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A holographic correspondence from tensor network states Sukhwinder Singh 1Institute for Quantum Optics & Quantum Information, Austrian Academy of Sciences, Vienna, Austria∗ and 2Center for Engineered Quantum Systems, Department of Physics & Astronomy, Macquarie University, 2109 NSW, Australia We introduce a toy holographic correspondence based on the multi-scale entanglement renormalization ansatz (MERA) representation of ground states of local Hamiltonians. Given a MERA representation of the ground state of a local Hamiltonian acting on an one dimensional ‘boundary’ lattice, we lift it to a tensor network representation of a quantum state of a dual two dimensional ‘bulk’ hyperbolic lattice. The dual bulk degrees of freedom are associated with the bonds of the MERA, which describe the renormalization group flow of the ground state, and the bulk tensor 7 network is obtained by inserting tensors with open indices on the bonds of the MERA. We explore 1 properties of copy bulk states—particular bulk states that correspond to inserting the copy tensor 0 onthebondsoftheMERA.Weshowthatentanglementincopybulkstatesisorganizedaccording 2 to holographic screens, and that expectation values of certain extended operators in a copy bulk n state, dual to a critical ground state, are proportional to n-point correlators of the critical ground a state. We also present numerical results to illustrate e.g. that copy bulk states, dual to ground J states of several critical spin chains, have exponentially decaying correlations, and that the corre- 7 lation length generally decreases with increase in central charge for these models. Our toy model 1 illustrates a possible approach for deducing an emergent bulk description from the MERA, in light of the on-going dialogue between tensor networks and holography. ] l e - CONTENTS I. INTRODUCTION r t s . I. Introduction 1 Inrecentyears,renormalizationgroupideashavebeen t a playing an important role in the efficient classical sim- m II. Boundary state 2 ulation of quantum many-body systems at low energies. - By properly identifying and renormalizing the high en- d III. An ansatz for the dual bulk state 3 ergy degrees of freedom one can build an efficient repre- n A. Copy bulk states 4 sentation of the ground state of the system. A promi- o nent example is entanglement renormalization—a real c IV. Holographic screens 5 [ space renormalization group (RG) transformation on a V. A bulk/boundary dictionary 7 lattice, which removes local entanglement before coarse- 1 graining the system [1]. Entanglement renormalization v 8 VI. Bulk entanglement vs boundary central charge 8 forms the basis of the multi-scale entanglement renormalization ansatz (MERA): an efficient tensor network 7 7 VII. Summary and Outlook 9 representation of ground states of local Hamiltonians on 4 a lattice [2]. 0 References 10 The MERA representation of the ground state of a . 1 local Hamiltonian, acting on an one dimensional (1D) A. Causal cone structure of a copy lifted MERA 11 0 quantum lattice, consists of a two dimensional (2D) hy- 7 perbolic tensor network. The 2D tensor network also 1 B. Area law entanglement in the bulk 12 encodes the RG flow of the ground state, and the ex- : v tra dimension in the tensor network description corre- C. Constrained MERA representations 13 i sponds to length scale in the 1D system. The MERA X 1. Higher order singular value decomposition 13 has been successfully applied to simulate ground states r 2. A matrix form of the MERA 14 a of several quantum lattice systems, and seems particu- D. Proof of Eq. (4) for holographic screens 14 larly well suited for critical systems, which are described by conformal field theories (CFTs) in the continuum [3]. E. Proof of the bulk/boundary dictionary 15 Ontheotherhand,theAdS/CFTcorrespondencesug- 1. 2-point correlators 15 gests an intimate relationship between RG and emer- 2. 3-point correlators 18 gent gravity in anti-deSitter (AdS) spacetimes. The 3. Reyni entanglement entropy 18 AdS/CFT correspondence [4, 5]—a concrete realization of the holographic principle—is an equivalence between certain quantum gravity theories in d + 1-dimensional AdSspacetimesandCFTsthatliveonthed-dimensional ∗ [email protected] boundary of the AdS spacetime. The correspondence es- 2 sentially translates an RG description of the CFT to a the entangement and correlation properties of the dual bulk gravity theory in AdS geometry. In particular, the bulk state. In particular, here we do not attempt to de- extra dimension of the AdS geometry is identified with duce a semi-classical bulk geometry from the MERA (or length scale in the CFT. even from the dual bulk states), or assume any partic- Recently, it has been conjectured [6–9], first in Ref. 6, ular spacetime interpretation of the MERA’s hyperbolic thattheMERArealizescertainfeaturesoftheAdS/CFT geometry. correspondence. In particular, it has been proposed that The bulk ansatz is described by a tensor network that theMERAtensornetworkcanbeviewedasadiscretiza- is obtained by inserting tensors with open indices on tioneitherofaspatialsliceofanemergentAdSspacetime the bonds of the MERA. We explore a particular sub- [6]oroftheabstractspaceofgeodesicsofaspatialsliceof set of bulk states—copy bulk states—which have inter- AdS[8]. However,whileitisunderstoodhowtheMERA esting properties. For example, entanglement in a copy encodes an 1D critical ground state, there is no conclu- bulkstateisorganizedaccordingto‘holographicscreens’. sive view on how it could also potentially encode a dual A holographic screen consists of sites of the bulk lattice description of an emergent 2D system. An even more thatarelocatedalonga1Dpathanchoredattwobound- basic problem is to identify the relevant bulk degrees of ary locations, such that the entanglement entropy of the freedom. sitesonthescreenisequal totheentanglemententropyof the 2D bulk region