Haag duality for Kitaev's quantum double model for abelian groups
aa r X i v : . [ m a t h - ph ] M a y Haag duality for Kitaev’s quantum double modelfor abelian groups
Leander Fiedler ∗ and Pieter Naaijkens † Institut f¨ur Theoretische Physik, Leibniz Universit¨at Hannover,GermanyOctober 3, 2018
Abstract
We prove Haag duality for conelike regions in the ground state rep-resentation corresponding to the translational invariant ground state ofKitaev’s quantum double model for finite abelian groups. This propertysays that if an observable commutes with all observables localised outsidethe cone region, it actually is an element of the von Neumann algebra gen-erated by the local observables inside the cone. This strengthens locality,which says that observables localised in disjoint regions commute.As an application we consider the superselection structure of the quan-tum double model for abelian groups on an infinite lattice in the spirit ofthe Doplicher-Haag-Roberts program in algebraic quantum field theory.We find that, as is the case for the toric code model on an infinite lat-tice, the superselection structure is given by the category of irreduciblerepresentations of the quantum double.
Kitaev’s quantum double model for finite groups is a spin model on a 2D latticethat exhibits anyonic excitations [25]. One of its main features is that it hascertain topological properties: the ground space degeneracy depends on thetopology of the underlying lattice. In addition, the model has (quasi-)particleexcitations with braid (anyonic) statistics. This can be exploited to performquantum computations. In fact, for certain groups it allows even for universal quantum computation [27, 28]. The computations are made possible by thebraid statistics of the anyons which are encoded in the superselection structureof the model.It turns out that even on a topologically trivial lattice, such as a square latticeon the plane, the excitations with anyonic statistics exist. One way to recoverthe properties of these excitations is by doing a Doplicher-Haag-Roberts (DHR)type analysis of the superselection sectors [14, 15]. This has been carried out for ∗ [email protected] † [email protected] G = Z in Kitaev’s quantum double model. Here we extend these results to generalfinite abelian groups G . In particular, we show that algebras of observableslocalised in cone-like regions fulfil Haag duality in the vacuum representation.This means that in the vacuum representation observables which commute withall observables outside the cone are exactly those which can be approximated(in the weak operator topology) by operators localised inside the cone. Moreprecisely, suppose Λ is a cone and A (Λ) is the algebra of quasi-local observableslocalised in Λ. Then we have in the ground state representation π of thequantum double model, that π ( A (Λ)) ′′ = π ( A (Λ c )) ′ , where the prime denotestaking the commutant, and Λ c is the set of all sites in the complement of thecone Λ. Note that one of the inclusions readily follows from locality, the otherone is non-trivial and is what the larger part of this paper is devoted to.For the proof of Haag duality we follow the ideas introduced in [31]. Inparticular, we first show that cone algebras leave certain subspaces of the vac-uum Hilbert space invariant. These subspaces can be shown to be generatedby the self-adjoint parts of the cone algebra and the algebra associated to thecomplement of the cone. Using a result by Rieffel and van Daele [37] we canthen conclude Haag duality for cone algebras. In essence the proof relies on athorough understanding of the ground state, or rather, the full excitation spec-trum of the model. This allows us to get a precise understanding of the Hilbertspaces describing all pairs of excitations in a certain region of the system. Aprecise understanding of how these states can be obtained by acting with lo-cal operators on the ground state vector allows us to use the aforementionedtheorem by Rieffel and van Daele.As a consequence of Haag duality and a property of the ground state weobtain the approximate split property, which implies that if we have two conesthat are removed from each other sufficiently far, then we can prepare normalstates on these two cones independently [16]. In this sense it is a form ofstatistical independence of the two cone regions. This is no longer true if wetake a cone Λ and its complement. In that case, the split property does nothold any more and one cannot find normal product states on the two regions,and they are not independent in the strong sense (c.f. [38]).Another application of Haag duality that we consider is the analysis of thesuperselection structure of the model for finite abelian groups. We show thatin this case the superselection structure is described by conelike localised en-domorphisms by explicitly constructing such endomorphisms that describe asingle excitation. In this way we can show that the superselection structure(including the braiding and fusion roles) is described by finite dimensional rep-resentations of Drinfeld’s quantum double D ( G ) of the underlying group. Thisresembles analogue results for Kitaev’s toric code model [29]. We do this byconstructing states describing a single charge. These are obtained by creatinga pair of excitations from the ground state, and move one of these to infinity.Because of the topological properties of the quantum double model, the direc-tion in which we do this cannot be observed. It follows that the correspondingrepresentations satisfy a superselection criterion: they are irreducible represen-tations that are unitarily equivalent to the ground state representation, but only In earlier work [29, 31] we called this the distal split property. However, we feel ap-proximate is more appropriate, since we will only need to assume a small separation of tworegions. v fe Figure 1: Geometric setting of the quantum double model. The black arrowson the edges indicate their orientation. v , e and f are a vertex, edge and face,respectively, and s = ( v, f ) is a site. The star at v is given by the four black edgesconnecting to v . The plaquette at f is defined by the four edges surrounding f .when one restricts to observables localised outside a cone. This resembles theBuchholz-Fredenhagen criterion in algebraic quantum field theory [11]. UsingHaag duality we can then restrict to endomorphisms of the observables, and dothe DHR superselection theory [14, 15].The paper is organised as follows. In Section 2 we review the geometric andalgebraic setting of the quantum double model for finite groups and introducenecessary notation. We also recall the main properties of the excitation spec-trum of the model. This is then used in Section 3 to show that there is a uniquetranslational invariant ground state in our setting. Section 4 contains the mainresult of this paper: the proof of Haag duality. The next two sections concernthe approximate split property and an analysis of the superselection structureof abelian quantum double models. We end with an outlook on the extensionto non-abelian groups of our results. We start with recalling the setting of the quantum double model for finitegroups. Most of the result in this section are not new, but since the nota-tion and properties we introduce here play an important role in the main partof this paper, we recall the essentials to make the paper more self-contained.For a more detailed introduction we refer to [25] and [6]. We will mainly followthe notation of [6].Consider a square lattice Z and let G be any finite group. Vertices of thelattice are denoted by v . Between nearest-neighbour vertices there are orientededges e . The set of all these edges (or bonds) is called Γ. For simplicity we fixthe orientation of the edges as in Figure 1: the edges point either right or up.If e is an edge,we write e for the edge with the opposite orientation. Faces ofthe lattice are denoted by f . Note that faces can be identified with vertices inthe dual lattice, and similarly vertices in the lattice correspond to faces in thedual lattice. The edges in the dual lattice connect two faces of the lattice. Theyinherit an orientation from the edges in the lattice in the following way. Givenan edge e in Γ then its orientation fixes a notion for the neighbouring faces tolie to the “left” or to the “right” of e in direction of the orientation. A dualedge e is then oriented in such a way that it point from the face to the right We later specialise to abelian groups, but for the definition of the model this is immaterial.
3o the face to the left of the associated oriented edge in Γ. A site s will meana tuple s = ( v, f ) of a vertex and a neighbouring face f . Finally, we refer tothe four edges enclosing a face as a plaquette (notation: plaq( f )), and the fouredges ending or starting at a vertex v as a star, notation star( v ).To each edge e of the lattice we associate a Hilbert space H e with a basislabelled by the elements of G . The orthonormal basis in H e is denoted by | g i with g ∈ G . For any edge e ∈ Γ denote A ( { e } ) := B ( H e ) for the algebra ofobservables acting on this edge. Similarly for any set O ∈ P f (Γ), where P f (Γ)is the set of all finite subsets of Γ, the local algebras are given by A ( O ) := N e ∈O A ( { e } ). We will also write A e for A ( { e } ). If A ∈ A ( O ) we say that A is localised in O .This construction gives rise to an isotonous net O 7→ A ( O ) of C ∗ -algebras.The corresponding embedding ∗ -isomorphisms are given by the natural embed-ding provided by the tensor product structure. That is, if O ⊂ O it is givenby the map ι O O defined by A A ⊗ I O \O . The inductive limit of this netis called the quasilocal algebra and denoted by A . It is the norm closure of the ∗ -algebra all observables localised in finite regions. Similarly one can define for infinite sets Λ ⊂ Γ the algebra A (Λ) ⊂ Λ as the norm closure of S O⊂ Λ A ( O ),where the union is over finite subsets.An important part of the model’s structure is most easily explained in termsof certain operators associated to triangles on the lattice. We recall the maindefinitions here. A direct triangle τ can be thought of connecting a face with twoneighbouring vertices which are connected by an edge. More specific, considera site s = ( v, f ) can be thought of as a line connecting a vertex v with a face f . A direct triangle τ is then a tuple ( s , s , e ) of the sites s , s and an edge e or its inverse such that the tuple lists the sides of the triangle τ in clockwiseorder.Similarly, a dual triangle τ ′ connects two neighbouring faces with a vertexover some dual edge. Again τ ′ is given by a tuple ( s , s , e ) where s , s aresites and e is an edge or its dual edge such that the tuple lists the sides of τ ′ in counterclockwise order. Given a triangle τ = ( s , s , e ) denote ∂ τ := s and ∂ τ := s . Two triangles are said to overlap if and only if the correspondingedges intersect. Any triangle inherits an orientation by the edge in the tuple.Note that it can either coincide with the orientation given by the lattice (ordual lattice) Γ, or be anti-parallel.Given a direct triangle τ and an element h ∈ G we can now associate anoperator T hτ ∈ A e by T hτ | k i = (cid:26) δ h,k | k i , if τ is oriented parallel to Γ δ h,k | k i , else , k ∈ G. For a dual triangle τ ′ we set for any element g ∈ GL gτ ′ | k i = (cid:26) | gk i , if τ is oriented parallel to Γ | kg i , else , k ∈ G. Here and in the remainder of the paper we will use the notation g for the inversegroup element of g ∈ G , to keep the sub and superscripts in the formula morereadable. If τ and τ ′ overlap the corresponding triangle operators act on the One can in fact regard it as the group C ∗ -algebra C [ G ]. More generally, this can be donefor any finite dimensional Hopf- ∗ algebra (c.f. the remark on p.13 of [25]). e and one can verify that the operators L gτ ′ T hτ , h, g ∈ G are matrixunits spanning A e .A crucial role is played by operators that act along a ribbon . A ribbon ρ isgiven by a tuple ( τ , τ , . . . , τ n ) of pairwise non-overlapping triangles such that ∂ τ i = ∂ τ i +1 , i = 1 , . . . , n −
