From Path Integral to Tensor Networks
for AdS/CFT
Kento Watanabe
(Center for Gravitational Physics, YITP, Kyoto U)
1609.04645v2 [hep-th] w/ Tadashi Takayanagi (YITP + Kavli IPMU)
Masamichi Miyaji (YITP)
Seminar @ Osaka U 2016/11/15
Toward Better Understanding
about Tensor Networks
as a Toy Model for AdS/CFT
“It” From “Qubit”
Holography, Spacetime & Entanglement
Quantum Many-body System (Gauge)
Holographic Principle or AdS/CFT argues:
Gravity(String)
A “Geometrization” of Quamtum States
[t’Hooft 93,Susskind 94] [Maldacena 99]
Symmetry, Partition Function, States, Operators…
“Entanglement”“Geometry(Spacetime)”
Futhermore, it suggests :
[Ryu-Takayanagi 06]
[van Raamsdonk 09,10]
[Maldacena-Susskind 13]
onon
AdS/MERA ( Tensor Networks ) Correspondence
Quantum States
MERA: An efficient variational ansatz for CFT ground states
MERA (Tensor Networks)
[Swingle 09]
[Vidal 05]
“Geometry(Spacetime)”
If naively combine these 2 correspondences….
Another “Geometrization” of Quamtum States
MERA (Tensor Networks)
Emergent
Hyperbolic Space??
(= a time slice of AdS)
Quantum Error Correction
Bulk Operator Reconstruction
Scrambling, Complexity,…BH &
&
As you (may) have seen
in Beni Yoshida’s lectures,
Surface/State Correspondence
If we assume AdS/MERA(TN),
Closed & Homologically Trivial
Quantum States
[Miyaji-Takayanagi 15]
Co-dim 2 Convex Space-like Surfaces
Pure
MixedOpen or Homologically Non-Trivial
Generalized Holography even for spacetimes without boundaries(e.g. Flat, dS, …)
Some Issues of AdS/MERA
Locality of the bulk AdS ?
(Special) Conformal invariance is not clear?
Why the EE bound is saturated ?
RG causal structure in MERA? dS instead of AdS ?
MERA is local only at AdS radius scale
Lattice artifact ?
(a)
(b)
(c)
(d)
[see e.g. Beny 11, Bao et.al. 15, Czech et.al. 15]
Kinematical SpaceIntegral
trf. AdS
dS
dS slice? Or hyperbolic slice?
(not Planck scale)
(Perfect TNs, Random TNs resolve some issues. But they do not give CFT vacuum…)
AdS/cMERA (cTN) from 2 Different Viewpoints
- A Time Slice of AdS3 from an “Optimization” of
- cTN formulation in CFT2 & Hyperbolic Slice of AdS3
Some Other Arguments
- Sub-AdSScale Bulk Locality for Holographic CFTs
- Bulk Locally Excited States in cTN & Perfect Tensors
Euclidean Path-Integral
Some More about Tensor Networks (MERA)
Contents [Miyaji-Takayanagi-KW 16] 1609.04645v2 [hep-th]
in CFT2
Some More about Tensor Networks (MERA)
Tensor Network Descriptions of Quantum States
Optimization of the Energywith
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,… ]
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
Entanglement Renormalization
[Vidal 05, 06]
MERA, Hyperbolic Space and Holographic EE
: size of
steps (a time slice of )
Hyperbolic space
Holographic EE saturates this bound
Entanglement Renormalization
Efficient Manipulation & “Causal Cone”
Tensors relevant to
“Causal Cone” forassociates the “causal structure” with[Beny 11], [Czech et.al. 15], ….cf )
(“Kinematic Space” of )
“Past”
“Future”
Ex)
Efficient Manipulation for Reduced Density Matrix
Links crossing the bdy of
relate to (minimally)
Holographic EE saturates this bound
steps
Outside of the “causal cone” for
Inside of the “causal cone” for
The tensors in this region only affect to
Similarly,Correlation functions can be reproduced
MERA for Thermo-Field Double State
2-sided AdS-Schwarzschild BHdual
[Matsueda-Ishihara-Hashizume 12]
[Maldacena 01]
[Hartman-Maldacena 13]
[Hosur-Qi-Roberts-Yoshida 15]
The Perfect Tensor Model
Flat Tiling of Unitary Tensors
BH Interior ?
