From Path Integral to Tensor Networks for AdS/CFT

Transcription

1 Osaka U 2016/11/15 From Path Integral to Tensor Networks for AdS/CFT Kento Watanabe (Center for Gravitational Physics, YITP, Kyoto U) v2 [hep-th] w/ Tadashi Takayanagi (YITP + Kavli IPMU) Masamichi Miyaji (YITP)

2 Toward Better Understanding about Tensor Networks as a Toy Model for AdS/CFT It From Qubit

3 Holography, Spacetime & Entanglement Holographic Principle or AdS/CFT argues: [t Hooft 93,Susskind 94] [Maldacena 99] Gravity(String) on Futhermore, it suggests : Quantum Many-body System (Gauge) on Symmetry, Partition Function, States, Operators Geometry(Spacetime) Entanglement [Ryu-Takayanagi 06] [van Raamsdonk 09,10] [Maldacena-Susskind 13] A Geometrization of Quamtum States

4 AdS/MERA ( Tensor Networks ) Correspondence MERA: An efficient variational ansatz for CFT ground states [Vidal 05] MERA (Tensor Networks) Quantum States Another Geometrization of Quamtum States If naively combine these 2 correspondences. [Swingle 09] Geometry(Spacetime) MERA (Tensor Networks) Emergent Hyperbolic Space?? (= a time slice of AdS) As you (may) have seen in Beni Yoshida s lectures, Bulk Operator Reconstruction & Quantum Error Correction BH & Scrambling, Complexity,

5 Surface/State Correspondence [Miyaji-Takayanagi 15] If we assume AdS/MERA(TN), Co-dim 2 Convex Space-like Surfaces Closed & Homologically Trivial Open or Homologically Non-Trivial Quantum States Pure Mixed Generalized Holography even for spacetimes without boundaries (e.g. Flat, ds, )

6 Some Issues of AdS/MERA [see e.g. Beny 11, Bao et.al. 15, Czech et.al. 15] (a) RG causal structure in MERA? ds instead of AdS? ds slice? Or hyperbolic slice? Kinematical Space Integral AdS trf. ds (b) (Special) Conformal invariance is not clear? Lattice artifact? (c) Why the EE bound is saturated? (d) Locality of the bulk AdS? MERA is local only at AdS radius scale (not Planck scale) (Perfect TNs, Random TNs resolve some issues. But they do not give CFT vacuum )

7 Contents [Miyaji-Takayanagi-KW 16] v2 [hep-th] Some More about Tensor Networks (MERA) AdS/cMERA (ctn) from 2 Different Viewpoints - A Time Slice of AdS 3 from an Optimization of Euclidean Path-Integral - ctn formulation in CFT 2 & Hyperbolic Slice of AdS 3 in CFT 2 Some Other Arguments - Sub-AdS Scale Bulk Locality for Holographic CFTs - Bulk Locally Excited States in ctn & Perfect Tensors

8 Some More about Tensor Networks (MERA)

9 Tensor Network Descriptions of Quantum States Efficient variational ansatz for ground state wave functions : Tensor Networks with Optimization of the Energy A Geometrization of Quamtum States ( respect the correct Quantum Entanglement ) Ex: MPS (Matrix Product States) [DMRG: White 92,, Rommer-Ostlund 95, ] good for gapped systems

10 MERA Network [Vidal 05, 06] For CFT ground states, a good TN is. MERA (Multi-scale Entanglement Renormalization Ansatz) network IR (Less Entangled) - 2 kinds of Tensors : - Isometry Coarse-graining - (Dis-) Entangler Add (Remove) Entanglement UV (More Entangled) - Layer by layer RG transformation Entanglement Renormalization Lattice scale changes exponentially

11 MERA, Hyperbolic Space and Holographic EE Entanglement Renormalization steps (a time slice of ) : size of Hyperbolic space Holographic EE saturates this bound

12 Efficient Manipulation & Causal Cone Ex) Past Future cf ) [Beny 11], [Czech et.al. 15],. associates the causal structure with ( Kinematic Space of ) Tensors relevant to Causal Cone for

