Entanglement renormalization and topological order
aa r X i v : . [ c ond - m a t . s t r- e l ] F e b Entanglement renormalization and topological order
Miguel Aguado
Max-Planck-Institut f¨ur Quantenoptik. Hans-Kopfermann-Str. 1. D-85748 Garching, Germany
Guifr´e Vidal
School of Physical Sciences. The University of Queensland. Brisbane, QLD, 4072, Australia (Dated: February 11, 2013)The multi-scale entanglement renormalisation ansatz (MERA) is argued to provide a naturaldescription for topological states of matter. The case of Kitaev’s toric code is analyzed in detail andshown to possess a remarkably simple MERA description leading to distillation of the topologicaldegrees of freedom at the top of the tensor network. Kitaev states on an infinite lattice are alsoshown to be a fixed point of the RG flow associated with entanglement renormalization. All theseresults generalize to arbitrary quantum double models.
PACS numbers: 05.30.–d, 02.70.–c, 03.67.Mn, 05.50.+q
Renormalization group (RG) transformations aim toobtain an effective description of the large distance be-havior of extended systems [1]. In the case of a sys-tem defined on a lattice, this can be achieved by con-structing a sequence of increasingly coarse-grained lat-tices {L , L , L , · · · } , where a single site of lattice L τ effectively describes a block of an increasingly large num-ber n τ ∼ exp( τ ) of sites in the original lattice L [2].Real-space RG methods can, in particular, be applied tostudy quantum systems at zero temperature, in whichcase each site of L τ is represented by a Hilbert space K τ [3]. There the goal is to identify the local degrees of free-dom relevant to the physics of the ground state and toretain them in the Hilbert space K τ , whose dimension d τ must be large enough to describe this physics. A severeproblem of such approach is that in D ≥ d τ must grow (doubly) exponentially in τ [4] as a resultof the accumulation of short-range entanglement at theboundary of the block. Entanglement renormalization [5] is a novel real-spaceRG transformation recently proposed in order to solvethe above difficulties. Its defining feature is the use of disentanglers prior to the coarse-graining step. These areunitary operations, acting on the interface of the blocksdefined by the RG procedure, that reduce the amountof entanglement in the system, see figure 1. A majorachievement of the approach is that, when applied to alarge class of ground states in both one [5] and two [6]spatial dimensions, the dimension d τ is seen not to growwith τ . A steady d τ is made possible by the disentan-gling step and has several implications [5, 6]. It meansthat, in principle, the resulting RG transformation canbe iterated indefinitely at a constant computational cost,allowing for the exploration of arbitrarily large lengthscales. In addition, the system can be compared with it-self at different length scales, and thus we can study RGflows in the space of ground state or Hamiltonian cou-plings. Finally, a constant d τ also leads to an efficientrepresentation of the system’s ground state in terms of atensor network, the multi-scale entanglement renormal-ization ansatz (MERA) [7]. site disentangler isometry coarse-grainedsiteblock FIG. 1: RG transformation based on entanglement renormal-ization. In order to build an effective site from a block offour sites, we first apply disentanglers between sites of theblock and surrounding sites. In this way part of the short-ranged entanglement between the block and its surroundingsis removed. Then we coarse-grain the four sites into one bymeans of an isometry that selects the subspace K ′ ⊆ K ⊗ to be kept. We show the case of a tilted square lattice inpreparation for the toric code where, in addition, each sitewill contain four qubits. At zero temperature, strongly correlated quantum sys-tems appear organized in a plethora of phases or orders ,including local symmetry breaking orders and topologi-cal orders [8]. Local symmetry breaking phases are de-scribed by a symmetry group and a local order param-eter, and they are associated with the physical mecha-nism of condensation of point-like objects. Transitionsbetween two such phases or orders involve a change inthe symmetry, as described by Landau’s theory. A sim-ple picture emerges from the perspective