1. We set ∂ ρ := ∂ τ and ∂ ρ := ∂ τ n . A ribbon ρ is called is called closed if ∂ ρ = ∂ ρ . Given a ribbon ρ and group elements g, h ∈ G an associated ribbon operator F g,hρ is defined recursively as follows: let τ be a direct, τ ′ a dual triangle and ǫ the trivial ribbon. In this case the ribbonoperators are defined as F g,hǫ := I, F g,hτ := T gτ , F g,hτ ′ := δ e,h L gτ ′ , where e ∈ G is the unit element. If ρ is any ribbon, we can decompose it into twopossibly smaller ribbons ρ and ρ , and write ρ = ρ ρ . The ribbon operatoron ρ is then defined recursively in terms of the ribbon operators on the smallerribbons by F g,hρ := X k ∈ G F g,kρ F hgh,khρ . (2.1)It can be checked that this is consistent and independent of the partition [6].We will sometimes refer to equation (2.1) as the ribbon decomposition rule .The algebra generated by the ribbon operators acting along a ribbon ρ will bedenoted by F ρ .The commutation relations for ribbon operators associated to some ribbon ρ are given by F g,hρ F k,lρ = δ h,l F gk,lρ and [ F g,hρ , F k,lρ ′ ] = 0 if ρ ∩ σ = ∅ . The casewhere ρ and σ overlap at some site will be discussed later in the context ofbraiding (see Section 2.2). Finally, the adjoint is given by ( F h,gρ ) ∗ = F h,gρ .Given a site s there are two distinct closed ribbons that start and end at s ,namely the smallest closed ribbon β s that consists just of direct triangles andthe smallest closed ribbon α s consisting only of dual triangles. For g, h ∈ G weset A gs := F g,eα s and B hs := F e,hβ s and define the star and plaquette operators by A s := 1 | G | X g ∈ G A gs B s := B es . (2.2)The definition definition of A s depends only on the vertex the site is located atand B s depends only on the face at s . The name star operator can be explainedby noting that it acts on the edges of star( s ). Similarly, the plaquette operatoracts on the corresponding plaquette.There is another convenient description of the operators A gs and B hs . It canbe obtained by choosing a basis vector in the tensor product of the Hilbert spaceson the edges, and describing the action on this basis vector. As an example theaction of A gs is visualized the following diagram (2.3): ❆❣s b (cid:0)✹(cid:0)✷(cid:0)✶ (cid:0)✸ ❂ b (cid:0)(cid:0)✹(cid:0)✷(cid:0)✁✶(cid:0)✶(cid:0)✁✶ (cid:0)(cid:0)✸ ✿ (2.3)5n a similar way one can see that B hs is a projection. In particular, choose abasis element corresponding to a choice of group elements g , . . . g , labelled byfollowing the edges around the plaquette in an anticlockwise direction, startingat the vertex v of the site s . The action of B hs on this vector is the identity if σ ( g ) σ ( g ) σ ( g ) σ ( g ) = h , and zero otherwise. Here σ ( g i ) = g i if the directionof the edge matches the anticlockwise path, and g i otherwise. The product ofthe group elements is also called the flux through the plaquette.Let s be a site. Using the definition above, it is not so difficult to work outthe commutation relations for operators A gs , B hs acting on the site s . One finds A gs A g ′ s = A gg ′ s , B hs B h ′ s = δ h,h ′ B hs , A gs B hs = B ghg − s A gs . (2.4)In particular this shows that for any pair of sites s, s ′ the plaquette and staroperators commute, i.e. [ A s , B s ′ ] = 0 = [ A s , A s ′ ] = 0 = [ B s , B s ′ ]. Remark . The operators A gs and B hs generate a finite dimenional algebra,that is in fact isomorphic to Drinfeld’s quantum double [17] of the group algebra C [ G ] regarded as a Hopf algebra. We write D ( G ) for this algebra. This explainsthe name “quantum double model”. The quantum double has been very wellstudied, and many of the properties that we will need in this paper can be tracedback to the representation theory of D ( G ). Good introductions can be foundin, for example, Ref. [39] for the quantum double and its representations in thecontext of C ∗ -algebras, or the textbook [24] for a more algebraic approach inthe language of category theory.With this notation we can introduce the dynamics of the quantum doublemodel. Recall that dynamics can be specified by local Hamiltonians, satisfyingcertain conditions that ensure that they lead to a time evolution on the entirequasi-local algebra of observables A [8]. These local Hamiltonians can be definedin terms of the operators A gs and B hs introduced above, or rather the sum A s andthe projection B s . Note that these operators mutually commute, even if theyboth act on the same site. We sometimes write B f or A v , where f is a face and v a vertex, instead of s = ( v, f ). Note that this does not lead to ambiguities.Concretely, let Λ ∈ P f (Γ). Then the corresponding local Hamiltonian isdefined by H Λ = − X star( s ) ⊂ Λ A s − X plaq( s ) ⊂ Λ B s . The summation is over all stars and all plaquettes (faces) whose bonds arecompletely contained in Λ. We will later see that the ground state is a stabilizerstate, that is stabilized by each A s and B p , and we can see the Hamiltonian asimplementing an energy penalty for violation of the “constraints” that A s Ω = B s Ω = Ω for a ground state (as we will see later).Ribbon operators are interpreted as creating excitations at the ending sitesof the ribbon. This interpretation is strengthened by the commutation relationswith star and plaquette operators (see [6]) A ks F h,gρ = F khk,kgρ A ks A ks F h,gρ = F h,gkρ A ks B ks F h,gρ = F h,gρ B khs B ks F h,gρ = F h,gρ B ghgks (2.5)for a ribbon ρ with s i = ∂ i ρ, i = 0 , g, h, k ∈ G . On the other hand thestars and plaquettes at sites different from s and s commute with ρ . Hence if6e act with a ribbon operator on the ground state, some of the constraints inthe Hamiltonian will be violated.For our purposes it will be convenient to consider a different basis for thespace of ribbon operators acting on a ribbon ρ . Let C = { c , . . . , c n } be aconjugacy class of G , r ∈ C some representative and π an irreducible unitaryrepresentation of Z G ( r ), the centraliser of r in G . Choose elements q , . . . , q n such that c i = q i rq i for i = 1 , . . . , n and set F C,π,i,i ′ ,j,j ′ ρ := X z ∈ Z G ( r ) π j,j ′ ( z ) F c i ,q i rq i ′ ρ (2.6)where j, j ′ ∈ { , . . . , | π |} label the matrix elements of π and i, i ′ ∈ { , . . . , n } .This relates the ribbon operators to irreducible representations of the quantumdouble D ( G ) of the group [13]. It can be shown (see [6]), that in case ρ consistsof both (direct and dual) types of triangles then these operators form a basis of F ρ , the algebra generated by the ribbon operators at ρ .Note that if the group G is abelian, these definitions somewhat simplify, es-sentially because we only have to deal with one dimensional representations. Inthat case, the centraliser is simply G , the conjugacy classes are single elements,and the irreducible representations are characters χ of G . We simply write F χ,cρ in that case, that is, F χ,cρ = X g ∈ G χ ( g ) F c,gρ . (2.7)It is not difficult to check that F χ ,cρ F χ ,dρ = F χ χ ,cdρ , where χ χ is the point-wise product of χ and χ .There is another useful property that is valid for these operators for abelian G , but not in general: if we decompose a ribbon ρ into two parts ρ and ρ ,the corresponding ribbon operator is just the product of the ribbon operatorsacting along the smaller ribbons. Lemma 2.2.
Let χ be a character and c ∈ G for some finite abelian group G .Suppose that ρ = ρ ρ is a ribbon. Then F χ,cρ = F χ,cρ F χ,cρ .Proof. With the help of equation (2.1) we find that F χ,cρ = X g,k ∈ G χ ( g ) F c,kρ F c,kgρ = X g,k ∈ G χ ( kg ) F c,kρ F c,gρ = F χ,cρ F χ,cρ , where we made the substitution g kg in the second equality, and used that χ ( kg ) = χ ( k ) χ ( g ). For later use we list some properties of ribbon operators that we will needlater. In particular, we are interested in the question how the action of theseoperators on the ground state depend on the ribbon itself. As will be outlinedin Section 3, in the present situation there is a unique translational invariantground state ω . The corresponding GNS representation will be denoted by( π , Ω , H ). If we talk about “the ground state” or “ground state vector”, wewill always mean the translational invariant ground state ω (resp. the GNSvector Ω). Since A is an inductive limit of simple algebras, it is simple, hence7 s ρ ′ ρ Figure 2: A deformation ρ ′ of a ribbon ρ connection sites s and s . The whitearrows indicates the orientation of the ribbons. π is a faithful representation. To simplify notation we therefore often writesimply A for π ( A ). An essential fact in proving the properties below is thatthe ground state vector Ω has the property that A gs Ω = Ω and B hs Ω = δ h,e Ω forany site s and group elements g, h ∈ G .As it turns out the action of a ribbon operator on the ground state onlydepends on the sites connected by the ribbon and not on the connecting ribbonitself. This allows to deform ribbons by fixing its endpoints and changing theshape in between. In the following G denotes any finite group. Lemma 2.3.
Let ρ, ρ ′ , σ, σ ′ be ribbons with ∂ i ρ = ∂ i ρ ′ and ∂ i σ = ∂ i σ ′ , i = 0 , .Then for all A ∈ A and all g, h, k, l ∈ G it holds ω ( F h,gρ AF l,kσ ) = ω ( F h,gρ ′ AF l,kσ ′ ) . Proof.
In [6] it is shown that for ribbons ρ, ρ ′ as above the ribbon operators F h,gρ , F h,gρ ′ map the ground state vector Ω to the same image, i.e. F h,gρ Ω = F h,gρ ′ Ω. Hence by noting that ω ( A ) = h Ω , A Ω i , A ∈ A and ( F h,gρ ) ∗ = F h,gρ theclaim follows.We refer to ρ ′ and σ ′ as deformations of ρ and σ . A more detailed definitionand description can be found in [6]. An example of a deformation of a ribbon isgiven in Figure 2. Later on we need to connect to ending sites of a ribbon ρ withanother ribbon ρ such that ρρ is a closed ribbon. The way ribbon operators of ρ and ρ relate to each other is given by the following lemma: Lemma 2.4.
Let ρ, ρ, σ, σ be ribbons with ∂ i ρ = ∂ − i ρ and ∂ i σ = ∂ − i σ, i =0 , . Then for all A ∈ A and all g, h, k, l ∈ G it holds ω ( F h,gρ AF l,kσ ) = ω ( F h,gρ AF l,kσ ) . For the proof we refer to [6]. The ribbons ρ and σ are referred to as inversions of ρ and σ . We can always choose such an inversion of a ribbon.From now on assume that G is abelian. Then the irreducible representationsof D ( G ) are given by elements c ∈ G and characters χ : G → C , as mentionedabove. Lemma 2.5.
Let ρ ⊂ Λ be a ribbon, c, k ∈ G , χ an irreducible character and s he starting site of ρ . Then ( F χ,cρ ) ∗ = F χ,cρ A ks F χ,cρ = X g ∈ G χ ( g ) A k F c,g = χ ( k ) F χ,cρ A ks B ks F χ,cρ = F χ,cρ B kcs Proof.
By direct calculation using equations (2.7) and (2.5).As mentioned earlier the commutation relations of ribbon operators with thestar and plaquette operators can be interpreted as the ribbon operators gener-ating excitations at the ending sites of their respective ribbons, when appliedto the ground state. The next lemma sheds more light on this interpretation.
Lemma 2.6.
Let ρ be an open ribbon and F χ,cρ an associated ribbon operator.Then, if s = ∂ ρ or s = ∂ ρ , it holds [ F χ,cρ , A s ] = 0 ⇐⇒ χ = id . Similarly [ F χ,cρ , B s ] = 0 ⇐⇒ c = e. Proof.
First note, that F χ,cρ = 0 as well as A s and B s , since the ground stateis not contained in the respective kernels. Note that the respective implicationsfrom the right hand side to the left hand side are true by Lemma 2.5. For thefirst statement we see, using Lemma 2.5 and [23, Theorem 27.15], A s F χ,cρ A s = 1 | G | X k ∈ G χ ( k ) F χ,cρ A s = δ χ, id F χ,cρ A s . Thus, if [ A s , F χ,cρ ] = 0 we have A s F χ,cρ = δ χ, id F χ,cρ A s which is only true, if χ = id. Using a similar derivation for the second statement we get[ F χ,cρ , B s ] = 0 = ⇒ δ c,e F χ,cρ B s = F χ,cρ B s and thus c = e .If ρ is an open ribbon, then the excitations created at its ends by applyingsome ribbon operator on the ground state can be detected by certain localoperators (see [6, Section B 9.] for detailed definitions). A particularly usefulexample is that of certain projections (compare [6, Section C 3.]). For this let s = ∂ ρ the initial site of ρ . Then D ξ,ds := 1 | G | X k ∈ G ξ ( k ) A ks B d (2.8)detects the charge created by F χ,cρ in the following sense: D ξ,ds F χ,cρ Ω = 1 | G | X g,h ∈ G ξ ( g ) χ ( gh ) F c,hρ A gs B cds Ω = δ ξ,χ δ c,d F χ,cρ Ω . ξ, d ) , ( χ, c ) denote irreducible representationsof D ( G ). Note, that by Lemma 2.3, D ξ,ds Ω = δ ξ, id δ d,e Ω and in particular theprojection onto the ground state is given by D id ,es . In subsequent sections wewill use the notion D s := D id ,es .Under some circumstances we can extend ribbons by triangles without chang-ing an associated ribbon operator. This will be of some use later. Lemma 2.7.
Let ρ be an open ribbon and denote s := ∂ ρ and s := ∂ ρ . Pick c ∈ G and an irreducible representation χ of G . If there is a direct triangle τ such that τ ρ is a ribbon the following holds: [ F χ,cρ , A s ] = 0 = ⇒ F χ,cρ = F χ,cτρ The analogue statement holds true if ρτ is a ribbon and the ribbon operatorcommutes with the star operator at s .If there is a dual triangle τ ′ such that τ ′ ρ is a ribbon then [ F χ,cρ , B s ] = 0 = ⇒ F χ,cρ = F χ,cτ ′ ρ and again an analogue statement holds true if ρτ ′ is a ribbon.Proof. By Lemma 2.6 [ F χ,cρ , A s ] = 0 implies χ = id. Hence F χ,cρ = F id ,cρ andtherefore F id ,cτρ = X g,k ∈ G T gτ F c,gkρ = F id ,cρ since P g ∈ G T gτ = I . Analogously the other case. For if τ ′ ρ is a ribbon[ F χ,cρ , B s ] = 0 = ⇒ c = e and F χ,eτρ = X g,k ∈ G χ ( k ) L eτ ′ δ g,e F e,gkρ = F χ,eρ and again analogously for the second case.Since G is abelian we also have that ribbon operators of closed ribbonscommute with all star and plaquette operators. Lemma 2.8.