13 Efficient Manipulation for Reduced Density Matrix Links crossing the bdy of steps relate to (minimally) Inside of the causal cone for Outside of the causal cone for The tensors in this region only affect to Holographic EE saturates this bound Similarly, Correlation functions can be reproduced

14 MERA for Thermo-Field Double State [Matsueda-Ishihara-Hashizume 12] 2-sided AdS-Schwarzschild BH dual [Maldacena 01] BH Interior? Flat Tiling of Unitary Tensors [Hartman-Maldacena 13] The Perfect Tensor Model [Hosur-Qi-Roberts-Yoshida 15]

15 Tensor Network Renormalization Yields MERA [Evenbly-Vidal 14, 15] 1D Quantum system Euclidean path-integral 2D Classical system Statistical partition function The Tensor Network Description: UV bdy 1 step of Tensor Network Repeat the step Renormalization (Refining & Coarse-graining tensors) MERA (+ IR bdy effect)

16 Continuous MERA (cmera) Associated with the RG property of MERA, We can describe a continuum limit of MERA - State at scale (Dis-)Entangler [Haegeman-Osborne-Verschelde-Verstraete 11] Coarse-graining (Space-like rescaling) IR UV - IR state No real space entanglement for any Dilatation [Canonical Choice] (Relativistic rescaling) This is Identified with Conformal Boundary State [Miyaji-Ryu-Takayanagi-Wen 14]

17 [Miyaji-Takayanagi-KW 16] v2 [hep-th] AdS/cMERA (ctn) from 2 Different Viewpoints - A Time Slice of AdS 3 from an Optimization of Euclidean Path-Integral in CFT 2 Another efficient description of states (wave functions) - ctn formulation in CFT 2 & Hyperbolic Slice of AdS 3

18 Euclidean Path-Integral for Ground State CFT 2 on : UV cutoff Free Scalar EOM & regular at At fixed,only modes with contribute in the -integral

19 Scale Dependent Wave Function with Cutoff Function Introduce a length scale dependent wave function with a cutoff function for Rescaled Reproduce at The Effective Wave Function at the length scale under a real space RG flow Simply set by retaking the cutoff function

20 Improve Cutoff Function in Non-Local Way We can improve this procedure by taking into account contributions from high momentum modes in a non-local way Ex) Suppressing factor for Reproduce High momentum modes Non-local For Symmetric Orbifold CFTs, More Layers are expected: (Large-c, Holographic?) Local Non- Local Non- Local Physical Non- Physical (Dirichlet bdy state) (Previous: Neumann one)

21 Interpretation : Optimization of Euclidean Path-Integral We can interpret this procedure to construct the vacuum wave function with the cutoff function as an optimization of the Euclidean path-integral (cf: Tensor Network RG) Optimize -dependent cutoff scale

22 Interpretation : Optimization of Euclidean Path-Integral This interpretation associates with the Euclidean path-integral Hyperbolic space with the cutoff function Efficient # of sites decays exponentially Size of UV cutoff grows linearly in invariance of the vacuum state Hyperbolic space (a time slice of ) Area measured by the metric # of sites

23 Some Other Backgrounds Finte T bdy2 Mass Gapped Removed TFD state (2-sided BH) bdy1 Confined Primary State broken not straightforward Primary op Qualitatively, Fine-grained again near the operator For deconfined states ( ) Large # of d.o.f. needed even at ( similar to BH horizon) Time Dependent No Criteria

24 Choice of Cutoff Function = Choice of Slice in the Bulk Actually, in general, we have choices of the interporation between the UV & IR states e.g. the cutoff function parametrized by Analytic continued to Lorentian AdS 3, we have 2 types of slices Hyperbolic slice For the vacuum, They are same (invariant under different RG flows) But for the excited states. slice

25 [Miyaji-Takayanagi-KW 16] v2 [hep-th] AdS/cMERA (ctn) from 2 Different Viewpoints - A Time Slice of AdS 3 from an Optimization of Euclidean Path-Integral in CFT 2 - ctn formulation in CFT 2 & Hyperbolic Slice of AdS 3 How to identify?