of entanglementrenormalization [5, 6]: under successive iterations of theRG transformation, ground states with local symmetrybreaking order progressively lose their entanglement andeventually converge to a trivial fixed point, namely anunentangled ground state. On the other hand, criticalground states describing transitions between these phasesare non-trivial —that is, entangled— fixed points of theRG transformation. In either case, the MERA providesan efficient, accurate representation of the ground state.Topological phases are fundamentally different from lo-cal symmetry breaking phases [8]. They do not stem from(the breakdown of) local group symmetries, but their topological order is linked to more complex mathemat-ical objects, like tensor categories, topological quantumfield theory, and quantum groups. Physically, topologicalphases exhibit gapped ground levels with robust degen-eracy dependent only on the topology of the underly-ing space. This, and the fact that excitations above theground level possess anyonic statistics, boosts the inter-est of these phases as scenarios for topological quantuminformation storage and processing. Condensation ofstring-like objects (in the so-called string-net models, see[9]) has been proposed as a general mechanism controllingtopological phases. As may be expected, such profounddifferences are also reflected in the way the ground stateis entangled. Specifically, the notion of topological en-tanglement entropy [10] (the subleading term in a large-perimeter expansion of the entanglement entropy of a sys-tem) has arisen as a quantitative measure of the groundstate entanglement due to topological effects. Systemswith topological order thus provide an unexplored sce-nario for entanglement renormalization techniques.The purpose of this Letter is to establish entanglementrenormalization and the MERA as valid tools also forthe description and investigation of topological phasesof matter. For simplicity, we analyze in detail Kitaev’storic code [11], a four-fold degenerate ground state widelydiscussed in the context of quantum computation andclosely related to Z lattice gauge theory [12] and to thesimplest of Levin-Wen’s models for string-net condensa-tion [9]. We show the following: ( i ) a MERA with finite,constant d τ can represent the toric code exactly ; ( ii ) ateach iteration of the RG transformation, entanglementrenormalization factors out local degrees of freedom fromthe lattice, while leaving the topological degrees of free-dom untouched; ( iii ) the MERA representation of thefour ground states is identical except in its top tensor,which stores the topological degrees of freedom; and ( iv )in an infinite system, the toric code is the fixed point ofthis RG transformation. All these results also hold formore complicated models, such as quantum double latticemodels, that we discuss in the appendix. We concludethat the MERA is naturally fitted to represent stateswith topological order, and the entanglement renormal-ization offers a new, useful framework for further studies.Following [11], we consider a square lattice Λ on thetorus, with spin-1 / H = − X + A + − X (cid:3) B (cid:3) (1)is a sum of constraint operators associated with vertices ‘+’ and plaquettes ‘ (cid:3) ,’ namely A + = Y i ∈ + X i , B (cid:3) = Y i ∈ (cid:3) Z i . (2)Stabilizers A + act as a simultaneous spin flip in allfour qubits adjacent to a given vertex. Stabilizers B (cid:3) yield the product of group assignments ± ±
1. Hamiltonian (1) isgapped, and states in the ground level (Kitaev states) aresimultaneous eigenstates of all A + , B (cid:3) with eigenvalue+1. The degeneracy of the ground level (i.e., the numberof Kitaev states) depends on the topology of the manifoldunderlying the lattice. If this manifold is a topologicallynontrivial Riemann surface, information is encoded innontrivial cycles, since operators Q i ∈C a,b Z i , where C a,b are nontrivial cycles along bonds of the lattice, commutewith all stabilizers. Besides, such operators along homo-logically equivalent nontrivial cycles C a , ˜ C a have the sameaction on Kitaev states. Hence, for a torus, two logicalqubits are encoded in the action of these operators. FIG. 2: Elementary moves adding plaquettes and vertices toa toric code. Arrows stand for CNOT operations.