enclosed between the screen and the In this paper, we construct a toy holographic cor- boundary of the geometry. We also consider copy bulk respondence from the MERA representation of ground statesdualtoan1Dcriticalgroundstate, andshowthat states of 1D local Hamiltonians to illustrate a possible thebulkexpectationvaluesofcertainextended operators wayinwhichtheMERAcouldencodeadualdescription. areproportionalton-pointcorrelatorsofthe(boundary) Specifically, we describe how to ‘lift’ a MERA represen- critical ground state. This caricatures the prescription tation of an 1D ground state to a quantum state of a to calculate boundary correlators in AdS/CFT by evalu- 2D quantum lattice. The two states are seen to live on ating Witten diagrams [5, 14]. the boundary and in the bulk of a 2D manifold respec- Thepaperisorganizedasfollows. InSec.II,webriefly tively. We refer to this quantum state correspondence, reviewtheMERArepresentationofgroundstatesofinfi- mediated by a tensor network, as holographic because it nite1Dquantumlatticesystems. InSec.IIIweintroduce isguidedbyandimplementscertainbasicfeaturesofthe an ansatz for the bulk state dual to a given 1D ground AdS/CFTcorrespondence. Namely,(i)thedualbulkde- stateandalsocopybulkstates—particularstatesbelong- grees of freedom describe the RG flow of the 1D system, ing to the ansatz. In Sec. IV we describe how copy bulk (ii) the 1D ground state is dual to a 2D quantum state states exhibit holographic screens. In Sec. V we describe in the bulk [10], which may reduce to a classical descrip- asimpleprescriptiontoobtainn-pointcorrelatorsofscal- tion in some limit, (iii) the dual bulk state is completely ing operators and the block Reyni entanglement entropy determined from boundary state (since the promise of ofan1Dcriticalgroundstatefromadualcopybulkstate. holography is that the physical states of gravity simply InSec.VI,wepresentnumericalresultspertainingtothe correspond to the states of dual CFT), and (iv) if the bulk entanglement and correlations in copy bulk states, ground state has a global internal symmetry described dual to ground states of several exactly solvable critical by a group G, the dual bulk state has a local gauge sym- spin chains. We conclude with a summary and outlook metry G. In this paper, we introduce an ansatz for the in Sec. VII. The appendices contain proofs and other dualbulkstatethatexhibitsthefeatures(i)−(iii)listed technical details that are not covered in the main text above,whileweextendtheformalismtoincorporatefea- for simplicity of presentation. ture (iv) in a following paper [11]. We make a few remarks pertaining to the feature (ii). The local Hamiltonians that we consider here may be II. BOUNDARY STATE critical and described by CFTs with a small central charge. (In fact, in practice the MERA has been mostly applied to simulate ground states of CFTs that have a Consider an infinite 1D lattice L, each site of which is small central charge.) In the AdS/CFT correspondence, described by a χ-dimensional Hibert space V. Lattice L a small boundary central charge (e.g. of order 1) gen- is equipped with the action of a local, translational in- erally corresponds to quantum gravity in the bulk [12]. variant Hamiltonian Hˆ, which may be gapped or critical For example, Ref. 13 presents a holographic description (gapless). We are interested in the ground state |Ψbound(cid:105) ofthe1DquantumcriticalIsingmodel,whichhascentral of Hˆ. The superscript ‘bound’ appears in anticipation charge equal to 1. Indeed, there the authors match the that the ground state will play the role of the boundary 2 partition function of the Ising model to a dual quantum state in our holographic correspondence. In this paper, gravity partition function, obtained by summing over all werepresent|Ψbound(cid:105)bymeansofaninfiniteMERAten- bulkgeometries(gravitationalfields)thatarecompatible sor network, depicted in Fig. 1. The MERA represen- with the asymptotic constraints imposed by the bound- tation of the ground state of a given local Hamiltonian ary theory. This motivates us to derive a dual quan- can be obtained by means of e.g. the variational energy tum statefromtheMERA,andsubsequentlywefocuson minimization algorithm [15]. 3 theHamiltonianHˆ (anditsgroundstate|Ψbound(cid:105))isalso scale-invariant,andthattheMERArepresentationofthe ground state is composed of copies of tensors uˆ and wˆ. The MERA tensors uˆ and wˆ are constrained to be isometries satisfying: (cid:88) (cid:88) (uˆ)ij(uˆ†)kl =δiδj , (wˆ)i (wˆ†)jkl =δi, (1) kl i(cid:48)j(cid:48) i(cid:48) j(cid:48) jkl i(cid:48) i(cid:48) kl jkl where i,j,k,l,i(cid:48),j(cid:48) ∈{1,2,...,χ}. Consequently, the re- FIG.1. (Coloronline)GraphicalrepresentationoftheMERA duced density matrix of any site on the lattice does not tensornetworkrepresentationofaquantummany-bodystate dependonalltheMERAtensors,butonlyonasubsetof of an infinite lattice L (∼= L0). Arrows in the figure indi- them; this subset of tensors is called the causal cone of catethatthetensornetworkextendsinfinitelyintheupward the site. The number of tensors in the causal cone that vertical and both left,right horizontal directions. Each open are counted at any given length scale is bounded (less index of the tensor network labels an orthonormal basis on than or equal to 3). Furthemore, the reduced density a different site (blue squares) of L. The dotted horizontal matrix of multiple sites depends only on tensors belong- lines separate the tensor network into layers of tensors, and ing to the union of the respective one-site causal cones, coincidewithasequenceofincreasingcoarse-grainedlattices: L0 → L1 → L2···. The vertical direction of the tensor net- which merge at a sufficiently large length scale. Thanks to these properties, expectation values can be efficiently work corresponds to length scale, namely, after discarding the bottom layers the residual tensor network describes the computed from an infinite scale-invariant