Path-Integral Complexity, Liouville Action and AdS/CFT Tadashi - - PowerPoint PPT Presentation
East Asia Joint Workshop on Fields and Strings 2017 KEK Theory workshop 2017 @ KEK, Nov.13-17, 2017 Path-Integral Complexity, Liouville Action and AdS/CFT Tadashi Takayanagi Yukawa Institute for Theoretical Physics (YITP), Kyoto Univ. Based on
East Asia Joint Workshop on Fields and Strings 2017 KEK Theory workshop 2017 @ KEK, Nov.13-17, 2017
Tadashi Takayanagi
Yukawa Institute for Theoretical Physics (YITP), Kyoto Univ.
Based on 1703.00456 [Phys.Rev.Lett. 119 (2017)071602]
1706.07056 [To appear in JHEP]
Collaborators: Pawel Caputa (YITP) Nilay Kundu (YITP) Masamichi Miyaji (YITP) Kento Watanabe (YITP) Poster Presentation
Motivation 1 What is the basic mechanism of AdS/CFT ? ⇒ One intriguing idea is the conjectured interpretation of AdS/CFT as tensor networks (TNs).
Tensor Network = Network of Quantum entanglement
``Emergent space from Quantum Entanglement’’
[After 20 years from Maldacena’s discovery]
[Swingle 2009,….]
Holographic Entanglement Entropy
N A A A A
G S 4 ) Area( log Tr Γ = − = ρ ρ
Holographic Entanglement Entropy
Minimal surface
[Ryu-TT 2006, Hubeny-Rangamani-TT 2007]
[Derivation: Casini-Huerta-Myers 2009, Lewkowycz-Maldacena 2013]
Entanglement Wedge
⇒ Entanglement Wedge MA (note: we took a time slice)
MA = A region surrounded by A and ΓA (on a time slice) AdS in CFT in
Bulk M CFT A
A
ρ ρ ⇔
[Czech et.al. 2012, Wall 2012, Headrick et.al. 2014, … ]
Some advertisements
[1] What is the CFT interpretation of EW cross section ? [2] Can we distinguish CFTs by behaviors of EE ? ⇒Time evolutions of EE can distinguish CFTs ! (i) Rational CFTs, (ii) Irrational but integrable CFTs, and (iii) Chaotic CFTs (Hol. CFTs).
MAB
?? 4 ) Area(
AB =
Σ
N
G
Koji Umemoto’s poster (on Wed.) Yuya Kusuki’s poster (on Wed.) We found an answer !
Planck length ~ 1 qubit
⇒ Emergent space via tensor networks (TNs) ?
Boundary =CFTd Bulk=AdSd+1
Spacetime in gravity = Collections of bits of entanglement
Area in the unit
d-1
TN = A geometrical description of wave functions (Ansatz for variation principle to find ground states)
Example of TN: Matrix Product State (MPS)
[DMRG: White 92,…, Rommer-Ostlund 95,..]
α β σ ) (σ
αβ
M
1
2
n
1
α
2
α
3
α
n
α
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
.
, 1,2,..., ↓ =↑ =
i i
σ χ α
Spin chain
CFTs 2D describe enought to not is nt Entangleme log 2 ] links [# Min ⇒ = ≤ χ
A
S
MERA [Vidal 05, …] [TN for AdS/CFT: Swingle 09,…]
=
! CFT 2d in results with agrees log ] links [# Min ⇒ ∝ ≤ L S A
Coarse-graining = Isometry
ad bcd abc T
T δ =
†
a b c d a d
Disentangler = Unitary trf.