Tensor Network Renormalization Yields MERA [Evenbly-Vidal 14, 15]
MERA
(+ IR bdy effect)
1D Quantum system
Euclidean path-integral Statistical partition function
2D Classical system
UV bdy
The Tensor Network Description:
Tensor Network Renormalization
1 step of Repeat the step
(Refining & Coarse-graining tensors)
“Continuous” MERA (cMERA)
We can describe a “continuum limit of MERA”
- State at scale (Dis-)Entangler
- IR state
Coarse-graining
(Space-like rescaling)
No real space entanglement
for any
Dilatation
(Relativistic rescaling)
Associated with the RG property of MERA,
UV
IR
[Canonical Choice]
[Miyaji-Ryu-Takayanagi-Wen 14]
[Haegeman-Osborne-Verschelde-Verstraete 11]
This is Identified with Conformal Boundary State
AdS/cMERA (cTN) from 2 Different Viewpoints
- A Time Slice of AdS3 from
- cTN formulation in CFT2 & Hyperbolic Slice of AdS3
an “Optimization” of Euclidean Path-Integral in CFT2
Another efficient description of states (wave functions)
[Miyaji-Takayanagi-KW 16] 1609.04645v2 [hep-th]
Euclidean Path-Integral for Ground State
CFT2 on
EOM & regular at
Free Scalar
At fixed ,only modes with
contribute in the -integral
: UV cutoff
Introduce a length scale dependent wave function
Scale Dependent Wave Function with Cutoff Function
with a cutoff function
Simply set
for
Rescaled
at
The Effective Wave Function at the length scale
under a real space RG flow
Reproduce
by retakingthe cutoff function
Improve Cutoff Function in Non-Local Way
We can improve this procedure by taking into account contributions
Suppressing factor for
Ex)
High momentum modes
Non-local
Reproduce
(Dirichlet bdy state)
(Previous: Neumann one)
For Symmetric Orbifold CFTs, (Large-c, Holographic?)
More Layers are expected:
Local Non- Local Non- Local
Non- Physical Physical
from high momentum modes in a non-local way
Interpretation : Optimization of Euclidean Path-Integral
We can interpret this procedure to construct
Optimize
as an “optimization” of the Euclidean path-integral
with the cutoff function
(cf: Tensor Network RG)
the vacuum wave function
-dependent cutoff scale
(a time slice of )
Interpretation : Optimization of Euclidean Path-Integral
This interpretation associates
Hyperbolic space
with the cutoff function
Efficient # of sites decays exponentiallyin
the Euclidean path-integral
Size of UV cutoff grows linearly
invariance of the vacuum state
Area measured by the metric # of sites
Hyperbolic space with
( )
Some Other Backgrounds
TFD state
Finte T
(2-sided BH)
Mass Gapped Removed
Primary StateFine-grained again near the operator
bdy1
bdy2
Primary op
Confined
broken
not straightforward
Qualitatively,
For deconfined states
Large # of d.o.f. needed even at
Time Dependent No Criteria…
( similar to BH horizon)
Choice of Cutoff Function = Choice of Slice in the Bulk
Hyperbolic slice
we have choices of the interporation between the UV & IR states
Actually, in general,
slice
Analytic continued to Lorentian AdS3, we have 2 types of slices
e.g. the cutoff function parametrized by
For the vacuum,
They are same
But for the excited states….
(invariant under different RG flows)
AdS/cMERA (cTN) from 2 Different Viewpoints
- cTN formulation in CFT2 & Hyperbolic Slice of AdS3
[Miyaji-Takayanagi-KW 16] 1609.04645v2 [hep-th]
How to identify?
- A Time Slice of AdS3 from
an “Optimization” of Euclidean Path-Integral in CFT2
General Formulation of cMERA in CFT2
In CFT2, we can identify with the dilatation from the sym.
: Conformal bdy state for the vacuum s.t.