Kitaev states are efficiently written in terms of theirstabilizers. The stabilizer formalism [13] also providesus with a useful language to analyse the action of oper-ators on Kitaev states, and has proved instrumental infinding an exact MERA. The key observation to this pur-pose is that there exist ‘elementary moves’ [14], minimaldeformations of the lattice and its Kitaev states, that re-spect the topological characteristics of the code. Thesemoves consists of addition or removal of faces and ver-tices together with qubits, and can be written in terms ofcontrolled-NOT (CNOT) operators, whose adjoint actionhas a very simple expression in terms of stabilizers: I ⊗ Z ↔ Z ⊗ Z, Z ⊗ I Z ⊗ I, (3) I ⊗ X I ⊗ X, X ⊗ I ↔ X ⊗ X. (4)Figure 2 depicts the construction of elementary moves.The creation of a face is achieved by introducing a newspin in a plaquette. Arrows stand for CNOT operatorsfrom control qubits (all qubits in one of the semiplaque-ttes) to the target qubit (the new qubit, introduced instate | i ). The following transformation of stabilizersholds (the new site is denoted as n ): Z Z Z Z Z Z Z Z Z Z , (5) Z n Z Z Z n , (6)which ensures plaquette constraints are obeyed. Simi-larly, the two relevant vertex constraints are extended tothe new qubit. The creation of a new vertex is achievedinstead by introducing a new qubit in state | + i . Thisqubit now plays the role of control for CNOTs acting onthe qubits adjacent to one of the split vertices. Stabilizerstransform as X X X X X X X X X X , (7) X n X X X n , (8)which is again compatible with the code constraints.Both final sets of operators are the correct stabilizersfor the code in the modified lattice (remember that X = Z = I .) Similarly, the two relevant paquetteconstraints are extended to the new qubit.These operations can be inverted to decouple qubits instates | i and | + i from the rest system. The disentanglersand isometries, defining both the RG transformation andthe MERA for the Kitaev states, are made of several ofthese decoupling moves. We regard the original squarelattice Λ, on which the toric code is defined, as a (tilted)square lattice L where each site contains four qubits.Then both disentanglers and isometries act on blocks offour sites of L as in figure 1 — equivalently, on blocksof 16 qubits in Λ. They consist of a series of CNOTs asspecified in figures 3 and 4.Upon applying the RG transformation, we obtain acoarse-grained lattice L which is locally identical to L and where, by construction, the toric code constraints arestill satisfied. This is quite remarkable. On the one hand,it is the first non-trivial example, in the context of entan-glement renormalization, where the RG transformation is exact [15], leading to the first non-trivial model that canbe exactly described with the MERA. On the other hand,if we consider an infinite lattice, the above observationimplies that Kitaev states are an explicit fixed point ofthe RG flow in the space of ground states, as induced bythe present RG transformation [16].Let us now consider a finite lattice L on the torus.The coarse-grained state carries exactly the same topo-logical information (values of Q Z along nontrivial cy-cles) as the original state, since the elementary moves pre-serve such information at each intermediate step. Thatis, different Kitaev states are not mixed during the RGtransformation. By iteration, we obtain a sequence ofincreasingly coarse-grained lattices {L , L , L , · · · , L T } for ever smaller toruses. The top lattice L T will containonly a few qubits. Recall that the MERA is made of allthe disentanglers and isometries used in the RG transfor-mations, together with a top tensor describing the stateof L T [7]. It follows that the MERAs for different statesof the toric code will contain identical disentanglers and (a) (b)FIG. 3: (a) The square lattice Λ for the toric code, with qubits(dots) on the links, is reorganized into a tilted square lattice L where each site is made of four qubits. The lattice constantis doubled (dotted lines dissapear) after the RG transforma-tion, which produces a new four-qubit site for lattice L fromevery block of sixteen qubits (the twelve light qubits in theblock are decoupled in known product states). (b) First stepof the RG transformation: Disentanglers. Arrows stand