MERA tensor many-body state at a coarser length scale. network [2, 15]. The open indices of the tensor network are associated III. AN ANSATZ FOR THE DUAL BULK STATE with the sites of L, namely, each open index labels an orthornormal basis on a different site of L. On the other In this section, we construct a 2D ‘bulk’ description hand, the bond indices—indices that connect the tensors dual to the 1D ground state |Ψbound(cid:105). Our construction inthenetwork—carrytheentanglementandcorrelations is motivated by the basic features of the AdS/CFT cor- inthequantumstate. Theprobabilityamplitudesforthe respondence listed (i)-(iv) in the introduction. state |Ψbound(cid:105), in the basis labelled by the open indices, To begin with, what are the dual bulk degrees of free- are formally obtained by contracting all the tensors of dom? In a tensor network representation of a quantum the infinite tensor network, which involves summing the many-bodystate, thebondindicesofthetensornetwork bond indices. are treated differently from the open indices. While the The MERA representation also describes the RG flow openindicesofthetensornetworkareassociatedwiththe of the ground state. Each layer of MERA tensors, sepa- degreesoffreedomofthestate,thebondindicescarrythe rated by dotted lines in Fig. 1, implements a real space entanglementandcorrelationsinthestate. Herewepro- RG transformation—known as entanglement renormal- posetoconstructadualemergentdescriptionofatensor ization—that maps a lattice Lk with L (→∞) sites to a networkstatebyassociatingthebondindiceswithemer- coarse-grained lattice Lk+1 with L/3 sites. The MERA gent degrees of freedom, and thus, figuratively speak- tensors are chosen so that the RG transformation pre- ing, treating the bond indices on an equal footing as the servesthegroundsubspaceateachstep. SubsequentRG open indices. The bond indices of the MERA, in partic- steps generate a sequence of increasingly coarse-grained ular, are also associated with the renormalized sites. In lattices: L0 →L1 →L2··· where L0 ∼=L is the ultravi- the AdS/CFT correspondence, the RG flow of the CFT oletlattice. Therefore,theextradimensionofthetensor plays an instrumental role in the dual bulk description. network corresponds to length scale. In particular, the This futher motivates associating the dual bulk degrees residual tensor network obtained after discarding one or of freedom with the bonds of the MERA. more bottom layers of the MERA furnishes a represen- Let us embed the MERA tensor network, which rep- tation of the ground state on a coarse-grained lattice. resents |Ψbound(cid:105), in a 2D manifold with a boundary, such If, in addition to translation invariance, the ground thattheopenindicesandbondindicesofthetensornet- state is also scale-invariant—namely, it remains in- workappearattheboundaryandinthebulkoftheman- variant under the RG (entanglement renormalization) ifold respectively. Construct a 2D quantum lattice M transformations—then its MERA representation is com- on the manifold by locating a site—described by the χ- posedofcopiesofthesametwotensorsuândwˆ through- dimensional Hilbert space V—on every bond of the (em- out the tensor network. This leads to a very compact bedded)tensornetwork, seeFig.2(a). LatticeMissim- description of the infinite ground state, namely, the en- ply a collation of the degrees of freedom that describe tire state is completely specified by the two tensors uˆ the RG flow of the boundary state, and also inherits the and wˆ. In the rest of the paper, we will assume that hyperbolic geometry of the tensor network [16]. 4 Next, let us insert a three index tensor (cˆ)i , where (a) i(cid:48)o i,i(cid:48),o ∈ {1,2,...,χ}, on each bond of the MERA as 2D manifold z shown in Fig. 2(b), and use the open index o to label x an orthonormal basis on the site of M located on the bond. The new tensor network, which is composed of the ground state tensors uˆ and wˆ and copies of the bond tensor cˆ, encodes a quantum state |Ψbulk(cid:105) of M. Analo- gous to the MERA, the probability amplitudes of |Ψbulk(cid:105) areobtainedbycontractingtogetherallthetensorsofthe newtensornetwork. Thus,wehave‘lifted’theMERAde- Boundary (z=0) scriptionofan1Dgroundstate|Ψbound(cid:105)toa2Dquantum (b) state |Ψbulk(cid:105) belonging to the lattice M. We refer to the new tensor network as the lifted MERA. Since we have embeddedtheMERAina2Dmanifoldwithaboundary, the two states |Ψbound(cid:105) and |Ψbulk(cid:105) are seen to live at the boundary and in the bulk of the hyperbolic lattice M. (HereweidentifythesiteslocatedattheboundaryofM with the sites of lattice L. For example, the site of M locatedat(x,0)isidentifiedwiththesiteofLlocatedat x. Note that the bulk state |Ψbulk(cid:105) also has support on the boundary sites of M.) FIG. 2. (Color online) (a) The dual 2D bulk lattice M, con- States |Ψbound(cid:105) and |Ψbulk(cid:105) constitute our holographic structed by embedding the MERA in a 2D manifold with correspondence, mediated by the MERA tensor network a boundary (at z = 0), and locating a site (red square) of composed of tensors uˆ and wˆ. For a more general con- M on every bond of the MERA. The green solid paths are struction, one may also pre-process the MERA by con- graph geodesics, namely, geodesic paths along the edges of tracting and/or decomposing some of its tensors before thegraphunderlyingtheMERA.Alsoshownisapath(green lifting it to a 2D quantum state as described here, see dashes)intheambientmanifoldthatintersectsonlytheedges Appendix C. Here, we have inserted copies of the same oftheMERAgraph. (b)TheliftedMERA,ouransatzforthe tensor cˆ on all the bonds of the MERA to build a lo- holographicdualstate,obtainedbyinsertinga3-indextensor cally uniform bulk tensor network, which ensures that (cˆ)ii(cid:48)o oneverybondoftheMERA.Byusingeachopenindex of the tensor network (e.g. index o) to label an orthonormal the corresponding bulk state |Ψbulk(cid:105) is