26 General Formulation of cmera in CFT 2 In CFT 2, we can identify with the dilatation from the sym. (for the vacuum) Virasoro IR state should be space-like rescaling invariant: In CFT 2, : Conformal bdy state for the vacuum s.t. Consistent with : Conformal sym. for the networks Below the UV-cutoff scale (momentum) for CFT 2 Also fix (for the excited states, different choices)

27 Continuous Tensor Network (ctn) from AdS 3 /CFT 2 We can construct a ctn which describes Global AdS 3 via the Surface/State correspondence Focus on the closed curve with at on the time slice Global AdS 3 : The isometry is generated by On the time slice, (in this case, fixing the hyperbolic slice )

28 Continuous Tensor Network from AdS 3 /CFT 2 Evaluate for Infinitesimally short interval : local dilatation Reproduce the state in cmera at the radius Below the UV-cutoff scale (momentum) For vacuum

29 [Miyaji-Takayanagi-KW 16] v2 [hep-th] Some Other Arguments - Sub-AdS Scale Bulk Locality for Holographic CFTs - Bulk Locally Excited States in ctn & Perfect Tensors

30 About sub-ads Scale Bulk Locality In generic CFTs, we can get AdS scale locality : Global AdS 3 Momentum : Near the center of global AdS Resolution ~ AdS radius However, in holographic CFTs, we can get finer resolutions : Ex: For Symmetric Orbifold CFTs, the long string sector vacuum gives Finer Resolution ~ Planck Sub-AdS Locality

31 Interpretation from MERA networks Long string sector vacuum with Single string sector vacuum with with UV cutoff Single string sector vacuum with with UV cutoff Folding 2πR0 2πR0 2πR0 2πR0 2πR0

32 Bulk Locally Excited State Construction favors Hyperbolic Slice in AdS (cross cap state) cf) HKLL By using (global) conformal boundary state with time shift in CFT 2, [Miyaji-Numasawa-Shiba-Takayanagi-KW 15] We can construct the bulk locally excited state in AdS 3 [Ooguri-Nakayama 15, 16] [Goto-Miyaji-Takayanagi 16] cf) [Verlinde 15, +Lewkowycz-Turiaci 16] No back-reaction on the hyperbolic slice Back-reaction should be treated on other slices including the ds slice cmera for the bulk locally excited state Actually, it has the same property as Perfect Tensor [Pastawski-Yoshida-Harlow-Preskill 15]

33 Summary [Miyaji-Takayanagi-KW 16] v2 [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 CFT 2 w/ a UV cutoff function - cmera (ctn) formulation for CFT 2 from AdS 3 /CFT 2 Both support a correspondence between Really New Approach Hyperbolic time slice H 2 in AdS 3 cmera We also found 2 more interesting properties of cmera : - Sub-AdS locality in (Folded) cmera based on Symmetric Orbifold CFT - Similarity of locally bulk excited states in cmera with perfect tensor networks

34 Future Works - Higher dimensional case - Explicit analysis of optimized Euclidean Path-Integral in specific lattice models - Modular Hamiltonian & Entanglement Wedge from optimized Euclidean Path-Integral - Explicit relation between the spacetime metric and the property of ctn especially the time-like component of the metric Thank you!!

35 Back-up Slides

36 Efficiency of Tensor Network Descriptions Generic States : Sites # of Parameters Exponential MPS : Tensors Polynomial!! Efficient!! MERA : Tensors Polynomial!! Efficient!!

37 MERA Diagrams for Two Intervals [Connected Case] measures entanglement between and Mutual Information measures entanglement between and

38 Example: cmera for a (d+1) dim. Free Scalar Theory Tadashi s slide Hamiltonian: H 1 dk d [ (k) ( k) (k 2 m 2 ) (k) ( k)]. Ground state IR state : i.e. 0 cmera: x 0 a x 0, x 2 : a k 0 0. a x S A 0. M (x) i M (x), i d dk K(u) (u) k / M a k a k (h.c.), 2 where (x) is a cut off function : (x) (1- x ). (s) 1 2u e 2, (for m 0, (u) 1/ 2.) e 2u m 2 / M 2

39 Efficient Manipulation for Reduced Density Matrix