Let ρ be any closed ribbon. Then for all h, g, k ∈ G [ F h,gρ , A k ] = 0 = [ F h,gρ , B k ] . The proof can be found in [6, Appendix B.5]. A somewhat weaker statementof this is also true if we remove one triangle from a closed ribbon.
Lemma 2.9.
Let ρ be an open ribbon such that there is a direct triangle τ with τ ρ is a closed ribbon. Then, with χ, c, s as above, [ A s , F χ,cρ ] = 0 = ⇒ [ B s , F χ,cρ ] = 0 Given instead that there is a dual triangle τ ′ such that τ ′ ρ is a closed ribbon.Then [ B s , F χ,cρ ] = 0 = ⇒ [ A s , F χ,cρ ] = 0 Proof. F χ,cρ = F χ,cτρ , and since τ ρ is a closed ribbon the claim follows.2.) The premises imply by Lemma 2.7, that F χ,cρ = F χ,cτ ′ ρ , and since τ ′ ρ is aclosed ribbon the claim follows. 10 .2 More on commutation relations In order to discuss statistics of superselection sectors later in Section 6.1 wehave to worry about commutation relations of ribbons. In particular we wantto know the commutation relations of ribbons which overlap at their ends and ofribbons that cross each other once, since then the commutation relations of as-sociated ribbon operators reflect the braiding and fusion structure of irreduciblerepresentations of D ( G ) (c.f. [25]).We start with some finite group G and two ribbons ρ, σ . We say that ρ, σ start at the same site s if there is a direct triangle τ , a dual triangle τ ′ , ribbons˜ ρ, ρ ′ , σ ′ such that ∂ ˜ ρ = s , ρ ′ ∩ σ ′ = ∅ and ρ = ρ ′ τ ′ ˜ ρ, σ = σ ′ τ ˜ ρ (in [6] this iscalled a left joint). The commutation relations of associated ribbon operatorsare then given by F p,qρ F s,tσ = F psp,ptσ F p,qρ . (2.9)Similarly, ρ, σ end at the same site s if ρ = ˜ ρτ ′ ρ ′ , σ = ˜ ρτ σ ′ and ∂ ˜ ρ = s (whichis called a right joint in [6]). The corresponding commutation relations are givenby F p,qρ F s,tσ = F s,tqpqσ F p,qρ . Note that in the remaining possible cases for ρ and σ , i.e. ρ ends at the samesite at which σ starts we have that the ribbon operators commute.We have particular interest in the commutation relations for finite abeliangroups G . Here, for instance, equation (2.9) becomes F p,qρ F s,tσ = F s,ptσ F p,qρ . Furthermore, with the notation ( χ, c ) , ( ξ, d ) for irreducible representations of D ( G ), the commutation relations for ribbons ρ, σ starting at the same site give F χ,cρ F ξ,dσ = ξ ( c ) F ξ,dσ F χ,cρ , and for ribbons ending at the same site we have F χ,cρ F ξ,dσ = ξ ( c ) F ξ,dσ F χ,cρ . Now consider two ribbons ρ, σ that cross each other , meaning there are ribbons ρ , ρ , σ , σ such that ρ = ρ ρ , σ = σ σ , ρ i ∩ σ i = ∅ , i = 1 , ∂ ρ = ∂ σ , ∂ σ = ∂ ρ . I.e. ρ , σ end at the same site as σ , ρ start at. Such a situationis illustrated in Figure 3. The commutation relations then are F p,qρ F s,tσ = F s,ptσ F p,sqρ . (2.10)Applied on ribbon operators labelled by irreducible representation of D ( G ) thisgives F χ,cρ F ξ,dσ = X g,h ∈ G χ ( dg ) ξ ( ch ) F d,hσ F c,gρ = χ ( d ) ξ ( c ) F ξ,dσ F χ,cρ , (2.11)where we used F χ,cρ = P g ∈ G χ ( g ) F c,gρ and with the usual notation ( χ, c ) and( ξ, d ) for irreducible representations of D ( G ).11 σ ρ Figure 3: Two ribbons ρ, σ crossing each other at site s .With the commutation relations at hand we can prove the following technicallemma, which will be used later in one of the proofs for Haag duality. It basicallystates that if at a site s there is an excitation of the ground state createdby multiple ribbons then it can be created by a single ribbon ending at s .The excitations at the remaining spots different from s are created by ribbonoperators connecting those sites with each other. Lemma 2.10.
Let ρ , . . . , ρ n be open ribbons and s be some site. Assume that ( ∀ i ∈ { , . . . , n } )( ∃ ! j ∈ { , } ) : ∂ j ρ i = s . This gives a map { , . . . , n } ∋ i j i ∈ { , } . Furthermore assume that for all i, i ′ ∈ { , . . . , n } it holds that ∂ − j i ρ i = ∂ − j i ′ ρ i ′ . Let χ i , i = 1 , . . . , n be irreducible representations of G andelements c i ∈ G , i = 1 , . . . , n . Set χ := χ · · · χ n and c := c · · · c n .Then there are ribbons σ , . . . , σ n − with { ∂ σ k , ∂ σ k | k = 1 , . . . , n − } = { ∂ − j i ρ i | i = 1 , . . . , n } , a ribbon γ with ∂ γ = s and ∂ γ = ∂ − j i ρ i for some i ∈ { , . . . , n } , and irreducible representations ξ , . . . , ξ n of G and elements d , . . . , d n − ∈ G such that F χ ,c ρ · · · F c n ρ n Ω = zF ξ ,d σ · · · F ξ n − ,d n − σ n − F χ,cγ Ω where z ∈ C and | z | = 1 .Proof. The proof works by induction over the number of ribbons. By means ofinversions of ribbons, i.e. Lemma 2.4, we can assume without loss of generalitythat j ( { , . . . , n } ) = { } for any n >
0. In other words we assume that allribbons involved have their starting point at s since otherwise we could invertthem due to the aforementioned lemma.If n = 1 the claim is trivial. We will elaborate on the case n = 2 sincethis illustrates the basic idea of the proof. Let ρ , ρ be ribbons as in theassumptions. Let χ , χ be irreducible representations of G and c , c ∈ G . Let ρ be an inversion of ρ such that ρ ρ is a ribbon. Then by Lemma 2.3 andLemma 2.4 we have F χ ,c ρ F χ ,c ρ Ω = F χ ,c ρ F χ ,c ρ F χ ,c ρ F χ ,c ρ Ω= zF χ χ ,c c ρ F χ ,c ρ F ρ ρ Ωwhere z is the factor given by the commutation relations in equation (2.9). Nowlet σ be a deformation of ρ ρ such that s = σ . We then have F χ ,c ρ F χ ,c ρ Ω = zF χ χ ,c c ρ F χ ,c σ Ω12s claimed.Now let ρ , . . . , ρ n be ribbons as in the preamble of the Lemma and assumethat the claim holds for all any n − χ , . . . , χ n be irreduciblerepresentation of G and c , . . . , c n ∈ G . Set ξ := χ · · · χ n and d := c · · · c n .Then F χ ,c ρ · · · F χ n ,c n ρ n Ω = zF χ ,c ρ F ξ ,d σ · · · F ξ n − ,d n − σ n − F ξ,dγ Ωwhere the ribbons γ, σ k , irreducible representations ξ k and c k ∈ G are corre-sponding to the claim. Let γ be an inversion of γ such that ρ γ is a ribbon. Let σ be a deformation of ρ γ . Again, using the same Lemmas as above we have F χ ,c ρ · · · F χ n ,c n ρ n Ω = ˜ zF ξ ,d σ · · · F ξ n − ,d n − σ n − F χ ,c ρ F ξ,dγ Ω= yF ξ ,d σ · · · F ξ n − ,d n − σ n − F χ ,c ρ F ξ,dγ F χ ,c γ F χ ,c γ Ω= ˜ yF ξ ,d σ · · · F ξ n − ,d n − σ n − F ξχ ,cc γ F χ ,c ρ γ Ω= ˜ yF ξ ,d σ · · · F ξ n − ,d n − σ n − F ξχ ,cc γ F χ ,c σ Ω= ˆ yF ξ ,d σ · · · F ξ n − ,d n − σ n − F χ ,c σ F ξχ ,cc γ Ω . The factors ˜ z, y, ˜ y and ˆ y are products with z and phase factors resulting fromthe commutation relations of the ribbon operators (c.f. Section 2.2). The lastexpression is of the form as in the claim. We now outline the proof that for each finite group G (not necessarily abelian!),the quantum double model has a unique translational invariant ground state. In case the model is defined by an oriented lattice on a compact surface it isknown that the ground space degeneracy is the number of flat G -connections upto conjugation [25], hence it is no surprise that in this infinite but topologicallytrivial setting we find a unique translational invariant ground state. The proofwe discuss here is based on the proof in [30], where the full details can be found,which in turn is partly based on ideas of [1].Each term in the local Hamiltonians only acts on the bonds of a star orof a plaquette. Moreover, in the present situation of a square lattice, there isan obvious action of the group Z by translations. It follows that the localHamiltonians H Λ are defined by a bounded, translation invariant interaction Φ.Since the interaction is of bounded range and translationally invariant there isa corresponding one-parameter group α t of automorphisms of A describing thetime evolution [8]. The next lemma is useful when discussing ground states withrespect to these dynamics. Lemma 3.1 ([1]) . Let ω be a state on a unital C ∗ -algebra A , and suppose X ∈ A satisfies X = X ∗ , X ≤ I , and ω ( X ) = 1 . Then ω ( XY ) = ω ( Y X ) = ω ( Y ) forany Y ∈ A . As was pointed out to us by Bruno Nachtergaele, the claim in [1] about uniqueness ofthe ground state is not entirely correct. By modifying finite volume boundary conditions andtaking the thermodynamic limit, it is possible to obtain additional (algebraic) ground states.These states however are not invariant with respect to translations. Examples of such statesare given by the “single anyon states”, cf. [29, Prop 3.2]. G = Z . Proposition 3.2.
There exists a ground state ω for the dynamics of the quan-tum double model, which has the property that ω ( A s ) = ω ( B s ) = 1 for eachsite s . Moreover, every translation invariant ground state has this property.Proof (sketch). The star and plaquette algebras generate an abelian subalge-bra of A . We can identify each star and each plaquette with a classical Isingspin, hence this algebra describes two copies of the Ising model. The state weare looking for is the state with all spins in the up position. By the Hahn-Banach theorem there exists an extension to a state ω of A . This is thestate we are looking for: using Lemma 3.1 it is straightforward to show that − iω ( X ∗ δ ( X )) ≥ X (and δ the derivation implement-ing the dynamics). Hence ω is a ground state.To show that any translational invariant ground state has this property,let ω be such a state. Since A s and B s are projections, it follows that 0 ≤ ω ( A s ) , ω ( B s ) ≤
1. Because ground states minimise the mean energy H Φ ( ω )by Theorem 6.2.58 of [8] one sees that we must have ω ( A s ) = ω ( B s ) = 1.To show that there is only one state on A with these properties, the ideais essentially to use Lemma 3.1 again, just as it was used in the proof of theuniqueness of the translational invariant ground state of the toric code model [1].The combinatorics, however, are much more involved. The proof consists of twosteps. First we calculate the value of a ground state on certain products ofprojections acting on an individual site. In the second step this result is tocalculate the expectation values of arbitrary local observables, showing that theground state is completely fixed.It was already remarked by Kitaev that the ground states of the quantumdouble model are related to so-called flat G -connections [25]. Here we have toconsider local observables, and hence it is enough to specify a G -connectionfor finite parts of the system. The precise definition is a slight adaption fromdiscrete gauge theory [34]: Definition 3.3.