Virasoro
: Conformal sym. for the networks
IR state should be space-like rescaling invariant:
Consistent with
In CFT2,
for CFT2 Also fix
Below the UV-cutoff scale (momentum)
(for the vacuum)
(for the excited states, different choices)
Continuous Tensor Network (cTN) from AdS3/CFT2
Global AdS3 :
Focus on the closed curve with at on the time slice
The isometry is generated by
On the time slice,
We can construct a cTN which describes Global AdS3
via the Surface/State correspondence
(in this case, fixing the hyperbolic slice )
Continuous Tensor Network from AdS3/CFT2
Evaluate for Infinitesimally short interval
: local dilatation
Reproduce the state in cMERA at the radius
For vacuum
Below the UV-cutoff scale (momentum)
Some Other Arguments
- Sub-AdSScale Bulk Locality for Holographic CFTs
- Bulk Locally Excited States in cTN & Perfect Tensors
[Miyaji-Takayanagi-KW 16] 1609.04645v2 [hep-th]
Finer Resolution ~ Planck
Resolution ~ AdS radius
Sub-AdS Locality
Momentum :
Near the center of global AdS
For Symmetric Orbifold CFTs,
About sub-AdS Scale Bulk Locality
However, in holographic CFTs, we can get finer resolutions :
the long string sector vacuum gives
In generic CFTs, we can get AdS scale locality :
Ex:
Global AdS3
Folding
2πR0 2πR0 2πR0 2πR0 2πR0
Long string sector vacuum with
Single string sector vacuum with
with UV cutoff
Single string sector vacuum with
with UV cutoff
Interpretation from MERA networks
Bulk Locally Excited State Construction
favors Hyperbolic Slice in AdS
No back-reaction on the hyperbolic slice
Back-reaction should be treated
By using (global) conformal boundary state with time shift in CFT2,
We can construct the bulk locally excited state in AdS3
[Miyaji-Numasawa-Shiba-Takayanagi-KW 15]
(cross cap state)
[Ooguri-Nakayama 15, 16] [Goto-Miyaji-Takayanagi 16]
cf) [Verlinde 15, +Lewkowycz-Turiaci 16]
on other slices including the dS slice
cMERA for the bulk locally excited state
cf) HKLL
Actually, it has the same property as “Perfect Tensor”
[Pastawski-Yoshida-Harlow-Preskill 15]
Summary [Miyaji-Takayanagi-KW 16] 1609.04645v2 [hep-th]
Toward better understanding Tensor Networks and Ad/CFT,
We discussed their connections from 2 different viewpoints :
- Euclidean Path-Integral for ground state in CFT2 w/ a UV cutoff function
- cMERA (cTN) formulation for CFT2 from AdS3/CFT2
Both support a correspondence between
Hyperbolic time slice H2 in AdS3 cMERA
- Sub-AdS locality in (Folded) cMERA based on Symmetric Orbifold CFT
- Similarity of locally bulk excited states in cMERA with perfect tensor networks
We also found 2 more interesting properties of cMERA :
Really New Approach
- Explicit analysis of “optimized” Euclidean Path-Integral in specific lattice models
- Explicit relation between the spacetime metric and the property of cTN
especially the time-like component of the metric
- Higher dimensional case
Future Works
- Modular Hamiltonian & Entanglement Wedge from “optimized” Euclidean Path-Integral
Thank you !!
Back-up Slides
Efficiency of Tensor Network Descriptions
Generic States :
Exponential …
Polynomial !!
Efficient !!
# of Parameters
MPS : Tensors
MERA :
Tensors Polynomial !!
Efficient !!
Sites
MERA Diagrams for Two Intervals [Connected Case]
Mutual Information
measures entanglement between andand
measures entanglement between
Example: cMERA for a (d+1) dim. Free Scalar Theory
Ground state
cMERA:
Hamiltonian: H 1 dk d [ (k) (k) (k 2 m2 )(k)(k)].
2
: ak 0 0.0
SA 0.i.e. 0x
(x),
M(x)ax 0,IR state :
ax
x
M
i
(u)k / M a adk (h.c.),
(x) is a cut off function : (x) (1- | x |).where
i
2K (u)
k k
d
, (for m 0, (u) 1/ 2.)m2 / M 2
(s) 1
2e2u
2ue
Tadashi’s slide
Efficient Manipulation for Reduced Density Matrix