invariant under basis on a different site of M, the lifted MERA encodes a translations along some directions. quantum state of M. Since we have still not fixed the bond tensor cˆ, the lifted MERA describes a set of states, one for each choice of the bond tensor, on the lattice M. The lifted glement scaling’ and is commonly exhibited by ground MERA is our ansatz for the holographic dual of the 1D statesoflocal,low-dimensionalquantumlatticesystems. groundstate|Ψbound(cid:105). Requiringthatthedualbulkstate Incontrast, oneexpectsthesubsystementanglementen- must completely derive from the ground state, as in the tropy of generic states of M to scale as the subsystem’s AdS/CFTcorrespondence,weimposethattensorcˆmust area. eitherbeaconstantorafunctiononlyofthegroundstate tensors uˆ and wˆ. Additional guidelines from AdS/CFT may be used to further fix the properties of cˆ. For ex- A. Copy bulk states ample, if the boundary state |Ψbound(cid:105) has a global symmetry G, we require cˆ to transform under the action of Intheremainderofthepaper,weexplorepropertiesof thesymmetryinaparticularway,inordertoensurethat particular bulk states belonging to the ansatz described thecorrespondingbulkstatehasalocalsymmetryG [11] above. Specifically, we consider bulk states obtained by (thus implementing feature (iv) of the AdS/CFT corre- choosing cˆsimply as the copy tensor: spondence listed in the introduction). The bulk states described by a lifted MERA contain (cid:40) 1, if i=i(cid:48) =o, only a limited entanglement. Given a subsystem of the (cˆ)i = (2) bulk lattice M, we define its perimeter and area as the oi(cid:48) 0, otherwise. numberofsitesthatlieatthesubsystem’sboundaryand inside the subsystem respectively. For bulk states de- (This corresponds to fixing a basis on the bulk sites as scribedbyaliftedMERA,theentanglemententropyofa discussed later in this section.) Colloquially, cˆ copies a sufficientlylargesubsystemscalesatmostastheperime- bond index of the MERA to an open index of the lifted ter of the subsystem, as proved in Appendix B. Such an MERA,andisperhapsthesimplestchoicetoliftatensor entanglement scaling is ubiquitous in condensed matter networkrepresentation. Theboundarystate|Ψbound(cid:105)can physics where it is called ‘(boundary) area law entan- be recovered from a dual copy bulk state |Ψbulk(cid:105), simply 5 as troduced thus far, is to constrain the intrinsic bond freedom in MERA representations by demanding that the χ |Ψbound(cid:105)=((cid:79)Pˆ )|Ψbulk(cid:105) Pˆ =(cid:88)|j(cid:105)(cid:104)j|, (3) tensors fulfill additional constraints. k k For example, in Ref. 11 we describe how the fomalism k j=1 presentedinthispapercanbeextendedtoimplementthe where Pˆ projects the bulk site located on bond k to the holographictranslationofaboundaryon-siteglobalsym- k state |+(cid:105) ≡ (cid:80)χ |j(cid:105). Analogous to the MERA, a copy- metry to a bulk local gauge symmetry. This is achieved j=1 by partially constraining the bond freedom, which illus- liftedMERAisalsoendowedwithacausalconestructure tratesthatconstrainingthebondfreedommayindeedbe thatallowsforefficientcomputationofexpectationvalues useful (or even necessary) to implement certain features in the bulk, as described in Appendix A. of the AdS/CFT in our holographic correspondence. In Given a MERA representation of a quantum many- AppendixCweintroducetwomodifiedMERArepresen- body state, one can obtain another equivalent MERA tations that have a significantly constrained bond free- representation of the state by inserting a resolution of identityMˆ Mˆ−1onbondk,andmultiplyingthematrices dom,ascomparedtothestandardMERArepresentation k k which is reviewed in Sec. II. One of these constrained Mˆ and Mˆ−1 respectively with the two tensors that are k k MERArepresentationswasusedtoobtainthenumerical connectedbythebond. Thisbond freedom isanintrinsic results presented in Sec. VI. However, these constrained property of tensor network representations of quantum MERA representations are not directly motivated from many-body states. The two MERAs are an equivalent AdS/CFT. Nonetheless, they illustrate a possible gener- representation of the state since the expectation value alization of our bulk construction to control the number of any observable is equal in both representations. (Ob- ofdifferentcopybulkstatesdualtoagivengroundstate. taininganexpectationvaluefromatensornetworkstate It is also possible that the different copy bulk states, involves contracting all the bond indices of the tensor dual to a ground state, obtained here are related to one network,andMˆ ismultipliedwithMˆ−1 intheprocess.) k k another by an unidentified (emergent) bulk symmetry, Clearly,insertingthecopytensordefinedinEq.(2)se- and are thus equivalent holographic duals of the ground lectsoutaparticularMERArepresentationoftheground state. Or that the relevant dual bulk state is a certain state—the one whose tensors are expressed in the basis superpositionofthedifferentcopybulkstatesthatisde- in which the copy tensor has the components of Eq. (2). termined by some holography inspired bulk conditions. Considercopybulkstates|Ψ(cid:105)and|Ψ(cid:48)(cid:105)thatareobtained We leave further exploration of these issues for future by lifting MERA tensor networks T and T(cid:48) respectively. work. Here MERA T(cid:48) is obtained by transforming the tensors ofT bymeansofnon-diagonal unitarybondtransforma- tions(Mˆ ’s)asdescribedabove. Inthiscase,eventhough k IV. HOLOGRAPHIC SCREENS the two MERAs T and T(cid:48) describe the same quantum many-bodystate,thetwocorrespondingcopybulkstates Let us parameterize the sites of the bulk lattice M |Ψ(cid:105) and |Ψ(cid:48)(cid:105) are generally different e.g. they have dif- bycoordinates(x,z)where,intheboundarydescription, ferent entanglement. This is because the copy tensor cˆ x labels spatial translations and z corresponds to the ‘commutes’ with only diagonal matrices, namely, a con- lengthscale(theboundaryislocatedatz =0). Consider tractionofcˆwithadiagonalmatrixonanyindexisequal two points P and P at the boundary of the ambient to a contraction of cˆwith the same diagonal matrix on 1 2 manifold, in which the (lifted) MERA is embedded. P a different index. This implies that |Ψ(cid:105) and |Ψ(cid:48)(cid:105) are not 1 islocatedonthelinesegmentbetweenbulksites(x−1,0) relatedtoeachotherbyone-siteunitaryrotationsonthe and(x,0),andP islocatedonthelinesegmentbetween bulk lattice, and therefore they have different entangle- 2 bulksites(x(cid:48),0)and(x(cid:48)+1,0),forsomex(cid:54)=x(cid:48). Consider ment. Thus, ourbulkconstructiongenerallyliftsagiven a path P between points P and P that intersects only ground state to a set of different copy bulk states. 