Let F be a finite collection of faces and let Λ ⊂ Γ be the setof bonds bounding any face f ∈ F . A G -connection c is a map c : Λ → G . Aconnection is called flat if the monodromy around each face is trivial. That is,let f ∈ F and list the edges j , . . . j n of f in counter-clockwise order. Then themonodromy is trivial if σ ( c ( j )) σ ( c ( j )) · · · σ ( c ( j n )) = e , where σ is as definedas follows: σ ( c ( j )) = c ( j ) if the direction of j coincides with the direction of thepath around f , and c ( j ) otherwise. The set of all G -connections on Λ will bedenoted by C G (Λ), whose subset of flat connections is called C fG (Λ).The constant map defined by c ( j ) = e is trivially a flat G -connection, hencethe set of flat connections is certainly non-empty. To each such a G -connectionwe can associate a projection, projecting on the basis vector | c ( j ) i at the site j .That is, P c = Y j ∈ Λ T c ( j ) τ ( j ) , where τ ( j ) is the direct triangle with edge j whose orientation matches. Now,if c is not a flat connection, there is a face f with non-trivial monodromy. Let14 be a site with face f . Then it follows that B s P c = 0 (since B s projects onthe subspace of trivial monodromy around the face f ). With Lemma 3.1 itfollows that ω ( P c ) = ω ( P c B s ) = 0 if c is not flat. Now suppose that c isa flat connection. Then one can show that A gs P c = P c ′ A gs , where c ′ is also aflat connection. In fact by a sequence of such moves one can go from any flatconnection c to any other flat connection c ′ . With the same Lemma as beforeone then deduces the following Lemma. For a detailed proof we refer to [30]. Lemma 3.4.
Let c ∈ C G (Λ) and suppose that ω is a ground state for thequantum double model. Then ω ( P c ) = 1 / | C fG (Λ) | if c is flat, and zero otherwise.Here | C fG (Λ) | is the number of flat G -connections. As remarked before the operators L g T h acting on the same edge form aset of matrix units for the local algebra. Hence every local observable can bewritten as a linear combination of operators of the form X = LP c , where c isa connection and L is a product of operators of the form L gj . By the argumentabove it follows that ω ( X ) = 0 if c is not flat. By systematically multiplying X on the left (right) by star operators (plaquette operators), one can “clean up”the observable X , and show that ω ( X ) is either zero, or equal to ω ( P c ′ ) forsome flat connection c ′ . This argument leads to the following theorem [30]: Theorem 3.5.
Kitaev’s quantum double model on a square lattice on the planehas a unique translational invariant ground state ω , completely determined by ω ( A s ) = ω ( B s ) = 1 . This state is pure. Purity follows because ω restricted to the abelian subalgebra generated byall A s and B p is multiplicative (hence pure). Since there exists a pure extensionto A and by the argument above, the state ω is completely determined bythe values on stars and plaquettes, it follows that ω must be pure. We willhenceforth only consider this translational invariant ground state, and just referto it as “the” ground state and will call the corresponding GNS representationthe vacuum representation .If one inspects the full proof of the theorem given in [30] carefully, one seesthat in fact ω ( AB ) = ω ( A ) ω ( B ) if A and B are local, and their supportsare sufficiently far removed from each other. This is related to the approximatesplit property, which will be discussed in Section 5. The main result in this paper is the proof that Haag duality holds in the GNSrepresentation of the translational invariant ground state for certain cone-likeregions. We first introduce some definitions to make clear what we mean witha “cone”. With these definitions we then discuss the proof. What is essentialin our proof is a good understanding of how one can build up the Hilbert spaceof the ground state representation from excitations of the ground state. Inparticular, how one can obtain those excitations that are localised in a cone,by acting with the appropriate ribbon operators. We use this to reduce theproblem to a commutation problem of algebras acting on a smaller Hilbertspace, consisting only of excitations inside the cone. The ground state vectoris cyclic for this Hilbert space, with respect to the cone algebra. This finally15akes it possible to apply a result by Rieffel and Van Daele [37], which relatesthe commutation properties of algebras to a density property of self-adjoint partsof algebras acting on a cyclic vector. In this way we circumvent the problemthat the Reeh-Schlieder property (which says that the ground state vector iscyclic and separating for local algebras) is not available, unlike for relativisticquantum field theories where it usually plays an important role in proving Haagduality [5, 12].
The main motivation to consider cone-like regions is given by the localisationregions of single excitations of the ground state. These will turn out laterto be suitably described by cones . How these cones are defined and whichproperties we need them to fulfill is described in the following. We will state alist of requirements as a definition and then give a family of regions which fulfillthis list. Some of these requirements originate in the localisation properties ofexcitations sitting at the end of ribbons. Others are motivated as a technicalrequirement for proving a weaker form of the split property. Most importantlycones should be “ribbon connected” in the sense that we can connect any siteinside the cone with ribbons without leaving the cone. Furthermore it shouldbe possible to translate any finite subset of the lattice into the cone using somelattice translation.First we discuss what we mean by the boundary of a subset of Γ. We regardedges as a pair of vertices which are connected by an oriented bond. If weremove one of those vertices we also discard the bond.
Definition 4.1.
Let Λ ⊂ Γ be a collection of edges and denote Λ c := Γ \ Λ.The interior int(Λ c ) of Λ c is defined by the collection of edges in Λ c obtainedby removing from Z all vertices contained in Λ and discarding the associatedbonds in Γ. The boundary ∂ Λ c of Λ c is then defined to be ∂ Λ c := Γ \ (Λ ∪ int(Λ c ))and we set ∂ Λ := ∂ Λ c .Note that the definition of ∂ Λ is symmetric under the exchange of Λ andint(Λ c ). Furthermore Λ ∪ int(Λ c ) is a proper subset of Γ. That is to say ∂ Λ isthe “gap” between Λ and the interior of Λ c . Definition 4.2.
Given a subset Λ ⊂ Γ, a triangle τ ⊂ Γ and a ribbon ρ ⊂ Γ.We say that τ belongs to or is contained in Λ if the edge of τ is in Λ. Similarlywe say ρ belongs to Λ if all triangles of ρ belong to Λ. If this is the case we write τ ⊂ Λ and ρ ⊂ Λ.An illustration of this definition can be found in Figure 4. As we saw inSection 2.1 excitations of the ground states are localised at sites and can bedetected by star and plaquette operators. Therefore, in order to distinguishwhether an excitation is contained inside an area or not we have to specifywhen a site is, which is rather obvious. Less clear on the other hand is thespecification of a site sitting at the boundary of an area. For our purposes andhaving an eye on Lemma 2.7 we use the following notion.
Definition 4.3.
Let Λ ⊂ Γ be again a subset and let s = ( v, f ) be any site.Then s is considered to be contained in Λ, writing s ∈ Λ, whenever for any edge e ∈ Γ with ∂e = v it holds e ∈ Λ. 16 τ ρ Figure 4: This image illustrates when triangles are contained in a region. Theblack lines indicate a collection Λ of edges. The triangle τ and the ribbon ρ are contained in Λ whereas τ is not.We say that s is contained in ∂ Λ, writing s ∈ ∂ Λ whenever s / ∈ Λ and thereare edges e ∈ Λ and e ′ ∈ Λ c which bound f or are neighbours of v .In other words s = ( v, f ) ∈ Λ if the star at v is contained in Λ, and s ∈ ∂ Λif the star or the plaquette has non-empty intersection with Λ and if s / ∈ Λ (c.f.Figure 5b).Unfortunately the definition of s ∈ ∂ Λ is not symmetric under swapping theroles of Λ and the interior int(Λ c ) of Λ c : There might be sites that are containedin ∂ Λ that have empty intersection with Λ c . Nevertheless this definition issufficient for our purposes since we just want to distinguish stars and plaquettesthat are contained in int(Λ c ) from those having non-trivial intersection with Λ.We will use this later on to move excitations that sit on the boundary of conesinto the interior of the respective cone. Lemma 4.4.
Let Λ ⊂ Γ be some subset and let s = ( v, f ) ∈ int(Λ c ) be somesite. Then for all edges e ending at v or bounding f it holds e ∈ Λ c .Proof. Assume that there was an edge e ∈ Λ ending at v or bounding f . Thenin case it ends in v we have s / ∈ int(Λ c ). In case e bounds f but does not end in v we have that both ∂ e, ∂ e ∈ Λ. But then there is at least one edge e ′ endingat v and one of ∂ e, ∂ e and hence e ′ ∈ ∂ Λ. But then s / ∈ int(Λ c ).Finally the straightforward definition of a ribbon ρ starting or ending at ∂ Λis given by requiring that the starting and ending sites ∂ / ρ are contained in ∂ Λ.With this definition we have that a ribbon ρ ⊂ Λ c with, say, ∂ ρ ∈ ∂ Λ, is atmost one triangle apart from Λ in the following sense. There is a ribbon ρ ⊂ Λ c with ∂ ρ ∈ ∂ Λ such that ρ ρ is a ribbon and ρ is either a single triangle ora trivial ribbon. (Here we have again Lemma 2.7 in mind.). This situation isdepicted in Figure 5b.We now come to the definition of cones. For any subset O ⊂
Γ and anypoint y ∈ Z we denote by y + O the subset in Γ obtained by translating allpairs of vertices corresponding to edges in O by y . Definition 4.5.
A subset Λ ⊂ Γ is called cone if it satisfies all of the followingcriteria.1. For any finite subset
O ⊂
Γ there is a point y ∈ Z such that y + O ⊂
Λ.17 int Λ c (a) ρs s Λ (b) Figure 5: In both pictures the grey shaded region Λ is a cone. (a): Edges thatare drawn black are either contained in Λ or int(Λ c ) The grey bonds form ∂ Λ.(b): Dotted lines indicate sites, especially s ∈ Λ and s ∈ ∂ Λ. The black lineshighlight the edges belonging to the stars and plaquettes at s and s . Theribbon ρ connects a site at Λ with a site in ∂ Λ2. For any pair of sites s , s ∈ Λ there is a ribbon ρ ⊂ Λ with ∂ / ρ = s / .3. For any pair of sites s , s ∈ ∂ Λ there are ribbons ρ , ρ ⊂ Λ c and ρ ⊂ Λsuch that ρ ρρ is a ribbon, ∂ i ρ i = s i , i = 0 , ρ i , i = 0 , s , s ∈ ∂ Λ there is a ribbon ρ ⊂ Λ c such that ∂ i ρ = s i , i = 0 , II ∞ or of Type III (see also Section 5 and reference [29]).The second and the third conditions express a kind of connectedness: Anypair of sites inside a cone Λ can be connected with a ribbon, and sites at theboundary can be connected by ribbons that are contained in Λ up to singletriangles at the ends. Both of them do not prohibit Λ of having holes insidethey just make sure that it is sufficiently connected in the aforementioned sense.The last condition ensures that that the complement Λ c is properly connectedso that there are no holes in Λ.As a result we can choose whether we want to connect sites at the boundaryof the cone by ribbons that run in the exterior or in the interior of the cone upto triangles at the endpoints of the ribbon. In particular for any ribbon ρ ⊂ Λ c with ∂ i ρ ∈ ∂ Λ , i = 0 , ρ , ρ ⊂ Γ and ˜ ρ ⊂ Λ such that ρ ˜ ρρ is a ribbon, ∂ ρ = ∂ ρ , ∂ ρ = ∂ ρ and ρ , ρ are trivial ribbons or singletriangles. Furthermore, by condition 1, any cone is an infinite set. Examples forcones can be generated by those in R : let l = l be two semi-infinite lines in R emanating from a common point in Z and enclosing an angle smaller than π . Denote by Λ the set of edges that are contained in area enclosed by or have18on-empty intersection with the two lines (see also Figure 5a). It can be easilychecked that Λ is a cone.In the following, and if not specified otherwise, Λ will be a cone. In this section we prove Haag duality of cone algebras in the vacuum represen-tation. The proof is based on ideas developed in [31] and subdivides into severalsteps. First we consider certain subspaces of the vacuum representation Hilbertspace that are invariant under the action of cone algebras and show that thecone algebras are completely determined by this restriction. Secondly we showthat these subspaces are also invariant under the commutants of the algebrasassociated to the complement of the cones. The last step consists in showingthat linear combinations of the self-adjoint parts of the restricted cone algebrasgenerate the above subspaces. Using these facts together with a result by Rieffeland van Daele [37] we can conclude Haag duality for the cone algebras.For the start consider a cone Λ ⊂ Γ, denote the associated cone algebraby A (Λ) and by A (Λ c ) the one of the complement. The ground state’s cyclic(GNS) representation is given by the tuple ( π , Ω , H ) where the state itself isreferred to as ω . For any region O ⊂
Γ we denote the weak closure of A ( O ) by R O := π ( A ( O )) ′′ . As sketched above we aim at finding a subspace H Λ ⊂ H such that Ω is cyclic for R Λ . Again, we will identify operators A ∈ A with theirimage under π .Let ρ be a ribbon and let again F ρ := { F h,gρ | h, g ∈ G } be the algebra linearlygenerated by all ribbon operators at ρ . Note that the inclusion F ρ ⊆ N e ∈ ρ A e is usually proper since F ρ can be viewed as the subset of elements of the righthand side singled out by the commutation relations given by equation (2.5)(c.f.[6, B.8]). For cones Λ we denote by F Λ := S ρ ⊂ Λ F ρ the algebra of ribbonoperators localised in Λ. Analogously we denote F Λ c the algebra of ribbonoperators localised in Λ c .The first observation is that products of operators in F Λ and F Λ c generatea norm-dense subspace of H when applied to Ω (compare also [31]). Lemma 4.6.
