral and AdS/CFT

[Miyaji-Watanabe-TT 2016 + work in progress]

How to find the ground state of a given quantum many-body system ?

⇒（１）Variational Method： First constrain the possible quantum states and next minimize its energy.

Ex. Look at behaviors of quantum entanglement

(e.g. Area law) → The idea of tensor networks

⇒（２）Path-integral： Path-integrate for a long time in

Euclidean theory:

lim

_β_→_∞

e

−βH

ψ

=

ψ

₀

⑤ Optimization of Path-Integral and AdS/CFT

Euclidean theory:

lim

_β_→_∞

e

−βH

ψ

=

ψ

₀

⑤ Optimization of Path-Integral and AdS/CFT

Euclidean theory:

lim

_β_→_∞

e

−βH

ψ

=

ψ

₀

Tensor Network Renormalization (TNR) : OPTIMIZATION of Tensor Networks

( ˜'(x)) =Z Y x Y ✏<z<1 D'(x, z) ! e SCF T(')Y x ('(✏, x) '(x))˜

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Discretized Path-integral and Its Optimization

Time

τ

Space

x

[

(

)

]

(

,

)

( )

(

)

(

,

0 )

)

0

=

−

⋅

=

Ψ

− ∞ < < ∞ −∞< < −

∫ ∏

φ

τ

δ

ϕ

φ

τ

ϕ

φ τ

x

e

x

D

x

SCFT x

[

ϕ

(x

)

]

Ψ

τ=0

τ=-∞

Optimize

High momentum modes

k>1/τ is not important !

= AdS !

（hyperbolic space）

[

(

)

]

Min

ϕ

x

Ψ

(4-3) AdS/MERA duality ? [Swingle 09]

0 =

u

=

−

1 A

A

Min[Area]

A

γ

. , ) ( ₂ 2 2 2 2 2 2 2 2 u _where _z _e u z x d dt dz x d dt e ds + − + = − + = ε ⋅ − ε 　 2 + d

AdS

1 + d

CFT

A

γ

Links]

Min[#

? Equivalent ) ( u_IR u = −∞ ≡ The idea: Tensor Network of MERA (∃scale inv.) = a time slice of AdS space Qualitative evidences (i) Real space RG = radial evolution in AdS → something we usually expect in AdS/CFT. (ii) The bound of EE in MERA: If this is saturated, it seems to agree with the HEE . log ] Min[ _Int A N S . ) Area( Min 4 1 A A N A G S

0 = u u = −1 A A A

Min[Area]

A

γ

AdS

1 + d

CFT

A

γ

Links]

Min[#

? Equivalent ) ( u_IR u = −∞ ≡

0 =

u

=

−

1 A

A

Min[Area]

A

γ

AdS

1 + d

CFT

A

γ

Links]

Min[#

0 = u u = −1 A A A

Min[Area]

A

γ

AdS

1 + d

CFT

A

γ

Links]

Min[#

0 =

u

=

−

1 A

A

Min[Area]

A

γ

AdS

1 + d

CFT

A

γ

Links]

Min[#

0 = u u = −1 A A A

Min[Area]

A

γ

AdS

1 + d

CFT

A

γ

Links]

Min[#

Path-Integral Wave-fn : it’s Discretization and Optimization

(Miyaji, Watanabe, Takayanagi-2016)

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Another way of looking at it : Free scalar Field

Euclidean Path-Integral for Ground State

CFT₂ on

EOM & regular at Free Scalar

At fixed ,only modes with

contribute in the -integral : UV cutoff x z= 0 0 0 0 0 0 0 0 0 0 Hyperbolic Space z Flat Space (Miyaji, Watanabe, Takayanagi-2016)

AdS/CFT and tensor network

⇥ log

S

_A

=

Area( )

4G

N

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

S

_A

_

[Evenbly, Vidal(2014)]

AdS/CFT and tensor network

⇥ log

S

_A

=

Area( )

4G

_N

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

S

_A

_

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.