1 2 thecopytensorsoftheliftedMERA,asillustrated(green In such a scenario, how can one compare bulk states dashes)inFig.3. PathP dividesthebulklatticeMinto corresponding to different boundary states? For exam- three parts: ple, one could be interested in probing if the bulk entanglement depends on the boundary central charge, see 1. an ‘interior’ composed of bulk sites enclosed be- Sec. VI. A possible approach to compare bulk states tween the path and the boundary, and including corresponding to different boundary states is to average sites located at (x,0),(x+1,0),...,(x(cid:48),0), thebulkpropertiesoverallthedifferentcopybulkstates that are dual to the ground state. Another possibility is 2. bulksitesassociatedwiththecopytensorsthatare to compare the statistics of the bulk properties by ran- intersected by P, and domlysamplingfromthedifferentcopybulkstates. (See 3. an ‘exterior’ composed of all remaining bulk sites. Ref.11forsomestatisticsofthebulkproperties). Onthe other hand, one may take the view that a given ground Letusdecoratetheindicesoftensorsuândwˆwitharrows state must lift to only one dual bulk state to begin with. as depicted (red) in Fig. 3(a). If the arrows on all the A possible way to achieve this, within the framework in- bond indices located in the immediate exterior of the 6 path are incoming to the interior then P can be viewed (a) as a holographic screen. It can be shown that P satisfies (see Appendix D) holographic screen ρˆscreen =Rˆ†(ρînterior)Rˆ, (4) EXTERIOR where ρˆscreen is the reduced density matrix of the bulk sitesintersectedbythescreen,ρînterior isthereducedden- INTERIOR sity matrix of the bulk sites located in the interior, and Rˆ is an isometry, namely, RˆRˆ† = Iˆ. Equation 4 implies thattheexpectationvalueofanobservableoînterior acting P1 BOUNDARY P2 in the interior is equal to the expectation value of the observable oˆscreen = Rˆ†(oînterior)Rˆ acting on the screen, (b) since Tr(ρˆscreenoˆscreen)=Tr(Rˆ†ρînteriorRˆRˆ†oînteriorRˆ), holographic screen (5) =Tr(ρînterioroînterior). EXTERIOR Here we used Eq. (4), the fact that RˆRˆ† = Iˆ, and the cyclic property of trace: Tr(AB)=Tr(BA). INTERIOR Thus, the expectation value of any observable supported in the 2D interior region can be calculated from BOUNDARY the 1D screen, which encloses the interior. Furthe- more, the expectation value of a local interior observ- (c) able equates to the expectation value of a local screen observable. Namely, an observable supported on a small number of interior sites maps to an observable that is alsosupportedonasmallnumberofscreensites. Thisis because Rˆ is a composition of isometries, each of which EXTERIOR act on a small number of sites. (In contrast, the expectation value of a local screen observable ωˆscreen generally equatestotheexpectationvalueofaninteriorobservable INTERIOR ωînterior =(Rˆ†Rˆ)ωˆscreen(Rˆ†Rˆ) that is smeared over all the interior degrees of freedom.) From Eq. (4) it also follows BOUNDARY that ρˆscreen and ρînterior have the same eigenvalues, which in turn implies e.g. that the entanglement entropy of all FIG.3. (Coloronline)(a,b)Examplesofholographicscreens the interior sites is equal to the entanglement entropy of inacopy-liftedMERA.Inthebox: adecorationoftheindices all the screen sites, namely, oftheMERAtensorswith(red)arrows. Holographicscreens are paths on the ambient manifold that extend between two −Tr(ρînteriorlog2 ρînterior)=−Tr(ρˆscreenlog2 ρˆscreen). (6) boundary locations (e.g. P1,P2), intersect only copy tensors, and the red arrows in the immediate exterior of the path are Thisentanglementfeatureiscompatible,butextendsbe- allincomingtotheinterior. Thereduceddensitymatrixofthe yond, the entanglement scaling proved in Appendix B. 2D interior (highlighted in yellow) transforms to the reduced Let us define the length of path P as the number of density matrix of the bulk sites located on the (1D) screen under conjugation by an isometry, Eq. (4). (a) A geodesic copy tensors that it intersects. It can be easily veri- holographic screen (green dashes). (b) A non-geodesic holo- fied that if P is a geodesic between the points P and 1 graphic screen (green dashes). (c) Example of a path (green P , namely, P intersects the smallest possible number of 2 dashes)thatdoesnot furnishaholographicscreen,sincesome copy tensors then it is necessarily a holographic screen of the red arrows are outgoing from the interior. [Fig.3(a)]. Thisisbecauseageodesicpathalwaysfulfills the arrow criterion stated previously and therefore satisfies Eq. (4). On the other hand, Fig. 3(b) illustrates a non-geodesic holographic screen. The presence of holographic screens described here is a generic property of copy bulk states, and seems to im- itate the holographic screens—a feature of (quantum) spacetime—that often appear in quantum gravity. We [9]. Thus,theprojectionfromabulkregiontoanenclos- also remark that the MERA tensors located in the in- ing holographic screen may be viewed as the bulk dual terior of a holographic screen considered here compose of the action of a local conformal transformations on the a local conformal transformation on the boundary state boundary state. 