forsimultaneous CNOT operators from control to target qubits.Disentanglers act on sixteen-qubit domains overlapping withfour blocks each (thick dashed line, cf. figure 1.) Four qubitsper block decouple in state | i . isometries, and will only differ in their top tensor, whereall the topological information is stored.All the above results automatically extend to the loopmodel considered by Levin and Wen as the simplest oftheir family of string-net models [9]. Indeed, the toriccode on a square lattice can be locally transformed, us-ing the decoupling moves depicted in figure 5, into atoric code on a triangular lattice, which is equivalent tothe ground state of the loop model defined on the dual(hexagonal) lattice. This local transformation shows thatthe topological order of both models are identical, a factalready pointed out in [17] and which can also be un-derstood in terms of the projected entangled-pair stateansatz (PEPS) [18].Finally, our construction generalizes almost straight-fowardly to quantum double models (see, e.g., [11]), bothfor Abelian and non-Abelian groups. This is achieved byreplacing CNOTs with controlled group multiplicationoperators and by paying due attention to the order ofthe operations (see appendix).In conclusion, we have shown that several models withtopological order can be exactly represented with theMERA, where topological degrees of freedom are nat-urally isolated in its top tensor. We have also seen thatsuch models are fixed points of the RG flow induced byentanglement renormalization. Our results are an unam-biguous sign that entanglement renormalization and theMERA, originally developed to efficiently simulate sys-tems with local symmetry-breaking phases, provide alsoa most natural framework to study topological phases. Acknowledgements:
We thank J. I. Cirac, A. Ki- (a) (b)(c) (d)FIG. 4: (a)–(c) Second step of the RG transformation: Isome-tries. (a) Two qubits per block decouple in state | + i . (b) Twomore qubits per block decouple in state | i . (c) One qubit peredge, four per block, decouple in state | + i . The isometry alsotraces out the twelve decoupled qubits. (d) State of the sys-tem after the RG transformation.(a) (b)FIG. 5: Local mapping between the toric code on a squarelattice (a) and on a triangular lattice (b). The dual modelin a honeycomb lattice (displayed for reference) is Levin andWen’s loop model. taev, D. P´erez-Garc´ıa, J. Preskill and F. Verstraete for re-lated discussions. M. A. thanks the University of Queens-land for hospitality and a stimulating working atmo-sphere during his visit. G. V. acknowledges financial sup-port from the Australian Research Council, FF0668731. APPENDIX A: EXACT MERA FOR QUANTUMDOUBLE MODELS
Here we generalize the above RG transformation andMERA to lattice quantum double models (see [11].)Local degrees of freedom are associated with oriented bonds of a lattice Λ and identified with the group al-gebra of a discrete, in general non-Abelian, group G ,i.e., the Hilbert space spanned by an orthonormal ba- sis {| g i , g ∈ G } . A change in the orientation of a bondcorresponds to the map S : | g i 7→ | g − i . The Hamil-tonian is a sum of mutually commuting projectors oververtices and plaquettes, H D( G ) = − X v A v − X p B p (A1)where vertex projector A v acts on edges incoming to ver-tex v by simultaneous right multiplication by each groupelement, A v = 1 | G | X h ∈ G O i → v R i ( h ) , (A2)right multiplication acts as R ( h ) | g i = | gh i , and plaquetteprojector B p selects configurations where the orderedproduct of group elements taken along an oriented cir-cuit C p around p is the unit element of G , B p = δ ( Y i along C p g i , e ) . (A3) (a)(b)FIG. 6: Elementary moves adding plaquettes and vertices toa quantum double model. Local degrees of freedom live in thegroup algebra of a discrete group G . The ancilla is initialisedin state | e i (unit element of G .) Thick arrows stand for con-trolled right-multiplication of the target element by the con-trol element. Orientation of the edges plays an important rˆole.