Holographic Geometries from Tensor Network States

Transcription

1 Holographic Geometries from Tensor Network States J. Molina-Vilaplana 1 1 Universidad Politécnica de Cartagena Perspectives on Quantum Many-Body Entanglement, Mainz, Sep 2013

2 1 Introduction & Motivation The AdS/CFT correspondence The connection between AdS/CFT and entanglement renormalization tensor networks (MERA) 2 Contributions and Results Hybrid MERA-MPS networks Holographic geometries for hybrid MERA-MPS networks Geometric Computation of Entanglement Entropy & Correlators Results

3 The AdS/CFT correspondence Outline 1 Introduction & Motivation The AdS/CFT correspondence The connection between AdS/CFT and entanglement renormalization tensor networks (MERA) 2 Contributions and Results Hybrid MERA-MPS networks Holographic geometries for hybrid MERA-MPS networks Geometric Computation of Entanglement Entropy & Correlators Results

4 The AdS/CFT correspondence The AdS/CFT is a concrete example of the holographic principle, which suggests that all the information about the interior of some region is actually contained on the boundary of that region. The AdS/CFT, connects the physics of a weakly coupled string theory living in an Anti de Sitter space with the physics of a strongly coupled QFT living on the boundary of AdS.

5 The AdS/CFT correspondence The AdS/CFT correspondence [Maldacena 98]: Z SUGRA [AdS D+1, J (x, z)] = DJ (x, z) e S SUGRA[J (x,z)] Z CFT [ AdS D+1, J(x)] = Dφ(x) e [S 0[φ(x)]+ J(x) O(x) dx] AdS/CFT correspondence Z SUGRA [AdS D+1, J (x, z)] exp ( J(x) O(x) dx ) CFT J (x, 0) = J(x) z is the extra dimension of the gravity theory in AdS, Oare operators of the CFT on the boundary of AdS and J (x, z) are SUGRA fields such that J(x) J (x, z) z 0 are the sources for correlation functions of the CFT.

6 The AdS/CFT correspondence AdS/CFT provides models to study non perturbative effects in QFT (confinement, quantum phase transitions...) However, still there is not a first principle derivation of the duality. It is widely accepted that the AdS/CFT is at heart a geometric formulation of the renormalization group (RG), such that the renormalization scale becomes the extra dimension z. Curvature of the holographic direction contains the RG flow information.

7 Outline 1 Introduction & Motivation The AdS/CFT correspondence The connection between AdS/CFT and entanglement renormalization tensor networks (MERA) 2 Contributions and Results Hybrid MERA-MPS networks Holographic geometries for hybrid MERA-MPS networks Geometric Computation of Entanglement Entropy & Correlators Results

8 Tensor Network States (TNS) efficiently describe the low energy physics of strongly correlated QMBS. TNS techniques drawed from RG methods + knowledge of the entanglement structure in the ground state of QMBS TNS may be classified into two categories according to the geometry of the underlying networks:

9 Tensor Network States (TNS) efficiently describe the low energy physics of strongly correlated QMBS. TNS techniques drawed from RG methods + knowledge of the entanglement structure in the ground state of QMBS TNS may be classified into two categories according to the geometry of the underlying networks:

10 Tensor Network States (TNS) efficiently describe the low energy physics of strongly correlated QMBS. TNS techniques drawed from RG methods + knowledge of the entanglement structure in the ground state of QMBS TNS may be classified into two categories according to the geometry of the underlying networks:

11 Physical Geometry Networks The network mimics the physical geometry of the system, as specified by the pattern of interactions in the Hamiltonian Ex. MPS (1D), PEPS (2D) Holographic Networks Tensors are connected so as to parametrize the physics at different length scales relevant to describe the QMBS wave function The network spans an additional dimension related with the RG scale Ex: MERA, TTN

12 The MPS anstaz is: Ψ MPS = d {s j }=1 T s 1,s 2,, s N s 1, s 2,, s N T s1,s 2,, s N = Tr [A s1 A s2 A sn ]

13 Expectation values with MPS ( ) E Oj = d s j s s j =1 j O j s j A sj A s j Ψ Θ Ψ = Tr [E O1 E O2 E ON ]

14 Correlation Functions in MPS C [l+1] = Ψ O(s i ) O(s j ) Ψ = Tr [ ] E [s 1] 1 E [s i ] O E[s i ] O E[s N ] 1 C [l+1] = c ν (λ ν ) l ν 2 where λ ν 2 are the eigenvalues of E 1 for which it holds that λ ν 2 < 1 Correlation functions are a superposition of exponentially decaying function with decay lengths given by ξ ν 1/ log λ ν