7 V. A BULK/BOUNDARY DICTIONARY (a) (b) string z operator x In this section, we describe a simple prescription to x obtaincorrelatorsandblockReynientanglemententropy of a critical ground state |Ψbound(cid:105) from a dual copy bulk (c) branched string (d) string operator state |Ψbulk(cid:105). This is schematically depicted in Fig. 4. operator For simpler presentation, here we only list the formulae B´ for calculating these boundary properties from the bulk, B while their derivation is presented in Appendix E. For the purpose of this section, we denote by x,x(cid:48),x(cid:48)(cid:48) FIG. 4. (Color online) A dictionary that translates be- the locations of special sites of the boundary lattice L, tween properties of a critical ground state |Ψbound(cid:105) and a dual copy bulk state |Ψbulk(cid:105). Here x,x(cid:48),x(cid:48)(cid:48) locate special namely, x locates a site associated with an open index sites on the lattice L, see main text. (a) A schematic de- of the MERA at the base of an arbitrarily long verti- piction of the MERA (represents |Ψbound(cid:105)) and a copy-lifted cal graph geodesic [see Fig. 2(a)], |x − x(cid:48)(cid:48)| = 3q, and MERA(represents|Ψbulk(cid:105)). (b)The2-pointboundarycorre- |x(cid:48)−x(cid:48)| = 3q(cid:48) where q,q(cid:48) are positive integers. Here by lator (cid:104)oˆ (x)oˆ (x(cid:48))(cid:105) is proportional to the bulk expecta- ‘graph geodesic’ we mean the shortest connected path tion valαue ofβa strinbgounodperator, Eq. (7). (c) The 3-point cor- between two locations along the MERA graph itself, in relator (cid:104)oˆ (x)oˆ (x(cid:48))oˆ (x(cid:48)(cid:48))(cid:105) is proportional to the bulk α β γ bound contrast with geodesics on the ambient manifold outside expectation value of a branched string operator, Eq. (8). (d) the lattice that were considered in the previous section. TheReynientropyofaboundaryregionB⊂Lisequaltothe Considertheone-sitescalingsuperoperator Sôbtained Reyni entropy (plus a constant) of a bulk region B(cid:48) ⊂M as calculatedfromtheprojectedbulkstate|Ωbulk(cid:105),Eq.(9). B(cid:48)is from the MERA tensor wˆ as comprisedofsitesenclosedbetweenthegeodesicthatextends (Sî)k(cid:48) =(cid:88)wî (wˆ†)jk(cid:48)l, between x and x(cid:48) and the boundary, including the boundary i(cid:48) k jkl i(cid:48) sites of M located at (x+1,0),(x+2,0),...,(x(cid:48)−1,0). jl and let oˆ and λ denote an eigenoperator and the corresponding eigenvalue of Sˆ, namely, Sˆ(oˆ) = λoˆ. Operator the bulk expectation value of the n scaling operators oˆ is identified with a scaling operator of the underlying tensor product with multi-branched string projection CFT with scaling dimension ∆ = −log λ [3]. Also, let operator with n−2 branch points. (Here x locates a 3 1 Gx,x(cid:48) denote the set of bulk sites that are located along site of L that is associated with an open index of the the graph geodesic extending between the bulk sites at MERA at the base of an arbitrarily long vertical graph (x,0) and (x(cid:48),0). geodesic, and |xi −xj| = 3qij for all i,j where qij is a The 2-point boundary correlator of scaling operators positive integer.) oˆα and oˆβ, applied at site locations x and x(cid:48), can be Thus,n-point correlatorsofan1Dcriticalgroundstate obtained from the bulk state as translate to the expectation value of extended operators in a dual 2D copy bulk state. This identification, (cid:104)oˆ (x)oˆ (x(cid:48))(cid:105) =f(uˆ,wˆ,oˆ ,oˆ )× α β bound α β (7) though extremely simple, appears to caricature the pre- (cid:104)Kˆx,x(cid:48)oˆα(x,0)oˆβ(x(cid:48),0)(cid:105) , scription of pertubatively calculating boundary correla- bulk tors in AdS/CFT by evaluating Witten diagrams [5, 14]. where function f(uˆ,wˆ,oˆα,oˆβ) is defined in Appendix E, (The latter is a type of Feyman diagram that involves andKˆ =(cid:78) Pˆ isastringoperator thatprojectseach integrating certain bulk degrees of freedom and propa- x,x(cid:48) i i bulk site i∈G to the state |+(cid:105) [Eq. (3)]. That is, the gating the boundary operators into the bulk. Note that x,x(cid:48) boundary correlator can be obtained by calculating the applyingtheprojectorPˆ, Eq.(3), onabulksiteineffect same correlator from the bulk state but after projecting sums the basis vectors at the site.) all the bulk sites in Gx,x(cid:48) to the state |+(cid:105). Note that the extended operators that appear in Analogously,the3-pointboundarycorrelator ofscaling Eq. (7) and Eq. (8) act on a finite number of bulk operators oˆα,oˆβ and oˆγ can be obtained from the bulk sites. On the other hand, any of the boundary corre- state as latorsconsidered abovecan beobtained exactly [namely, without the multiplicative factors f(uˆ,wˆ,oˆ ,oˆ ) and (cid:104)oˆ (x)oˆ (x(cid:48))oˆ (x(cid:48)(cid:48))(cid:105) =g(uˆ,wˆ,oˆ ,oˆ ,oˆ )× α β α β γ bound α β γ (8) g(uˆ,wˆ,oˆα,oˆβ,oˆγ)] by computing the same correlator in (cid:104)Tˆx,x(cid:48),x(cid:48)(cid:48)oˆα(x,0)oˆβ(x(cid:48),0)oˆγ(x(cid:48)(cid:48),0)(cid:105) , the bulk state but after projecting an infinite number bulk of bulk sites—those located in the joint causal cone of where function g(uˆ,wˆ,oˆα,oˆβ,oˆγ) is defined in Appendix thesitesonwhichthescalingoperatorsact—tothestate E, and Tˆ = (cid:78) Pˆ is a branched string operator |+(cid:105). (This can be seen by matching the tensor network x,x(cid:48),x(cid:48)(cid:48) i i that projects all bulk sites i ∈ {G ∪ G } to the contractionequatingtotheboundarycorrelatorwiththe x,x(cid:48) x(cid:48),x(cid:48)(cid:48) state |+(cid:105). tensor network contraction equating to the same corre- More generally, the n-point correlator lator in the bulk after projecting all the bulk sites in the (cid:104)oˆ (x ),oˆ (x ),··· ,oˆ (x )(cid:105) is proportional to joint causal cone.) The point here is that in order to ob- α 1 β 2 ν n bound 8 tain the critical exponents, part of the underlying CFT data, from the bulk it suffices to consider the bulk ex- 100 Ising pectation value of extended operators that act only on a BC Potts finite number of bulk sites. 