A Basic Key Idea: Tensor Network of MERA = a time slice of AdS space
[Perfect TN: Pastawski-Yoshida-Harlow-Preskill 15] [Random TN: Hayden-Nezami-Qi-Thomas-Walter-Yang 16]
Questions [see e.g. Beny 2011, Bao et.al. 2015, Czech et.al. 2015] (a) Special Conformal invariance ? (b) Non-isotropic tensors ? (EW is not properly realized) (c) Why the EE bound is saturated ? (d) How to derive Einstein eq. ? (Sub AdS Scale Locality) Recent developments in lattice models
・Improved TN models: ⇒(a),(b),(c) ⇒Refer also to Arpan Bhattacharrya’s talk
Some of these problems may be due to lattice artifacts. Moreover, we want to eventually understand the genuine AdS/CFT in the continuum limit. We propose a new alternative approach based on path-integrals, related to a continuum limit of TNs.
Our guiding principle 1
Eliminating unnecessary tensors in TN for a given state = Creating the most efficient TN (= Optimization of TN) Solving the dynamics of Gravity (Einstein eq. etc.)
Motivation 2 How can we define complexity in CFTs ?
A given N qubit state: Quantum Circuit
= min[ #(Gates) ]
Computational Complexity of State
Computational Complexity of State
⋅ ⋅ ⋅ = Ψ
⋅ ⋅ ⋅ = Ψ
Acting (Local) Unitary Gates
Ψ
Recently, holographic formulas of complexity have been proposed and actively studied: (i) Complexity = Max. volume in AdS
[Stanford-Susskind 14]
(ii) Complexity= Gravity action in WDW patch
[Brown-Roberts-Susskind-Swingle-Zhao 15]
(iii) Information Metric = Max. volume in AdS
[Miyaji-Numasawa-Shiba-Watanabe-TT 15]
For tensor network descriptions, Complexity = Min [# of Tensors] in TN
This motivates us to consider its QFT counterpart We introduce ``Path-integral Complexity’’ . Our guiding principle 2 ・Lattice (tensor) structures 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
] [
ab
g IΨ
ab
g
(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
dz dx e ds
z x
+ =
φ
). (
2 2 2 2
dz dx 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]
In CFTs, owing to the Weyl invariance, we have
Our Proposal (Optimization of Path-integral for CFTs):
Minimize w.r.t with the boundary condition
)] , ( [ z x I φ
. |
2 2 − = = ε ε φ z
e
) ( )] , ( [ exp ) (
Flat
2
x z x I x
UV e g UV
ab ab
Φ Ψ ⋅ = Φ Ψ
=
φ
δ
φ
[ ]
( )
) , ( ) ( ) , ( ) (
) (
= Φ − Φ ⋅ Φ = Φ Ψ
Φ − ∞ < < ∞ − ∞ < <
z x x e z x D x
CFT
S x z g UV
δ
The wave functional for CFT vacuum is given by
gab(x,z): background metric Original wave func. Optimized wave func.
A Reason for Minimization
The normalization N estimates repetitions of same
⇒ Our Complexity Formula:
) , (
z x
φ Ψ Ψ =
Ψ ≡
Ψ
state quantum the
complexity nal computatio C
. 1 : Minimum term) surface ( ) ( 24 ) ( ) ( 24 ] [ ], [ Log ] [
2 2 2 2 2 2 2
2
z e e dxdz c e dxdz c S S I
z x z x L L g e g
ab ab
= ⇒ + + ∂ + ∂ = + ∂ + ∂ = = Ψ Ψ =
= = φ φ φ δ δ
φ φ π φ φ π φ φ φ
φ
Hyperbolic plane (H2) = Time slice of AdS3
# of Unitaries # of Isometries
(2-2) Liouville Action as Complexity in 2D CFTs
. / ) (
2 2 2 2
z dz dx ds + =
[Czech 17] [Caputa-Kundu-Miyaji-Watanabe-TT 17]
Liouville Action ε φ
φ
L c S C
L
⋅ =
Ψ
~ ]] [ [ Min
A Sketch: Optimization of Path-Integral
Time
Space x
) (
Flat
x
UV Φ
Ψ
Optimize UV modes k>1/z are not important !