S

_A

=

Area( )

4G

_N

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

S

_A

_

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x z= 0 0 0 0 0 0 0 0 0 0 Hyperbolic Space z Flat Space

What we learn so far:

In the language of Tensor networks ==> Eliminating extra tensors in TN is creating most efficient TN

==> The algorithm at work for this, is the “OPTIMIZATION” of TN for a given state. For free fields ==> Introducing momentum dependent cut-off.

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Our first Key Insight :

Rearrangement of the lattice-structure for the tensors in TN, can be engineered by changing the background metric in the Euclidean Path Integral, keeping the boundary condition for the fields at UV unchanged.

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Rearrangement of the lattice-structure for the tensors in TN, can be engineered by changing the background metric in the Euclidean Path Integral, keeping the boundary condition for the fields at UV unchanged. What we are still lacking :

What is the counterpart of the OPTIMIZATION algorithm in TN, that will be the guiding principle to determine the changed (optimized)

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This takes us to the “COMPLEXITY” part of our story .. Recently an interesting question, that people are after : How to define complexity of a state in CFT’s ..

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Computational complexity of a quantum state = Min. no. of quantum gates required to

prepare it from a given “simple” reference state

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Computational complexity of a quantum state = Min. no. of quantum gates required to

prepare it from a given “simple” reference state

= Min. no. of tensors in the TN description .

In the literature holographic formulas for computing complexity is already proposed :

1) Complexity = Max. Volume in AdS (Standford-Susskind .. )

2) Complexity = Gravity action in WDW patch of AdS (Brown, Roberts, Susskind, Myers …)

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“PATH-INTEGRAL COMPLEXITY”

= We define the CFT analogue of the Complexity

This motivates us to consider its QFT counterpart

We introduce

``Path-integral Complexity’’

.

Our guiding principle 2

・Lattice structure (= arrangement of tensors) in TN

Background metric g

ab

in Euclid path-integral

・Optimization of TN for a state Ψ

Minimizing Path-integral Complexity

w.r.t the metric

]

[

g

_ab

I

_Ψ ab

g

This motivates us to consider its QFT counterpart We introduce ``Path-integral Complexity’’ .

Our guiding principle 2

・Lattice structure (= arrangement of tensors) in TN Background metric gab in Euclid path-integral

・Optimization of TN for a state Ψ

Minimizing Path-integral Complexity w.r.t the metric

]

[

g

_ab

I

_Ψ ab

g

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This motivates us to consider its QFT counterpart

We introduce

``Path-integral Complexity’’

.

Our guiding principle 2

・Lattice structure (= arrangement of tensors) in TN

Background metric g

ab

in Euclid path-integral

・Optimization of TN for a state Ψ

Minimizing Path-integral Complexity

w.r.t the metric

]

[

g

_ab

I

_Ψ ab

g

]

[

g

_ab

I

_Ψ ab

g

Rearrangement of the lattice-structure for the tensors in TN, can be engineered by changing the background metric in the Euclidean Path Integral, keeping the

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This motivates us to consider its QFT counterpart

We introduce

``Path-integral Complexity’’

.

Our guiding principle 2

・Lattice structure (= arrangement of tensors) in TN

Background metric g

ab

in Euclid path-integral

・Optimization of TN for a state Ψ

Minimizing Path-integral Complexity

w.r.t the metric

]

[

g

_ab

I

_Ψ ab

g

]

[

g

_ab

I

_Ψ ab

g

boundary condition for the fields at UV unchanged. Our second Key Insight (This is our proposal) :

Optimization of TN for a given state is equivalent to Minimizing the path-integral complexity with respect to the back-ground metric.

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This motivates us to consider its QFT counterpart

We introduce

``Path-integral Complexity’’

.