10-2 XXZ, =0 The Reyni entanglement entropy of a block B of sites XXZ, =0.3 n XXZ, =0.4 in a quantum many-body state |Ψ(cid:105) is: o Rα(B)=−log2 TrρˆαB, α=2,3,..., itamrofn 10-4 XXXXXXXXZZZZ,,,, ====0001...589 where ρˆB is the reduced density matrix of the block B, i lau 10-6 obtainedbytracingoutallsitesbelongingtothecomple- tu M ment of B in state |Ψ(cid:105). The boundary Reyni entanglement entropy Rbound(B) of a block B ⊂L of sites located 10-8 α at x,x+1,...,x(cid:48) can be also be obtained from the bulk in a simple way in the limit of large block size |x−x(cid:48)|. 10-10 We have 0 2 4 6 8 10 Distance between the two sites Rbound(B)≈Rbulk(B(cid:48))+h(uˆ,wˆ), large |x−x(cid:48)|. (9) α α FIG.5. (Coloronline)ThemutualinformationIbulk[Eq.(11)] z,z(cid:48) in copy bulk states, dual to the ground state of each of the Here B(cid:48) ⊂ M [see Fig. 4(d)] is the set of bulk sites en- critical spin chains listed in Eq. (10), between two bulk sites closedbetweenthegraphgeodesicextendingbetweenlo- located at (x,z) and (x,z(cid:48)) respectively (separated by dis- cations(x,0)and(x(cid:48),0)andtheboundary,andincluding tance |z−z(cid:48)|) as illustrated in Fig. 11(c). thebulksiteslocatedat(x+1,0),(x+2,0),...,(x(cid:48)−1,0). Rbulk(B(cid:48)) is the Reyni entanglement entropy of B(cid:48) calcu- α lated from the projected bulk state |Ωbulk(cid:105)=Kˆ |Ψbulk(cid:105), we probe for any potential dependence of the entangle- x,x(cid:48) and h(uˆ,wˆ) is a known function of the MERA tensors ment/correlations in copy bulk states, dual to critical defined in Appendix E. Note that Eq. (9) differs from boundary states, on the central charge. To this end, we the equality Eq. (6) between the entanglement entropy considered the ground states of the following 1D critical ofsiteslocatedonandinsideaholographicscreenrespec- spin models with different central charges: tively. While Eq. (9) relates a boundary Reyni entangle- Hîsing =(cid:88)σîσî+1+σî, ment entropy to a bulk Reyni entanglement entropy and z z x holds approximately in the limit of large |x−x(cid:48)|, Eq. (6) i is an exact property of a copy bulk state. Hˆbc =(cid:88)−XîXî+1+α(Xî)2+β(Zî)2, i (10) Hˆpotts =−(cid:88)Pî(PˆT)i+1+(PˆT)iPî+1+Mî, VI. BULK ENTANGLEMENT VS BOUNDARY i CENTRAL CHARGE Hˆxxz =(cid:88)σîσî+1+σîσî+1+∆σîσî+1, x x y y z z i CertainbulkfeaturesintheAdS/CFTcorrespondence depend on the central charge of the CFT. For example, where i labels sites of an 1D lattice on which the Hamil- correlations due to quantum fluctuations in the bulk are tonianacts,σˆ ,σˆ ,σˆ arePaulimatrices,Xˆ,Zˆ arespin-1 x y z generally suppressed for large central charge [12]. An- operators, and Pˆ,Mˆ are 3×3 Potts matrices: other example is the Ryu-Takayanagi formula, which     holds when the bulk is described by classical gravity 0 1 0 2 0 0 [17]. For instance, for an 1+1 dimensional CFT that Pˆ =0 0 1; Mˆ =0 −1 0  has a classical gravity dual, the Ryu-Takayanagi formula 1 0 0 0 0 −1 equates (in appropriate units) the von Neumann entan- Hîsing has central charge 1, Hˆbc is the spin-1 Blume glement entropy S(cid:96)bound of an interval of length (cid:96) in the Capel model which is criti2cal for α = 0.910207,β = CFT vacuum to the length L of the geodesic that ex- geo 0.415685 with central charge 7 , Hˆpotts has central tends between the end points of the interval through the 10 dual bulk spacetime, Sbound ≈L . (An analogous holo- charge 4, and Hˆxxz is critical for −1 ≤ ∆ < 1 with (cid:96) geo 5 graphicinterpretationofReynientanglemententropyhas central charge 1. (The scaling dimensions of the CFT also been proposed [18].) On the other hand, we have underlying Hˆxxz vary continuously with ∆.) For the re- Sbound = c log(cid:96), where c is the central charge [19]. Thus, sults presented in this section, we considered values of (cid:96) 3 Sbound, and therefore L determined in the dual bulk ∆∈{0,0.3,0.4,0.5,0.8,0.9,1}. (cid:96) geo geometry, increases with the central charge c. TheMERArepresentationofthegroundstatesofthese In this paper, we have not derived a (semi-)classical models was obtained using the variational energy mini- bulk geometry from the boundary state. So instead mization algorithm [15] keeping bond dimension χ=12. 9 isthevonNeumannentanglemententropyofthebulksite 180 locatedat(x,z(cid:48)),andSbulk isthejointvonNeumannen- Ising z,z(cid:48) 160 BC tanglement entropy of the two sites, see Fig. 11(c). Potts The results are plotted in Fig. 5. We find that the 140 XXZ, =0 mutual information decays exponentially, which implies XXZ, =0.3 that the bulk state has a finite correlation length along y 120 XXZ, =0.4 p o XXZ, =0.5 this direction. (The mutual information gives an upper rtn 100 XXZ, =0.8 boundforall2-pointcorrelators.) Theplotalsosuggests e inye 80 XXXXZZ,, ==01.9 awittrhenindcrtehaaste tinhebocournrdealartyiocnenlternagltchhagregneerfaolrlythdeesecrmeaosdes- R 60 els(withtheexceptionoftheIsingmodel). Ontheother hand, asmentionedpreviously, correlationsduetoquan- 40 tumfluctuationsinthebulkarealsogenerallysuppressed 20 forlargecentralchargeintheAdS/CFTcorrespondence. We also computed the second Reyni entanglement en- 0 0 20 40 60 80 100 tropy Rbulk =−log Tr(ρˆbulk)2. Here ρˆbulk is the reduced number of sites on the geodesic x,x(cid:48) 2 x,x(cid:48) x,x(cid:48) density matrix of the bulk sites that are located on a geodesic holographic screen that is anchored next to the FIG. 6. (Color online) Second Reyni entanglement entropy bulk sites located at (x,0) and (x(cid:48),0) [see Sec. IV and in copy bulk states, dual to the ground state of each of the also Fig. 11(c)] . The results are plotted in Fig. 6. We criticalspinchains listedinEq.