15 Entanglement Entropy in MPS S A 2 log W where W is the bond dimension

16 The MERA ansatz [Vidal07]

17 The MERA ansatz [Vidal07] The MERA representation of implements an effcient real space RG procedure through a layered tensor network labelled by τ Each layer of MERA defines a RG transformation: prior to the renormalization of a block of 2 sites at layer τ into a single site by means of a Λ τ tensor, SRE between the sites is removed by means of the disentangler χ τ Disentanglers χ / χ χ = I and isometries Λ/ Λ Λ = I For 1D systems, the MERA computation of an observable Θ requires the contraction of a 2D tensor network

18 Correlation Functions in MERA The total contraction of the tensor network proceeds by mapping an effective observable Θ in a sequential way Θ τ Θ τ+1 = S[Θ τ ] Ψ Θ Ψ = C Θ h C h = log 2 N S( ) = µ α φ α Tr(φ α, ) α = log µ α

19 Correlation Functions in MERA φ α (x) φ β (y) = Ψ φ α (x) φ β (y) Ψ = δ αβ = C S τ (φ α (x)) S τ (φ β (y)) C = x y 2 α

20 Entanglement Entropy in MERA S A Ω hol A log W = Length(γ A) log W S A = k log L

21 The connection between AdS/CFT and entanglement renormalization tensor networks (MERA) Outline 1 Introduction & Motivation The AdS/CFT correspondence The connection between AdS/CFT and entanglement renormalization tensor networks (MERA) 2 Contributions and Results Hybrid MERA-MPS networks Holographic geometries for hybrid MERA-MPS networks Geometric Computation of Entanglement Entropy & Correlators Results

22 The connection between AdS/CFT and entanglement renormalization tensor networks (MERA) The AdS/MERA duality [Swingle 09] The local RG MERA circuit happens to be a realization of the AdS/CFT From the entanglement structure of a quantum critical QMBS is possible to define a discrete higher dimensional geometry The discrete geometry emerging at the critical point is a discrete version of AdS

23 The connection between AdS/CFT and entanglement renormalization tensor networks (MERA) The AdS/MERA duality The Big Questions are: Are some systems of strongly interacting qubits, secretly theories of quantum gravity in an AdS emergent spacetime? Is this intimately related with the structure of entangle ment in QMBS? What is the role of large N / strong coupling in AdS/MERA? The AdS/MERA is established by analyzing the computation of Entanglement Entropy with MERA and in the AdS/CFT

24 The connection between AdS/CFT and entanglement renormalization tensor networks (MERA) The AdS/MERA duality Entanglement Entropy in the AdS/CFT: The Ryu-Takanayagi conjecture S A = 1 4G (d+2) N Area(γ A )

25 The connection between AdS/CFT and entanglement renormalization tensor networks (MERA) The AdS/MERA duality Ryu-Takanayagi: Entanglement AdS geometry TNS : Entanglement Tensor properties & connectivity Can through AdS/MERA duality? Tensor properties & connectivity AdS geometry

26 The connection between AdS/CFT and entanglement renormalization tensor networks (MERA) Depth of Entanglement in MERA and extra dimensions: Entanglement as the fabric of spacetime

27 The connection between AdS/CFT and entanglement renormalization tensor networks (MERA)

28 Hybrid MERA-MPS networks Outline 1 Introduction & Motivation The AdS/CFT correspondence The connection between AdS/CFT and entanglement renormalization tensor networks (MERA) 2 Contributions and Results Hybrid MERA-MPS networks Holographic geometries for hybrid MERA-MPS networks Geometric Computation of Entanglement Entropy & Correlators Results

29 Hybrid MERA-MPS networks We consider the ground state Ψ of gapped 1D-Hamiltonian H. The correlations decay exponentially for distances l ξ, while typically keep power-law decaying for distances l ξ. We set ξ = 2 z 0

30 Hybrid MERA-MPS networks A suitable TNS ansatz to reproduce these features is a tensor network state with an MPS at the top of a finite number τ 0 log ξ of MERA layers. The τ 0 layers correspond to those in the scale invariant MERA describing a neighbouring critical point. The top MPS describes the LRE in the gapped phase Hybrid MERA-MPS