Space H2
2 2 2 2
z dz dx ds + =
) (
2
x
e g UV
Φ Ψ
=
φ
MERA
The TFD state at T=1/β is described as the path-integral
[ ]
) 4 , ( ) ( ) 4 , ( ) ( ) , ( ) ( ), (
2 1 ) ( 4 / 4 / 2 1
− = Φ − Φ = Φ − Φ ⋅ Φ = Φ Φ Ψ
Φ − ∞ < < ∞ − < < −
∫ ∏
β δ β δ
β β
z x x z x x e z x D x x
CFT
S x z g
( ).
/ 2 cos 1 4 )] , ( [
Minimizati
2 2 2 ) ( 2
β π β π φ
φ
z e z x S
z L
⋅ = ⇒
+β/4 CFT1 CFT2
) (
1 x
Φ
) (
2 x
Φ
= Time slice of BTZ black hole. (i.e. Einstein-Rosen Bridge）.
(2-3) Thermofield Double of 2D CFT
Optimization
] [φ
L
S
(2-4) Primary States and Back-reactions
Vacuum state on a circle We optimize the path-integral
The solution of Liouville equation |w|=1
. ) | | 1 ( 4
2 2 2
w w dwd ds − =
= Hyperbolic Disk (=Time slice of Global AdS3) Primary state on a circle We insert an operator at w=0. It has conformal dim. hL=hR=h.
O(0)
|w|=1
) , ( w w O
2 φ ⋅ −
h
Thus we minimize
. /
) ( 2 ] [ 1
2
φ φ
φ
h S g e g
e e
L
− = =
⋅ ∝ Ψ Ψ
. ) ( 6 4 1
2 2
= + ∂ ∂ w c h e
w w
δ π φ
φ θ
ζ ζ ζ ζ
i a
re w d d ds = ≡ − = , ) | | 1 ( 4
2 2 2
Solution: . 2 ~ a π θ θ + ⇒ Deficit angle:
). / 12 1 ( c h a − ≡
Note: the AdS/CFT predicts Interestingly, if we consider the quantum Liouville CFT, then ⇒ We get
. / 24 1 c h a − =
). / 2 ( , 3 1 .), 2 / ( 4
2
γ γ αγ γα + ≡ + = − = Q Q c Q h
. / 24 1 c h a − =
Heuristic Summary Time
Optimize
A fine graining is needed ⇒ The metric gets larger ! Local excitation (energy source） This provides the back-reaction mechanism as in general relativity ! Map
[Goto-TT 17]
(3-1) Entanglement Wedge from Optimization
Consider an optimization of reduced density matrix . We decompose the geometry into two halves.
A
A
Optimize
(Squeezing)
Boundary ∂Σ Satisfies
K=0
⇒ Geodesic
∂Σ Entanglement wedge (EW) as opposed to MERA ! ∂Σ
[ ]
[ ]
. : condition Bdy . 12 2 ) ( ) ( 24 2
2 2 2
= ∂ + = ⇒ ⋅ × + + ∂ + ∂ × =
∫ ∫
Σ ∂ Σ
φ φ π φ φ π
φ φ n z x L
K K e K ds c e dxdz c S
A=an interval
[ ] [ ]
. 3 ) ( ) ( 6 ] [
2 2 2 ) (
Σ ∂ Σ
+ + + ∂ + ∂ =
φ φ
µ φ π φ φ π φ e K ds c e dxdz c S
B z x n L
Replica Method
n A
) 1 ( n − = π δ
. 6
1 ) ( φ
e ds c S n S
n n L A
∂ =
= ∂ ∂ − =
Reproduce the correct HEE！ ) 1 ( n
B
− = π µ ∂Σ
(3-2) Hol. Ent. Entropy from Optimization
which measures complexity of the corresponding TN.
An optimization of path-integral of a CFT state = Minimizing the complexity ⇔ a time slice of AdS
⇒ A factor mismatch with ``Complexity = Gravity Action’’ proposal. Future Problems
Massive QFTs [Bhattacharrya-Caputa-Das-Kundu-Miyaji-TT, work in progress]
Time dependent states ?, Sub AdS locality ?, dS/CFT ? [For details, refer to Kento Watanabe’s poster]