(10), ofbulksiteslocatedon find that the Rbulk increases linearly with the number of a geodesic holographic screen [such as the one illustrated in x,x(cid:48) Fig. 11(c)]. bulksiteslocatedonthescreen(proportionaltop),which is consistent with the area law scaling of bulk entanglement derived in Appendix B. Interestingly, the slope of In these simulations, the error in the ground state en- thescalingofRbulk alsoappearstoincreasemontonically x,x(cid:48) ergy density for the Ising model was O(10−8) while the with the central charge for these models. relative error in the central charge was 0.4%. For the Furthermore,thevalueofReynientanglemententropy remaining models, the error in the ground state energy Rbulk foragivenblocksizeislargerforaspinchainwitha x,x(cid:48) densitywasO(10−5)andthe relativeerrorinthecentral largercentralcharge. Ontheotherhand,asdescribedat charge was at most 1.2%. The relative error in the first the beginning of this section, in AdS/CFT the geodesic fewscalingdimensionsforallthemodelswasatmost4%. length L also increases with boundary central charge, geo The Hamiltonians listed in Eq. (10) are not scale- inaccordancewiththeRyu-Takayanagiformula. Inview invariant, but flow to a scale-invariant fixed point af- of this, it may be interesting to explore whether a bulk ter possibly several RG (entanglement renormalization) metric can be deduced from the bulk entanglement en- steps. We considered the renormalized scale-invariant tropy. For example, is the entanglement entropy of the ground state of each model, described by retaining only bulk sites intersected by a geodesic, plotted in Fig. 6, a thescale-invariantpartoftheMERAtensornetwork[3]. legitimatemeasureofthegeodesic’slength,L ≈Rbulk? geo x,x(cid:48) Wefirsttranslatedthescale-invariantpartoftheMERA Weleavethisasanopenquestionforfuturework. (Com- to a constrained MERA form based on higher order sin- puting the von Neumann entanglement entropy, which gular value decomposition, and then lifted the modifed appears in the Ryu-Takayanagi formula, incurs a higher MERA by inserting copy tensors on the bonds to obtain computational cost. But for short geodesics we verified a dual copy bulk state, as described in Appendix C. See that the slopes of the scaling of von Neumann entangle- discussion in Sec. III. mententropy,analogoustoRbulk,alsoincreasemonoton- x,x(cid:48) LetusparameterizethebulklatticeMbycoordinates ically with increase in central charge for these models.) (x,z)wherezcorrespondstothelengthscaleandxlabels spatial translations (in the boundary description). For the purposes of this section, let (x,z),(x,z(cid:48)) and (x(cid:48),z) VII. SUMMARY AND OUTLOOK locate bulk sites along an arbitrarily long vertical graph geodesic [illustrated in Fig. 2(a)] such that z (cid:54)= z(cid:48) and We introduced a toy holographic correspondence for |x−x(cid:48)|=3p where p is a positive integer. quantum lattice systems that is both motivated by and ForthegroundstateofeachcriticalHamiltonianlisted incorporates some general features of the AdS/CFT cor- in Eq. (10), we estimated the correlation length in the respondence, listed (i)-(iv) in the introduction. To sum- dualcopybulkstatealongthezdirectionfromthescaling marize, we lifted the MERA representation—which also of the mutual information Ibulk given by z,z(cid:48) describestheRGflow—ofan1Dgroundstatetoatensor networkrepresentationofadual2Dquantumstate,such Ibulk =Sbulk+Sbulk−Sbulk. (11) z,z(cid:48) z z(cid:48) z,z(cid:48) that the two states are seen to live on the boundary and Here Sbulk =−Tr(ρˆbulklog ρˆbulk) is the von Neumann en- inthebulkofa2Dmanifold. Weachievedthisbyembed- z z 2 z tanglemententropyofthebulksitelocatedat(x,z),Sbulk ding the MERA in a 2D manifold and inserting tensors z(cid:48) 10 with open indices on the bonds of the MERA. The open a global on-site symmetry at the boundary [11]. In the indices of the MERA and the lifted MERA are associ- presenceofon-sitesymmetriesattheboundary,itisalso ated with the boundary and the emergent bulk degrees morenaturaltoassociatetwobulksiteswitheverybond of freedom respectively, and the lifted MERA represents oftheMERA,whichcorrespondstoliftingtheMERAby the dual bulk state. inserting a 4-index bond tensor with each index taking We explored parallels between this tensor network χ values. (Or equivalently, a 3-index bond tensor, as in statecorrespondenceandtheAdS/CFTcorrespondence, this paper, but whose open index takes χ2 values.) in light of the ongoing dialogue between the MERA and In conclusion, we have tried to make a case for ob- holography. For example, we described how copy bulk taininganemergent2DdescriptionfromtheMERArep- states exhibit the presence of holographic screens, and resentation of an 1D ground state by associating dual also a simple prescription to obtain boundary correla- degreesoffreedomwiththebondsofthetensornetwork. torsfromexpectationvaluesofextendedoperatorsinthe Theformalismintroducedinthispaperillustratesapos- bulk. BoththesepropertiesremindoftheAdS/CFTcor- sible way in which the MERA could implement hologra- respondence. phy, andmoregenerallyhowatensornetworkwithopen The bulk construction described in this paper can be indices intrinsically encodes a correspondence between generalized in several ways. Motivated by further guide- two quantum many-body states (after a lifting action is lines from holography, one may fix the bond tensors dif- defined for the bond indices). ferently, or pre-process the tensor network in some way Acknowledgements.—Most of this research was before lifting it (e.g. see Appendix C). It may also be completed while SS was employed at the Australian interesting to consider inserting bond tensors on a fewer Research Council’s Center of Exellence for Engineered number of bonds e.g. only those that are output from Quantum Systems in Macquarie University. SS thanks the wˆ tensors since, strictly speaking, only these are as- GavinBrennenformanyimportantdiscussions,andalso sociatedwithrenormalizedsitesintheMERA,seeFig.1. Guifre Vidal, Juan Maldacena, Nathan McMahan, and Thiscorrespondstoafewernumberofbulksites. Onthe Giandemenico Palumbo for stimulating discussions. 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