Given that Λ ⊂ Γ is a cone we have with the notation from above: F Λ F Λ c Ω k·k = H . Proof.
Single triangle operators are contained in F Λ and F Λ c . Since they forma basis of the edge algebras, operators in A loc (Λ) and A loc (Λ c ) are containedin F Λ and F Λ c , respectively. But those are norm-dense in A (Λ) and A (Λ c ),respectively, and together with cyclicity of Ω we arrive at the claim. Definition 4.7.
Let Λ ⊂ Γ be a cone. We set H Λ := F Λ Ω k·k ⊂ H and write P Λ for the projection onto H Λ .This subspace turns out to be left invariant by observables localised in thecone. Furthermore such observables are completely determined by their restric-tion to this space. The proof of this is the same as in [31, Lemma 3.5] and wewon’t repeat it here. 19 emma 4.8. For any cone Λ ⊂ Γ the subspace H Λ ⊂ H is invariant under A (Λ) , i.e. A (Λ) H Λ ⊂ H Λ . Furthermore any element A ∈ R Λ is completelydetermined by its restriction to H Λ . As a consequence we have that P Λ ∈ R ′ Λ . One basic observation in theproof is that F Λ is dense in A (Λ). The next step consists of showing thata similar but less obvious statement holds true for the operators commutingwith those localised in Λ c . The main idea is to show that we can characterise H ⊥ Λ by certain ribbon operators in F Λ c namely those which create non-trivialexcitations in int(Λ c ).Next we show that observables in the commutant of A (Λ c ) leave this vectorspace invariant. The basic idea is the same as that of the proof of [31, Lemma3.6]: We can characterize vectors of the form F · · · F n Ω to lie either in H Λ or in H ⊥ Λ where F , . . . , F n ∈ A are ribbon operators. Namely if F · · · F n Ω containsnon-trivial excitations in int(Λ c ) then it is contained in H ⊥ Λ . If there are noexcitations in int(Λ c ) contained in this vector then it belongs to H Λ . The nexttwo lemmas show this in a stronger sense, namely that the orthogonal relationin the first case holds even if we apply any operator from A (Λ c ) ′ to the vector.The idea is to detect excitations with star and plaquette operators acting onthe ending sites of the corresponding ribbons. For this recall the definition ofthe projections D χ,cs in equation (2.8) acting at a site s . To say that there is acharge in int(Λ c ) created by some ribbon operator amounts to seeing that thereis some site s ∈ int(Λ c ) such that D id ,es does not commute with this operator.Note that this follows from the discussion in Section 2.1, especially the partaround equation (2.8).The following three lemmas are essential in gaining a better understandingof the Hilbert space H Λ . Lemma 4.9.
Let ˆ F := F · · · F n ∈ F Λ c be a product of ribbon operators associ-ated to ribbons in Λ c . Then the following holds: (cid:16) ∃ s ∈ int(Λ c ) : [ A s , ˆ F ] = 0 ∨ [ B s , ˆ F ] = 0 (cid:17) = ⇒ (cid:16) ( ∀ F, C ∈ F Λ )( ∀ X ∈ A (Λ c ) ′ ) : ( ˆ F F Ω , XC Ω) = 0 (cid:17) . (4.1) Especially the left hand side implies ˆ F Ω ∈ H ⊥ Λ .Proof. First note that because of Lemma 4.4 s ∈ int(Λ c ), implies A s , B s ∈ F Λ c .The proof works by repeated use of the lemmas of the discussion in Section 2.1.It is sufficient to work with ribbon operators labelled by irreducible repre-sentations of D ( G ) as defined in equation (2.7). Consider arbitrary such ribbonoperators ˆ F , . . . , ˆ F n ∈ F Λ c and let C, F ∈ F Λ be some operators. By definitionof F Λ the operators C and F are sums of products of ribbon operators localisedin Λ. For convenience we set η := ˆ F · · · ˆ F n F Ω ∈ F Λ c F Λ Ω and ζ := C Ω ∈ H Λ .Now for the proof of equation (4.1), namely that if there are excitations in η created by ˆ F , . . . , ˆ F n ∈ F Λ c then η is orthogonal to Xζ for all C, F ∈ F Λ and X ∈ A (Λ c ) ′ .Assume there exists a site s ∈ int(Λ c ) whose star operator A s does not20ommute with ˆ F · · · ˆ F n . Then, by Lemma 2.5 and locality, we have( η, Xζ ) = 1 | G | X k ∈ G ( ˆ F · · · ˆ F n F A ks Ω , Xζ )= 1 | G | X k ∈ G ˆ χ ( k ) · · · ˆ χ n ( k )( η, Xζ )where ˆ χ j ( k ) either coincides with the corresponding term of the non-trivialrepresentation of ˆ F j if it doesn’t commute with A s , or ˆ χ j ( k ) = 1. Since forabelian groups the product of irreducible representations is again irreducible(they are all 1-dimensional), the right hand side equals 0 since the appearingproduct representation is non-trivial. If the product representation was triv-ial then [ A s , ˆ F · · · ˆ F n ] = 0 and hence would contradict the assumptions (seeLemma 2.6). Thus we arrive at ( η, Xζ ) = 0.Assume that there is a site s ∈ int(Λ c ) such that the associated plaquetteoperator B s does not commute with ˆ F · · · ˆ F n . Then there is at least one j ∈{ , . . . , n } with [ B s , ˆ F χ,cρ j ] = 0 implying c = e due to the commutation relations,see Lemma 2.6. More general there is a k ∈ G with k = e such thatˆ F · · · ˆ F n B s = B ks ˆ F · · · ˆ F n giving ( η, Xζ ) = ( B ks ˆ F · · · ˆ F n C Ω , ζ ) = ( η, XF B ks Ω) = 0 . This completes the proof.
Lemma 4.10.
Let ˆ F := F · · · F n ∈ F Λ c be a product of ribbon operators asso-ciated to ribbons in Λ c . Then the following holds: (cid:16) ∀ s ∈ int(Λ c ) : [ A s , ˆ F ] = 0 ∧ [ B s , ˆ F ] = 0 (cid:17) = ⇒ ˆ F Ω ∈ H Λ (4.2) Proof.
Again, as in the previous proof, it is sufficient to work with ribbon oper-ators labelled by irreducible representations of D ( G ). First some remarks aboutsome general simplifications we are are going to assume. In case two ribbons ρ, σ have the same starting and ending sites then, by Lemma 2.3, one of themcan be deformed into the other, givingˆ F χ,cρ ˆ F τ,dσ Ω = ˆ F χτ,cdρ Ω . (4.3)We always can assume that there are non-trivial and non-closed ribbons in theproduct ˆ F · · · ˆ F n . If ribbon operators associated to closed ribbons appearedthen we simply could commute them past the other operators in C to Ω wherethey leave Ω invariant. This can be seen by noting that if ρ is a closed ribbonand ρ = ρ ρ is a partition into ribbons then by Lemma 2.4 and Lemma 2.3we have ˆ F χ,cρ Ω = ˆ F χ,cρ ˆ F χ,cρ Ω = Ω. Here ρ is an inversion of ρ which, byconstruction, starts and ends at the same sites as ρ . Due to the commutationrelations of ribbons, see the discussion in Section 2.2, we may pick up somephase factors which will not be important here.In case that there are two ribbon operators ˆ F , ˆ F associated to open ribbons ρ , ρ such that ρ ρ is a closed ribbon we can write them as a product of a21 Figure 6: The two main cases in Lemma 4.10 depicted in one image: On theleft hand side of the cone Λ is the case where only ribbons occur that connectsites of ∂ Λ whith each other. On the right hand side is the case with ribbonshaving ending sites in ∂ Λ.ribbon operator of a closed ribbon and an operator associated to an open ribbon.To see this we move ˆ F and ˆ F to each other using the commutation relationsof ribbons. Then we use Lemma 2.2 and the remark after equation (2.7) to findˆ F χ,cρ ˆ F ξ,dρ = ˆ F χ,cρ ˆ F ξ,dρ ˆ F ξ,dρ ˆ F ξ,dρ = ˆ F χξ,cdρ ˆ F ξ,dρ ρ . (4.4)We also can always assume that ribbons just appear at most once in each productby the remark following equation (2.7).Now we turn to the claim of the lemma, equation (4.2). We are performingan induction over the number of ribbon operators in ˆ F · · · ˆ F n , i.e. over thenumber of ribbon operators outside Λ. Let’s start with n = 1 and let ˆ F ∈ F Λ c be a ribbon operator. Then we have that the ribbon ρ ⊂ Λ c , to which ˆ F isassociated to, is either of one of the following forms: It connects two sites in ∂ Λor at least one ending site of ρ is contained in int(Λ c ).Consider the case that ρ connects two sites in ∂ Λ. Taking a look at Defini-tion 4.3 we see that there are at most two triangles τ, ˜ τ ⊂ Λ c such that τ ρ ˜ τ ⊂ Λ c is a ribbon. By assumption and Lemma 2.7 we have that ˆ F Ω = ˆ F τρ ˜ τ Ω. Butthen we can invoke Lemma 2.4 and Lemma 2.3 to obtain a ribbon ρ ⊂ Λ withˆ F τρ ˜ τ Ω = ˆ F ρ Ω and ˆ F ρ ∈ A (Λ). In case that ρ has at least one ending site con-tained in int(Λ c ) Lemma 2.6 (or an analogue calculation with equation (2.5))implies that ˆ F = I . Hence in either case the vector is contained in H Λ .Now let n > F , . . . , ˆ F n − ∈ A (Λ c ). Let therefore ˆ F , . . . , ˆ F n ∈ A (Λ c ) be ribbonoperators associated to ribbons in Λ c and set η := ˆ F · · · ˆ F n Ω. The remainder ofthe proof can be subdivided into different cases corresponding to the differentconfigurations ribbons. We will relate some of them to each other and proofthe remaining cases. The two main cases are the following: Firstly, there couldbe k ≤ n ribbons that start and end at ∂ Λ. Secondly, there could be severalribbons having at least one end in int(Λ c ). See also Figure 6.The first main case can be handled as follows. Assume that there is a ribbonthat connects two sites at ∂ Λ, say ρ k , 1 ≤ k ≤ n . Then we can commute theassociated ribbon operator ˆ F k in η to the right in front of Ω thereby possiblyobtaining a phase factor due to equation (2.11). But then, by using the argumentfrom above, we can replace ˆ F k with some operator F k ∈ A (Λ) leaving a productof n − A (Λ c ) in front of F k Ω.22he second main case is a bit more involved. Consider that there is no suchribbon as in the first main case. If there is a ribbon ρ having at least one endingsite inside int(Λ c ) the following scenarios are possible. Firstly, one ending site of ρ which is contained in int(Λ c ) does not coincide with an ending site of anotherribbon occurring in η . Secondly, ρ connects a site on ∂ Λ with a site in int(Λ c )at which k ≥ F ρ must be the identity operator. This reduces the product ˆ F · · · ˆ F n ⊂ A (Λ c )in η to a product of n − A (Λ c ).In the second case we can assume that every of these k ribbons connectsto ∂ Λ, since otherwise, we can just pick one of them that doesn’t and use theprevious procedure to remove it. Remember that we don’t have to considerclosed ribbons any more as well as open ribbons forming a closed loop. Nowconsider the ribbon operator ˆ F ρ associated to ρ . We can safely assume that ∂ ρ is the site of interest. The other case can be treated in complete analogy. Ifthere is a ribbon ρ l with ∂ ρ l = ∂ ρ then we first can deform ρ into a ribbon˜ ρ such that ˜ ρρ l is a ribbon. On the level of ribbon operators this means firstcommuting the associated ribbon operator ˆ F ρ in η to the right in front of Ω andthen using Lemma 2.3 to replace it with an operator ˆ F ˜ ρ . After that we use thecommutation relations of ribbon operators again to move ˆ F ρ to ˆ F ˜ ρ . We thencan invoke equation (4.4) to obtainˆ F χ,c ˜ ρ ˆ F ξ,dρ l = ˆ F χξ,cd ˜ ρ F ξ,d ˜ ρρ l . The ribbon ˜ ρρ l connects two sites at ∂ Λ and we can use a previous argumentto replace ˆ F ˜ ρρ l in η by a ribbon operator in A (Λ).If there is no ribbon ρ l with ∂ ρ l = ∂ ρ we pick one ribbon ρ l and applyLemma 2.4 to replace it with a ribbon operator associated to a ribbon ρ l with ∂ ρ l = ∂ ρ . But then we can proceed as before. Note that we also could haveapplied Lemma 2.10 instead to conclude the same for the second case.By induction we now can conclude that for any n ∈ N and any productof ribbon operators ˆ F , · · · , ˆ F n ∈ A (Λ c ) the relation in equation (4.2) holdstrue. Lemma 4.11.