31 Hybrid MERA-MPS networks Entanglement Entropy in the hybrid TNS S TNS A = S MERA A + S MPS A log 2 z log W

32 Hybrid MERA-MPS networks Correlators in the hybrid TNS C TNS (l) = Ψ Φ( l/2) Φ( l/2) Ψ = ξ 2 C MPS Φ( l/2ξ) Φ(l/2ξ) C MPS Effective correlation length = ξ 2 c ν (λ ν ) l/ξ ν 2 ξ TNS = 1 log λ Γ = ξ log λ 2 = 2 z 0 log λ 2

33 Holographic geometries for hybrid MERA-MPS networks Outline 1 Introduction & Motivation The AdS/CFT correspondence The connection between AdS/CFT and entanglement renormalization tensor networks (MERA) 2 Contributions and Results Hybrid MERA-MPS networks Holographic geometries for hybrid MERA-MPS networks Geometric Computation of Entanglement Entropy & Correlators Results

34 Holographic geometries for hybrid MERA-MPS networks The geometric ansatz for the hybrid tensor network (JHEP 05 (2013) 24, arxiv: ), corresponds to an asymptotically AdS 3 spacetime, with a capping region at the IR region located at z 0 The geometry remains approximately AdS for values of the radial coordinate z z 0

35 Holographic geometries for hybrid MERA-MPS networks AdS ansatz metric ds 2 t=0 = dz2 A(z) 2 + B(z)2 dx 2 with, A(z) 2 = z2 f (z), L 2 AdS f (z) = 1 + Q ( z z 0 B(z) 2 = L 2 f (z) AdS z 2 ) ( ) ( ) z z 2 log z 0 z 0 with f (z) 0 and with no singularities for 0 Q 2. The metric asymptotes to AdS when z 0 as f (z) 1

36 Geometric Computation of Entanglement Entropy & Correlators Outline 1 Introduction & Motivation The AdS/CFT correspondence The connection between AdS/CFT and entanglement renormalization tensor networks (MERA) 2 Contributions and Results Hybrid MERA-MPS networks Holographic geometries for hybrid MERA-MPS networks Geometric Computation of Entanglement Entropy & Correlators Results

37 Geometric Computation of Entanglement Entropy & Correlators We apply the RT prescription = compute minimal length curves for E.E & correlators Length(γ) = 2L AdS ɛ z max z max dz z z 2 maxf (z) z 2 f (z max ) S A = 1 4 G (3) Length(γ) N C holog (l) = Γ exp [ m Length(Γ)] exp [ m Length(γ)]

38 Results Outline 1 Introduction & Motivation The AdS/CFT correspondence The connection between AdS/CFT and entanglement renormalization tensor networks (MERA) 2 Contributions and Results Hybrid MERA-MPS networks Holographic geometries for hybrid MERA-MPS networks Geometric Computation of Entanglement Entropy & Correlators Results

39 Results Geodesic length Length(γ) = log l max (δ) ɛ l max (δ) = 4 z 0 δ δ = 2 Q Entanglement Entropy & Two point Correlators S A = S UV A + SIR A (δ) SUV A log 2 z 0 ɛ S IR A (δ) log 2 δ [ C holog (l) = ɛ 2 exp 2 l ] max (δ) = l max (δ) 2 ɛ ξ holo = l max (δ)

40 Results Comparing EE in TNS and in the ansatz geometry W MPS 2 δ and connecting the effective correlation lengths of the TNS and the geometry ξ holo = κ ξ TNS λ 2 = exp ( κ ) 2 δ

41 New insights on the connection between MERA states and holography have been provided. This proposal could be extendable to higher dimensions. e.g with 2D MERA and PEPS, TNS of LW string-net states. Useful to study topological phases. Holographic duals : ABJM models and /or D3-D7 brane constructions? Entanglement and TNS as new tools for the study of gauge/gravity dualities. Outlook Role of large N in AdS/TNS Time evolution and dynamics e.g quenches

42 Appendix For Further Reading J.M Maldacena. The Large N Limit of Superconformal Field Theories and Supergravity. Adv. Theor. Math. Phys. 2, [hep-th/ ] B. Swingle Entanglement Renormalization and Holography. Phys. Rev. D 86, (2012) , [arxiv: ] G. Vidal Entanglement Renormalization. Phys. Rev. Lett. 99, (2007) [cond-mat/ [cond-mat.str-el]]