Our guiding principle 2

・Lattice structure (= arrangement of tensors) in TN

Background metric g

ab

in Euclid path-integral

・Optimization of TN for a state Ψ

Minimizing Path-integral Complexity

w.r.t the metric

]

[

g

_ab

I

_Ψ ab

g

]

[

g

_ab

I

_Ψ ab

g

boundary condition for the fields at UV unchanged. Our second Key Insight (This is our proposal) :

Optimization of TN for a given state is equivalent to Minimizing the path-integral complexity with respect to the back-ground metric.

The final crucial element :

Do we have a good answer for how to define the path-integral complexity given one CFT ..

(43)

FOR a 2D Euclidean CFT We have a very good answer !! In a 2D CFT with coordinates z (Euclidean time), x (space) we can always write any metric in the conformally flat form

② AdS from Optimization of Path-Integrals

(2-1) Formulation

A Basic Rule: Simplify a path-integral s.t.

it produces the correct UV wave functional.

Consider 2D CFTs for simplicity. ( z=- Euclidean time, x=space)

Deformation of discretizations in path-integral

= Curved metric such that one cell (bit) = unit length.

Note: The original flat metric is given by (ε is UV cutoff):

).

(

2 2 ) , ( 2 2

_e

_dx

_dz

ds

=

φ x z

+

). ( 2 2 2 2 _dx _dz ds =

ε

− ⋅ +

Remember, the unoptimized metric is flat space (with UV cutoff given by “epsilon”)…

② AdS from Optimization of Path-Integrals

(2-1) Formulation

A Basic Rule: Simplify a path-integral s.t.

it produces the correct UV wave functional.

Consider 2D CFTs for simplicity. ( z=- Euclidean time, x=space)

Deformation of discretizations in path-integral

= Curved metric such that one cell (bit) = unit length.

Note: The original flat metric is given by (ε is UV cutoff):

).

(

2 2 ) , ( 2 2

_e

_dx

_dz

ds

=

φ x z

+

). ( 2 2 2 2 _dx _dz ds =

ε

− ⋅ +

Optimization of Path-Integral

Tensor Network

Renormalization

(TNR) = Optimization of TN

Optimization of

Path-integral

Euclidean

Time (-z)

Space (x)

Hyperbolic Space = Time slice of AdS3

ε

Lattice Constant

[Evenbly-Vidal 14, 15] [Miyaji-Watanabe-TT 16]

A Sketch: Optimization of Path-Integral

Time

-z

Space x

[

( )

]

Flat _x UV Φ Ψ

z=0

z=∞

Optimize UV modes k>1/z are not important !

= Hyperbolic Space H2 2 2 2 2 z dz dx ds = +

∝

ΨUVg e2

[

Φ(x)

]

= φ 0 0 0 0 0 0 0 0 0 0 MERA

Time

-z

Space

x

[

( )

]

Flat _x UV Φ Ψ

z=0

z=∞

=

Hyperbolic Space H2 2 2 2 2 z dz dx ds = +

∝

ΨUVg e2

[

Φ(x)

]

= φ 0 0 0 0 0 0 0 0 0 0 MERA

Time

-z

Space

x

[

( )

]

Flat _x UV Φ Ψ

z=0

z=∞

=

∝

ΨUVg e2

[

Φ(x)

]

= φ 0 0 0 0 0 0 0 0 0 0 MERA

Time

-z

Space

x

[

( )

]

Flat _x UV Φ Ψ

z=0

z=∞

=

∝

ΨUVg e2

[

Φ(x)

]

= φ 0 0 0 0 0 0 0 0 0 0 MERA

Time

-z

Space

x

[

( )

]

Flat _x UV Φ Ψ

z=0

z=∞

=

∝

ΨUVg e2

[

Φ(x)

]

= φ 0 0 0 0 0 0 0 0 0 0 MERA

Time -z Space x [ ( )] Flat _x UV Φ Ψ z=0 z=∞ Optimize UV modes k>1/z are not important !

= Hyperbolic Space H2 2 2 2 2 z dz dx ds = + ∝ Ψ_UVg=e2φ [Φ(x)] 0 0 0 0 0 0 0 0 0 0 MERA