For any cone Λ ⊂ Γ it holds A (Λ c ) ′ H Λ ⊂ H Λ , hence P Λ ∈ R Λ c .Proof. Let ˆ F := ˆ F · · · ˆ F n be a product of ribbon operators ˆ F , . . . , ˆ F n ∈ F Λ c .Furthermore let F, C ∈ F Λ and X ∈ A (Λ c ) ′ be any, non-zero, operators. Forconvenience set η := ˆ F F
Ω and ξ := C Ω. Recall the definition of D s in equa-tion (2.8).By Lemma 4.9 we have that if ( η, Xξ ) = 0 holds for all F, C ∈ F Λ and X ∈ A (Λ c ) ′ then for any s ∈ int(Λ c ) the operator ˆ F commutes with D s , i.e.[ F, D s ] = 0. Now by Lemma 4.10 this implies η ∈ H Λ . To see this note thatˆ F Ω ∈ H Λ ⇐⇒ (cid:16) ∀ F ∈ F Λ : ˆ F F Ω ∈ H Λ (cid:17) since ˆ F F = F ˆ F and F Λ H Λ ⊆ H Λ . The other direction of this equivalence canbe seen by assuming that the right hand side was true while the left hand wasnot which immediately leads to a contradiction since I ∈ F Λ . Summarizing thiswe obtain ( η, Xξ ) = 0 = ⇒ η ∈ H Λ (4.5)23or all η = ˆ F F Ω , ξ = C Ω and ˆ
F , F, C, X as above.By definition F Λ c contains all matrix units of the edge algebras A e for e ∈ Λ c since the former are products of triangle operators. Hence products of ribbonoperators ˆ F , . . . , ˆ F n form a generating system of F Λ c . Thus, by Lemma 4.6,the linear span of the set { ˆ F · · · ˆ F n F Ω | ˆ F , . . . ˆ F n ∈ F Λ c ribbon operators , F ∈ F Λ , n ∈ N } is a dense subspace of H . From this we conclude that equation (4.5) holds forany η ∈ H and ξ ∈ H Λ . Therefore( ∀ ψ ∈ H ) : ψ ∈ H ⊥ Λ = ⇒ (( ∀ φ ∈ H Λ )( ∀ X ∈ A (Λ c ) ′ ) : ( ψ, Xφ ) = 0)and we arrive at A (Λ c ) ′ H Λ ⊥ H ⊥ Λ .As the next step we want to consider the restrictions of the von Neumannalgebras R Λ and R Λ c to H Λ . By [40, Proposition II.3.10] both restrictions areagain von Neumann algebras. Definition 4.12.
For any cone Λ ⊂ Γ we write A Λ := P Λ R Λ P Λ ↾ H Λ and B Λ = P Λ R Λ c P Λ ↾ H Λ as subalgebras of B ( H Λ ).By using similar techniques as in the proof of the lemmas 4.9 and 4.10 weshow that elements of the form A s + iB s with A s ∈ A s and B s ∈ B s alreadygenerate H Λ when applied on the ground state vector. Here A s is the self-adjointpart of A Λ , and similarly for B s . Lemma 4.13.
Let A s be the self-adjoint part of A Λ and B s that of B Λ . Thenthe set A s Ω + i B s Ω is dense in H Λ .Proof. First note that since both A s and B s are real vector spaces it suffices toshow for operators F ∈ F Λ that F Ω and iF Ω are contained in A s Ω + i B s Ω. Inorder to do so we first show this to hold if F is a finite product of ribbon operatorsin F Λ and then conclude for general operators F ∈ F Λ by a density argument.Essential now is the structure of the vector space H Λ that we elaborated onearlier in Lemma 4.6 and in the proofs of Lemma 4.9 and 4.10. This is to saythat finite products of ribbon operators in F Λ applied to the vacuum vector Ωsufficiently describe H Λ and certain ribbon operators in F Λ c map Ω to vectorsin H Λ and can be expressed as the images of Ω of certain elements of F Λ .Throughout the proof we consider ribbon operators labelled by irreduciblerepresentations of the quantum double model and we can assume that the label isnontrivial for if it was trivial we just obtain the identity operator. Again we willuse the charge projections D χ,cs introduced in equation (2.8) which project ontothe excitation given by ( χ, c ) at site s . Especially recall that D s = D id ,es = A s B s .Now let F , . . . , F n ∈ F Λ be ribbon operators with n > F := F · · · F n .The idea is to construct self-adjoint elements of A s and B s by taking linearcombinations of products of projections A s , B s and products of ribbon operatorsin F Λ and F Λ c . These self-adjoint operators are chosen in such a way that theymap the state vector to the same vector as F . Again, as in previous proofs, we24ill work with an induction over the number of ribbon operators in F . Withthe same argument as in the proof of Lemma 4.10 we can assume that in F there are no ribbon operators associated to closed ribbons or trivial ribbons.Let n = 1 and let ρ denote the corresponding ribbon. In case that both thestar and the plaquette at least one of the ending sites of ρ , denoted by s , arecontained in Λ we set˜ F := F D s + D s F ∗ and ˆ F := i ( F D s − D s F ∗ ) . Obviously these operators are selfadjoint hence contained in A s and it can easilybe checked that ˜ F Ω = F Ω and ˆ F Ω = iF Ω. Therefore F Ω and iF Ω belong to A s Ω.Assume that at both ends of ρ are contained in Λ but the plaquettes atboth sites are not contained in Λ. Then the stars are still contained in Λ, bydefinition (c.f. Definition 4.3 and the discussion after) and the star operatorsare elements of A s . In case [ F, A s ] = 0, with s = ∂ ρ or s = ∂ ρ , it suffices totake ˜ F := F A s + A s F ∗ and ˆ F := i ( F A s − A s F ∗ ) , since then ˜ F Ω = F Ω + δ χ, id Ω = F Ω an analogously ˆ F Ω = iF Ω where χ ispart of the label of F . These operators are selfadjoint and ˜ F , ˆ F ∈ A s hence F Ω , iF Ω ∈ A s Ω.If, however, [
F, A s ] = 0 we can use Lemma 2.7 to extend ρ with triangles τ, ˜ τ such that ˜ ρ := τ ρ ˜ τ is a ribbon, and ∂ ˜ ρ, ∂ ˜ ρ ∈ ∂ Λ. Furthermore we thenhave F ˜ ρ Ω = F Ω. But now we can invoke Lemma 2.4 and Lemma 2.3 to find aribbon ρ ⊂ Λ c such that F Ω = F ∗ ρ Ω. Now we can set˜ F := 12 ( F + F ∗ ) + i (cid:18) i F ρ − F ∗ ρ ) (cid:19) and it can easily be checked that the “real part” of ˜ F is an element of A s and the “imaginary part” one of B s , hence ˜ F ∈ A s + i B s . By construction F Ω = ˜ F Ω ∈ A s Ω + i B s Ω. Similarlyˆ F := i F − F ∗ ) + i F ρ + F ∗ ρ )and iF Ω = ˆ F Ω ∈ A s Ω + i B Ω .We now proceed by induction. Let n > F , . . . , F n − ∈ F Λ . Let F , . . . , F n ∈ F Λ be anynon-trivial ribbon operators. If one of them was trivial then we could remove itand obtained n − n − ρ i , ρ k with 1 ≤ i, k ≤ n such that theystart and end at the same site. Then either ∂ j ρ i = ∂ j ρ k , j = 0 , ρ i ρ k is aclosed ribbon. In either case in the product F · · · F n we can bring F i and F k tothe right by using the commutation relations of ribbon operators. Then we canuse equation (4.3) and the remark after equation (2.7) to replace F i F j in frontof Ω with a single ribbon operator. If ρ i ρ k is closed then we have F χ,ci F ξ,dj Ω =25 χξ,cdρ i Ω. In case ∂ j ρ i = ∂ j ρ k , j = 0 , F χ,ci F ξ,dj Ω = F χξ,cdi Ω. Again( χ, c ) and ( ξ, d ) are irreducible representations of D ( G ). That is, in both caseswe end up with a product of n − F · · · F n each ribbon involved thereis appearing exactly once.The rest of the proof can be divided into three main cases. Let again F · · · F n be the product of non-trivial ribbon operators in F Λ . Assume that there are nosuch ribbons as in the previous case. Then there are three possibilities: eitherthere exists a ribbon ρ involved in the product such that D ∂ i ρ ∈ F Λ for at leastone i = 0 ,
1, or all ribbons end at ∂ Λ, or neither of both, i.e. ∂ i ρ / ∈ ∂ Λ and D ∂ i ρ / ∈ F Λ .Consider the first main case, namely that D ∂ i ρ ∈ F Λ for i = 0 or i = 1 forat least one ribbon involved in F · · · F n . We set s := ∂ ρ and without loss ofgenerality we can assume that F ρ = F n and i = 0. If the ribbon operator wasnot F n we could use the commutation relations of ribbon operators to move thisoperator to the last place in the product. We can divide the treatment of thiscase into two different cases. The first case is that there is a site s ∈ Λ suchthat [ F · · · F n , D s ] = 0. In the other case we have that for all sites s ′ ∈ Λ with D s ′ ∈ F Λ it holds [ F · · · F n , D s ′ ] = 0.Now for the first subcase of the first main case. If there is a site s ∈ Λ with[ F · · · F n , D s ] = 0 we can set˜ F := F · · · F n D s + D s F ∗ · · · F ∗ n and ˆ F := iF · · · F n D s − iD s F ∗ · · · F ∗ n . Then ˜
F , ˆ F ∈ A s and it holds F · · · F n Ω = ˜ F Ω and similarly iF · · · F n Ω = ˆ F Ω.The case that for all sites s ′ ∈ Λ with D s ′ ∈ F Λ it holds [ F · · · F n , D s ′ ] = 0can be treated as follows. Since we assumed that there is at least one ribbon ρ involved in the product, the corresponding ribbon operator is either trivial, byLemma 2.6, or there is at least one additional ribbon ending or starting at oneof the endpoints of ρ . We excluded the first case by assumption so we have totreat the second one. Therefore consider the situation where there are k ribbons ρ n − k , . . . , ρ n in F · · · F n ending at s . By Lemma 2.6 the condition that theoperators commute with the charge projector is equivalent to χ n − k · · · χ n = idand c n − k · · · c n = e where χ i are irreducible representations of G and c i ∈ G with i = n − k, . . . , n . But by Lemma 2.10 we have that there are ribbons σ n − k , . . . , σ n − such that they do not cross the site s , a ribbon γ having s asan ending site, irreducible representations ξ n − k , . . . , ξ n − of G and elements d n − k , . . . , d n − ∈ G such that F χ ,c ρ · · · F χ n ,c n ρ n Ω = zF χ ,c ρ · · · F χ n − k − ,c n − k − ρ n − k − F ξ n − k ,d n − k σ n − k · · · F ξ n − ,d n − σ n − F χ,cγ Ωwhere z ∈ C , | z | = 1 and χ = χ n − k · · · χ n and c = c n − k · · · c n . The commutationrelation with the charge projection now tells us that ξ = id and c = e , hence F χ,cγ = I . This gives an expression with n − F · · · F n ∈ F Λ thereare only ribbons ρ i , i ∈ { , . . . , n } involved whose ending sites are containedin ∂ Λ. By definition, c.f. Definition 4.3, it holds for all i ∈ { , . . . , n } that D ∂ k ρ i = F Λ , k = 0 , c and which connect sites on ∂ Λ,26o ribbon operators of ribbons which are contained in Λ and which connect thesame sites, without changing the image of Ω under these operators. Of course,this works the other way round, too. So choosing˜ F := 12 (cid:0) F ρ · · · F ρ n + F ∗ ρ n · · · F ∗ ρ (cid:1) + i (cid:18) i F ˜ ρ · · · F ˜ ρ n − F ∗ ˜ ρ n · · · F ∗ ˜ ρ ) (cid:19) and ˆ F := i (cid:0) F ρ · · · F ρ n − F ∗ ρ n · · · F ∗ ρ (cid:1) + i (cid:0) F ˜ ρ · · · F ˜ ρ n + F ∗ ˜ ρ n · · · F ∗ ˜ ρ (cid:1) will do the job. We used the notation F ρ i instead of F i , i = 1 , . . . , n to indicatethe dependence on the ribbon. As above ˜ ρ i indicates the ribbon obtained byextending ρ i by triangles corresponding to Lemma 2.7 if necessary, and invertingit using Lemma 2.4. Then ˜ F , ˆ F ∈ A s + i B s and it can be easily be verified that˜ F Ω = F · · · F n Ω and ˆ F Ω = iF · · · F n Ω.It remains to treat the third main case. Consider there is no ρ involved inˆ F such that it falls under the two previous main cases. I.e. for any ρ ⊂ Λappearing in ˆ F at least one of the ending sites s i := ∂ i ρ, i = 0 , D s i / ∈ F Λ and s i / ∈ ∂ Λ. Then, by construction of Λ and by the Definition 4.3,the star operators at s i are still contained in Λ, i.e. A s i ∈ F Λ . Furthermorefor each such s i there are triangles τ i ∈ Λ such that τ i ρ or ρτ i is a ribbon and ∂ i τ i ∈ ∂ Λ. There are two cases appearing here: [ A s i , F ] = 0 for any such s i or[ A s i , F ] = 0 for some s i . In case [ A s i , F ] = 0 for any s i we simply set˜ F := F · · · F n A s i + A s i F ∗ · · · F ∗ n and ˆ F := iF · · · F n A s i − iA s i F ∗ · · · F ∗ n . Then ˜
F , ˆ F ∈ A s and it holds F · · · F n Ω = ˜ F Ω and similarly iF · · · F n Ω = ˆ F Ω.In case there is a s i such that [ A s i , F ] = 0 we first deform or invert any ribbon σ involved in F , using Lemma 2.3 and 2.4, such that any of them has s i asfinal site, i.e. ∂ σ = s i and any of them stays in Λ. This gives an expression F Ω = F ′ Ω where F ′ is again a product of ribbon operators in F Λ together witha possible phase factor from the commutation relations. More importantly,[ F ′ , A s i ] = 0. Let σ ′ denote these possibly deformed or inverted ribbons. Then,by Lemma 2.7 there is a triangle τ ∈ Λ such that ∂ τ = s i and for any ribbon σ ′ it holds τ σ ′ ⊂ Λ is a ribbon. Furthermore ∂ τ ∈ ∂ Λ. If we apply this procedureto any of the ending sites s i of ribbons in F for which [ F, A s i ] = 0 we end up atthe situation in the second main case from where we can proceed accordingly.This completes the third main case and also the proof of the claim.With these preparations we are finally in a position to prove the main theo-rem. In particular, the last lemma allows us to use the result of Rieffel and VanDaele mentioned before. Theorem 4.14.