(a) For plaquette addition, operations must be performed in aprescribed order (e.g., after application of the arrows in coun-terclockwise order (I, II), the new element becomes g g .) (b)For vertex addition, the new element is initialised in state | e e i ,the equal-weight superposition of all elements in G . Here alloperations can be performed simultaneously. Elementary moves are analogous to their counterpartsfor the toric code. The operations generalising CNOTsare controlled multiplications by the control element(CMs). Figure 6 shows how to create plaquettes andvertices using the controlled right multiplication A | h, g i = | h, gh i , (A4)where the first element is the control and the secondelement is the target. To cover the case of differentbond orientations, we also consider the transformations B = ( S ⊗ A ( S ⊗ C = (1 ⊗ S ) A (1 ⊗ S ), and D = ( S ⊗ S ) A ( S ⊗ S ); explicitly: B | h, g i = | h, gh − i ,C | h, g i = | h, h − g i ,D | h, g i = | h, hg i . (A5) (a) (b)(c) (d)FIG. 7: RG transformation for a quantum double model. Ar-rows stand for controlled multiplications (CM) from controlto target elements. Arrow labels denote the type of CM (seeequations (A4) and (A5)), as well as the order in which theyare applied within each step. The fiducial lattice orientation(horizontal bonds pointing to the right, vertical bonds point-ing upwards) is assumed. (a) Disentanglers. Four elementsper block decouple in state | e i . (b)–(c) Isometries. In (b),two elements per block decouple in state | e e i . In (c), anothertwo elements per block decouple in state | e i , and the latticebecomes a doubled square lattice with two elements per edge.One element per edge, four per block, then decouple in (d) instate | e e i , completing the MERA ansatz. By means of these operations, new edges initialised instates | e i and | e e i = 1 p | G | X h ∈ G | h i (A6)are incorporated into the code, creating new plaquettesand vertices. Of course, the inverse elementary moves removing plaquettes and vertices from the code, neededfor the MERA construction, are in general not identicalto those adding plaquettes and vertices. Note that op-erations leading to plaquette addition (or removal) can-not be performed simultaneously for non-Abelian groups,since the order of multiplication of the elements is impor-tant.The RG transformation corresponding to a quantumdouble model associated with group G and defined on asquare lattice proceeds along the same lines as for thetoric code, but there are qualitative differences. To fixthe setting, we work with a fiducial orientation of thebonds: horizontal bonds are oriented from left to rightand vertical bonds are oriented upwards. Then: • Operations within a plaquette cannot be performedsimultaneously and must be applied in a certain or-der. Hence, disentanglers must be applied in threesteps, while isometries demand another step withrespect to the toric code RG. • Which of the controlled operations A , B , C , D isneeded at each step depends on the bond orienta-tions.The explicit form of the RG leading to a MERA de-scription of the quantum double model is shown in fig-ure 7. The basic properties of the toric code MERA(bounded causal cone, topological degrees of freedom atthe top of the tensor network, ER fixed point in the infi-nite lattice limit) generalise to the quantum double set-ting. [1] M. E. Fisher, Rev. Mod. Phys. , 653 (1998).[2] L. P. Kadanoff, Physics , 263 (1966).[3] K. G. Wilson, Rev. Mod. Phys. , 773–840 (1975).S. R. White, Phys. Rev. Lett. , 2863 (1992),Phys. Rev. B48 , 10345 (1993).[4] In a D dimensional lattice the ground state typicallyobeys a boundary law S l ∼ l D − for the entanglemententropy of a block of l D sites. In this case the dimension d τ for a site of L τ must at least scale doubly exponen-tially in τ , d τ ∼ exp(exp( τ )). Indeed, one the one handthe dimension of an effective space for a block of l D sitesmust be at least d ∼ exp( S l ) = exp( l D − ). On the other, l τ ∼ exp( τ ) after τ iterations of the RG transformation,where n τ = l Dτ is the number of sites of L effectivelydescribed by a single site of L τ .[5] G. Vidal, Phys. Rev. Lett. , 220405 (2007), arXiv: cond-mat/0512165v2 [cond-mat.str-el] .[6] G. Evenbly, G. Vidal, arXiv:0710.0692v2 [quant-ph] ;G. Evenbly, G. Vidal, arXiv:0801.2449v2 [quant-ph] .[7] G. Vidal, arXiv:quant-ph/0610099 .[8] X.-G. Wen, Quantum Field Theory of Many-Body Sys-tems , Oxford University Press (2004).[9] M. A. Levin, X.-G. Wen, Phys. Rev.
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