Cone algebras of the quantum double model for finite abeliangroups on the infinite square lattice satisfy Haag duality in the vacuum repre-sentation.More precisely, if Λ ⊂ Γ is a cone then π ( A (Λ c )) ′ = π ( A (Λ)) ′′ . roof. The argument is exactly the same as that given in reference [31]. Forthe convenience of the reader, we will restate it here.It remains to prove A (Λ c ) ′ ⊂ A (Λ) ′′ since, by locality, the other directionalready holds. By construction it holds that A Λ ⊂ B ′ Λ (as sub-algebras of B ( H Λ )) and both, A Λ and B Λ , are von-Neumann algebras on the same Hilbertspace H Λ . Hence, by [37, Theorem 2], the statement of Lemma 4.13 is equivalentto A Λ = B ′ Λ .Furthermore, by [40, Proposition II.3.10], it holds that B ′ Λ = P Λ R ′ Λ c P Λ ↾ H Λ .Now let B ∈ R ′ Λ c and denote B Λ := P Λ BP Λ ↾ H Λ ∈ B ′ Λ . Then B Λ ∈ A Λ and, byLemma 4.8, there exists a unique element A ∈ R Λ such that B Λ = P Λ AP Λ ↾ H Λ .To proof the claim it suffices to show that B = A . Pick any ˆ F ∈ F Λ c and F ∈ F Λ . Then B ˆ F F
Ω = ˆ
F BF
Ω = ˆ
F B Λ F Ω = ˆ
F AF
Ω = A ˆ F F
Ωgiving A = B , by Lemma 4.6, and consequently B ∈ R Λ . One can ask the question if the observable (von Neumann) algebra actuallyis isomorphic to R Λ ⊗ R Λ c if Λ is a cone, so that we can see the cone partand the outside as two separate, independent systems without any correlationsbetween them. This turns out not to be the case, because R Λ is not a factorof Type I (remark that if this was the case Haag duality would follow readily).The proof that these factors are not of Type I given in [29, Thm 5.1] works forgeneral finite groups G . Nevertheless, a slightly weaker condition is true. Ifwe separate the cone Λ from the complement of a slightly bigger cone Λ ′ , theresulting von Neumann algebra is a tensor product of the observable algebrasin the two disjoint regions. This follows from the approximate split propertyFor the convenience of the reader we first recall the precise definition. Definition 5.1.
We say that π has the approximate split property if for eachpair Λ ≪ Λ there is a Type I factor N such that R Λ ⊂ N ⊂ R Λ .The notation Λ ≪ Λ means that Λ ⊂ Λ and that the edges of Λ andΛ are sufficiently far removed. For the models that we study in this paper itis sufficient to demand that there is no star or plaquette that has a non-emptyintersection with both Λ and Λ .The approximate split property is a variant of the split property as it ap-pears in algebraic quantum field theory [9, 10] and in operator algebra [16].The approximate split property also plays a role in the definition of a coneindex that tells us something about the number of superselection sectors thetheory has [32]. There are nice physical consequences of the approximate splitproperty: it implies a certain statistical independence of the regions Λ and Λ c .In particular one can find normal product states across these regions, so it ispossible to find states which do not violate Bell’s inequality [38]. In fact onecan locally (in the sense that one acts only with operators in R Λ or R Λ c ) suchproduct states [43]. We note again that in previous work we called this the distal split property. ω is actually a product state, when one restricts to regions that are sufficientlyfar away. More concretely, let Ω be the GNS vector for ω . We will write ω again for the state on R Λ ∨ R Λ c induced by the vector Ω. Note that it isnormal, since it is a vector state for the von Neumann algebra. Note that weremarked before that ω is actually a product state for A (Λ ) and A (Λ c ) if theboundaries of Λ and Λ are sufficiently far apart. This is precisely guaranteedby the condition Λ ≪ Λ . One can then show what ω ( AB ) = ω ( A ) ω ( B ) if A ∈ R Λ and B ∈ R Λ c . The approximate split property then follows from thesame proof as given in [29, Thm. 5.2].Another way to prove the approximate split property is to explicitly con-struct a unitary that as in [31]. We do not attempt a proof along these lineshere, although we believe that using the techniques developed above for theproof of Haag duality, the proof carries over to the present situation withoutmuch changes. Indeed, the main idea behind the proof is to remove some ofthe ambiguity in the description of a vector in the form F · · · F n Ω due to theinvariance of states under ribbon deformations. This can be done using thesame techniques as employed above. This explicit construction can be helpfulin the calculation of the cone index in concrete examples [32], but for our presentpurposes it is not necessary.
As an application of Haag duality we outline the sector theory for abelian groups G , in the spirit of the Doplicher-Haag-Roberts programme in algebraic QFT [14,15]. The goal here is to retrieve all properties of the superselection sectors (orcharges) in the theory, from a few basic principles. We will construct equivalenceclasses of such sectors for quantum double models for abelian groups G , andshow explicitly how one can obtain the fusion and braiding rules. The techniquesthat we will use here were developed in [29], which essentially deals with thecase G = Z . The main ideas are the same in the case of general abelian G ,hence we will focus here on those steps that are different.The goal is to characterise “single charge” representations. These represen-tations describe how the observables of the system change in the presence ofa single charge (or quasi-particle excitation) in the background. The differ-ent superselection sectors or charges correspond to equivalence classes of irre-ducible representations of A [21]. This implies that vector states in inequivalent“charged” representations can not be coherently superposed. Alternatively onecan see that by local operations one cannot transform a vector state in onesuch irreducible representation into a vector state of another (inequivalent) ir-reducible representation. Physically this means that one cannot change thetotal “charge” of the system with local operations. This is exemplified in thequantum double model by the property that the ribbon operators always createa pair of conjugate charges, hence they do not change the total charge of thesystem.There are very many equivalence classes of irreducible representations, mostof which do not carry any reasonable physical interpretation. It is therefore29ecessary to restrict the representations of interest. Recall that in the class ofmodels that we are interested in, excitations or charges can be obtained fromthe ground state by applying a ribbon operator. Note that this always gives us a pair of excitations if the model is defined on the plane. The idea is then to moveone end of the ribbon (or physically, one of the charges) to infinity. For a relatedconstruction of charged states in Z N Higgs models, see for example [3, 18]. Thecharge at the fixed endpoint can only be detected by measuring a “Wilson loop”that encloses the charge. Hence if we disallow operators that form a loop aroundthe charge, it cannot be detected and the state will look like the ground statefor such measurements.What does this mean for the corresponding representations, obtained viathe GNS construction for example? One can choose any cone, and restrict tomeasurements outside such a cone. By the argument above this should look likethe ground state representation. We therefore restrict to those representationsthat satisfy (c.f. [11, 29]) π ↾ A (Λ c ) ∼ = π ↾ A (Λ c ) (6.1)for any cone Λ. That is, the representation π is unitary equivalent to theground state representation, but only when one restricts to observables outside a cone. Equation (6.1) is called a selection criterion . The construction of suchrepresentations that we will outline below will make clear why this is a physicallyreasonable criterion. We stress that equation (6.1) should hold for all cones Λ(where the unitary setting up the equivalence may depend on the cone).By finding all representations that satisfy this criterion one finds a list of allcharges that the system supports. But one can recover much more structure,and this is the point where Haag duality comes in: using Haag duality we caninstead look at maps of A into a slightly bigger algebra, and in fact these mapscan be extended to endomorphisms of this bigger algebra. To see this, fix acone Λ and let V be the unitary such that π ( A ) = V π ( A ) V ∗ for all A ∈ A (Λ c ).Then define α ( A ) = V π ( A ) V ∗ for all A ∈ A . Then we have that, for A ∈ A (Λ)and B ∈ A (Λ c ), π ( B ) α ( A ) = V π ( BA ) V ∗ = V π ( AB ) V ∗ = α ( A ) π ( B ) . Hence by Haag duality it follows that α ( A ) ∈ R Λ . As mentioned, α can beextended to a proper endomorphism. That is, one can introduce an auxiliaryalgebra A Λ a (where Λ a is a fixed cone), such that the map ρ can be extendedto an endomorphism of A Λ a [11, 29]. This is mainly a technical issue whichwe will large suppress here. In the explicit construction of such maps α below,it turns out that we can even restrict to automorphisms of A , although forthe construction of braiding operators the extension to the auxiliary algebra isnecessary. In addition, with Haag duality it follows that all the results thatwe show for these automorphisms are true for any representative in the sameequivalence class, even if it cannot be restricted to an automorphism of A .The advantage of using automorphisms or endomorphisms is that these canbe composed, unlike representations. That is, we can define α ⊗ β := α ◦ β . Theinterpretation is that we first add a charge β , then a charge α . In addition, if S is an intertwiner from α to α , meaning Sα ( A ) = α ( A ) S for all A ∈ A , and T is an intertwiner from β → β , it follows that S ⊗ T := Sα ( T ) is an intertwinerfrom α ⊗ β to α ⊗ β . Using Haag duality and the extension of α to the30uxiliary algebra one can show that this is well-defined. This makes the categoryof localised and transportable (which we will discuss below) endomorphisms,with as morphisms the intertwiners. Studying the superselection sectors is thenstudying the properties of this category. In this case this amounts to showingthat it is in fact the representation category of the quantum double of the group G . This is a modular tensor category [2], as is appropriate for applicationsto quantum computing [26, 42]. However note that we only consider abelianmodels at the moment, which from a quantum computation point of view areless interesting. We comment briefly on this point at the end of this paper. The first task is to construct different equivalence classes of representations sat-isfying the selection criterion. We already mentioned that the ribbon operatorscreate a pair of excitations. We will use this fact to first construct “chargedstates”, from which the representations can be obtained straightforwardly. Asexpected, to each element c ∈ G and irreducible representation χ of G (that is,a character), we can associate an equivalence class of representations. To thisend, fix a cone Λ and consider a semi-infinite ribbon ρ inside Λ. That is, oneend of ρ is fixed, the other end is thought of to be sent to infinity. The ribbonconsisting of the first n triangles will be denoted by ρ n . We associate an en-domorphism (in fact, since the model is abelian this will be an automorphism)to the each pair ( χ, c ) and semi-infinite ribbon ρ . In the next sub-section wewill show that the choice of ribbon is not important, in the sense that anotherchoice will lead to a unitarily equivalent automorphism.The operators F χ,cρ defined in equation (2.7) create a pair with charge ( χ, c )at the start of ρ and its conjugate at the other end. Therefore one can thinkof the following map as describing the effect of the presence of this pair on anobservable A : α χ,cρ ( A ) := (Ad F χ,cρ )( A ) = F χ,cρ A ( F χ,cρ ) ∗ . Note that since F χ,cρ is unitary this map is an automorphism. The idea is totake the limit in which we extend ρ to infinity. The next proposition shows thatthis indeed works. Proposition 6.1.
Let ρ be a ribbon extending to infinity, and denote ρ n for theribbon consisting of the first n triangles of ρ . Suppose that ( χ, c ) is as above.Then for each A ∈ A the limit α ( A ) := lim n →∞ α χ,cρ n ( A ) (6.2) converges in norm and this defines an automorphism α : A → A . This map hasthe following properties:(i) α ( A ) = A for A ∈ A with supp( A ) disjoint from ρ ;(ii) If A ∈ A loc , then α ( A ) = α χ,c b ρ ( A ) for any ribbon b ρ ⊂ ρ such that supp( A ) ∩ ρ ⊂ b ρ .The last property says that it is enough to move one end of the ribbon far enoughaway so that it is disjoint from the support of a local observable A . roof. Let A ∈ A be a local operator. Then since ρ goes to infinity, there issome N such that supp( A ) ∩ ( ρ \ ρ n ) = ∅ for all n > N . In addition, fromLemma 2.2 it follows that F χ,cρ n = F χ,cρ N F χ,cρ n \ ρ N . Because of locality and because F χ,cρ N \ ρ n is unitary, it is clear that the limit converges in the operator norm,since the sequence α χ,cρ n ( A ) is eventually constant. Note that this is essentiallyproperty (ii). The maps are clearly bounded, hence by continuity they can beextended to a map α of A . An easy check along the lines above shows that α ( AB ) = α ( A ) α ( B ) and α ( A ∗ ) = α ( A ) ∗ .We still have to show that α is an automorphism. The easiest way to do thisis by constructing an inverse. Consider ( χ, c ) where χ is the complex conjugateof the character χ , which again is a character. A simple calculation showsthat F χ,cρ F χ,cρ = I , and the same holds with the order reversed. It followsthat α which is defined in the same way as α , but with the pair ( χ, c ) satisfies α ◦ α ( A ) = α ◦ α ( A ) = A for all local A , hence it is the inverse of α .Property (i) immediately follows from locality.As we will see later the automorphisms constructed above give us repre-sentatives of the equivalence classes of representations satisfying the selectioncriterion. Anticipating this, we will also write α χ,c for an automorphism definedin such a way, or even α χ,cρ if we want to emphasize the ribbon to infinity. Notethat each pair ( χ, c ) gives rise to many different automorphisms, since one canchoose many different ribbons. If the ribbon is not important, we sometimesrefer to any representative of this class of automorphisms by ( χ, c ). The fol-lowing proposition shows that the automorphisms associated to different pairs( χ, c ) belong to different superselection sectors, as expected. The idea behindthe proof is that one can always detect the total charge in any finite region bypulling a charge and its conjugate from the vacuum, moving one charge aroundthe region, and fusing again. Proposition 6.2. If ( σ, c ) = ( χ, d ) then the corresponding localised automor-phisms belong to different superselection sectors.Proof. Write ( π , H , Ω) for the GNS triple corresponding to the ground state ω . Note that since ω is pure it follows that π is irreducible. Because α σ,c is an automorphism, π ◦ α σ,c is also irreducible and ( π ◦ α σ,c , H , Ω) is a GNStriple for the state ω ◦ α σ,c . A similar statement is of course true for the state ω ◦ α χ,d . To prove the claim it therefore suffices to show that the two states canbe distinguished by an operator localised outside some arbitrary finite region O by Corollary 2.6.11 of [7]. This is true because quasi-equivalent irreduciblerepresentations are unitarily equivalent.Now let O be any finite set. Then we can find a closed rotationally invariantribbon b ρ encircling the region O and such that the endpoint of the ribbon ρ that extends to infinity lies in the bounded area encircled by b ρ . To this ribbonwe associate the projection K σc , projecting onto the subspace of charge ( σ, c )in the region enclosed by b ρ . It is defined as follows (c.f. equation (B.75) of [6]): K σc b ρ = 1 | G | X g ∈ G σ ( g ) F g,c b ρ . If ( σ, c ) = ( χ, d ) it follows that (by the discussion in Appendix B.9 of [6]) (cid:12)(cid:12)(cid:12) ω ◦ α σ,c ( K σ,c b ρ ) − ω ◦ α χ,d ( K σ,c b ρ ) (cid:12)(cid:12)(cid:12) = | − | ≥ (cid:13)(cid:13)(cid:13) K σ,c b ρ (cid:13)(cid:13)(cid:13) . σ, c ) belong to the same sector. Suppose that we have an automorphism α as defined above such that α islocalised in a cone Λ. Then α is said to be transportable if for any cone Λ ′ ,there is an automorphism β localised in Λ ′ such that α is unitarily equivalentto β . This unitary does not need to be in A (and generally also is not), but byHaag duality it follows that if b Λ is a cone containing both Λ and Λ ′ , then anyunitary V setting up such an equivalence is contained in R b Λ . Such a unitarywill also be called a charge transporter . We first show that the automorphismsare indeed transportable, and then give an explicit sequence that converges inthe weak operator topology to a charge transporter. The proof largely followsthe proof in the toric code case (up to some subtleties) [29], but since we needthe construction to calculate the statistics, we recall the main line of argument.Fix a pair ( χ, c ) and two semi-finite ribbons ρ i , i = 1 ,
2, with correspondingautomorphisms α i . First consider the case that both ribbons start at the samesite. With Lemma 2.3 one can show that the states ω ◦ α i are equal, by firstshowing equality on the dense subset of local observables. On the other hand,as was remarked in the proof of Proposition 6.2, both representations π ◦ α i are GNS representations for this state. Hence by the uniqueness of the GNSrepresentation, the two are unitarily equivalent. Note that in addition we mayassume that such a unitary intertwiner V satisfies V Ω = Ω. Requiring this willfix an irrelevant phase.Suppose now that the two ribbons do not start at the same site and thatwe consider a charge ( χ, c ). Then we get corresponding automorphisms α and α . We can then extend the ribbon ρ by a ribbon ρ , such that ρ and ρ startat the same site. This gives us an automorphism α ρρ , defined in terms ofthe extended ribbon, that is unitarily equivalent to α , by the argument in theprevious paragraph. The claim follows by noting that α and α ρρ are unitarilyequivalent. This can be seen because F χ,cρ is a unitary operator, and F χ,cρ α ρρ ( A )( F χ,cρ ) ∗ = α ( A ) . This can be seen by noting that if a ribbon ρ coincides with the first part ofa ribbon b ρ , then F χ,cρ F χ,c b ρ = F χ,c b ρ \ ρ . This equality can be easily verified usingequation (2.1).For the calculation of the braiding rules of the anyons, which we will out-line below, it is useful to have a more explicit description of the intertwinerssetting up the equivalence. To this end we construct a sequence V n of unitariesconverging to V in the weak operator topology. For simplicity we again assumethat the two semi-infinite ribbons ρ and ρ start at the same site. With ρ ni wemean the finite ribbon consisting of the first n triangles of the ribbon ρ i . Foreach n , choose a ribbon b ρ n from the site at the end of ρ n to the site at the endof ρ n , in such a way that ρ n b ρ n is a ribbon and the distance of b ρ n to the (fixed)starting point of ρ i goes to infinity as n → ∞ . This ensures that the ribbons b ρ n n is large enough. By Lemma 6.1it also follows that for n large enough, α n ( A ) = F χ,cρ n AF χ,cρ n for A strictly local.Define V n = F χ,cρ n F χ,cρ n b ρ n . The claim is that the sequence V n converges to V . Using the remark above about strictly local observables, a straightforwardcalculation shows that V n α ( A ) = α ( A ) V n if A is local and n is big enough.Another remark is that using the techniques that we employed in the proof ofHaag duality, it follows that V n Ω = Ω. To see this, note that F χ,cρ ′ n and F χ,cρ n create opposite charges at the endpoints of the ribbons. Since all charges areabelian, these opposite charges fuse to the vacuum. This can be seen explicitlyby using that F h,gρ Ω only depends on the endpoints of ρ , hence we can use thisto change the path ρ ′ n to ρ n when acting on the ground state. Since F χ,cρ n F χ,cρ n = F id ,eρ n , the claim follows.With these observations, we find for A and B strictly local operators and n large enough, that h α ( A )Ω , V α ( B )Ω i = h α ( A )Ω , α ( B ) V Ω i = h α ( A )Ω , α ( B ) V n Ω i = h α ( A )Ω , V n α ( B )Ω i . Since α is an automorphism, it follows that the set α ( A ) for local operators A is dense in the Hilbert space. Because the sequence V n is uniformly bounded, itfollows that V n indeed converges to V . Note that if Λ is a cone containing bothribbons ρ and ρ we can choose V n ∈ A (Λ) and consequently V ∈ A (Λ) ′′ = R Λ ,as also follows from Haag duality.The discussion so far can be summarised as the following theorem. Theorem 6.3.
Let G be a finite abelian group and let π be the ground staterepresentation of the quantum double model for G . Then for each pair ( χ, c ) where χ is a character of G and c ∈ G , there is an equivalence class of rep-resentations satisfying the selection criterion (6.1) . The representation π ◦ α ,where α is localised in some cone Λ and constructed as above, is a representativeof such an equivalence class. The equivalence classes corresponding to distinctpairs ( χ, c ) are disjoint. Fusion rules tell us what happens if we combine (“fuse”) to charges. Moreprecisely, they give a decomposition of the tensor product α ⊗ β of irreducibleendomorphisms as a direct sum of such endomorphisms. The fusion rules areindependent of the chosen representatives. Hence it suffices to fix a cone Λ anda path ρ to infinity inside this cone. We can then consider automorphisms α χ,c defined as above, acting along the ribbon ρ . Note that by Proposition 6.1(ii), forlocal observables it is enough to consider only finite parts ρ n of the path ρ . Notethat for any finite ribbon ξ we have F χ,cρ F σ,dρ = F χσ,cdρ as was remarked afterequation (2.7), where χσ is the character obtained by pointwise multiplication.Hence we find the fusion rules α χ,c ⊗ α σ,d ∼ = α χσ,cd . Note that in particular we see that the conjugate charge of ( χ, c ) is ( χ, c ).To study the statistics we have to relate α ⊗ β to β ⊗ α . For the constructionwe need to be able to talk about the relative position of two charges localised in34 ˆ ρ ρ Λ Λ ˆ Λ Λ a Λ Figure 7: The choice of auxiliary cone Λ a , as well as ribbons ρ and ρ that weuse in the calculation of the braiding operators. The idea is to move the chargeat the end of the ribbon ρ to the end of the ribbon b ρ , while the other chargestays in place.cones. That is, we want to say that one cone is to the left of the other one. Thiscan be done unambiguously by fixing an auxiliary cone: for convenience onecan take the cone Λ a briefly mentioned above. Then we can define a relationΛ < Λ for two disjoint cones (see [29] for details). This singling out of aparticular direction is analogous to the technique of puncturing the circle in,for example, conformal field theory. Alternatively one can cover the lattice bydifferent “charts” as in [20].Now suppose that we have two charges α and β localised in cones Λ andΛ . To construct a unitary ε α,β intertwining α ⊗ β and β ⊗ α , first choose a cone b Λ to the left of Λ (and disjoint of it), see Fig. 7. Then there is an intertwiner V transporting the charge β in Λ to a charge b β in the cone b Λ . Since b Λ andΛ are disjoint, it follows by the localisation properties of the automorphismsthat α ⊗ b β = b β ⊗ α . Finally, the charge b β can be transported back to the coneΛ . Note that the physical picture is precisely what one would think of as abraiding operation. This procedure leads to the following expression, which onecan show depends only on the position of the cone b Λ relative to Λ , not on thespecific choice of V (c.f. [19]): ε α,β = V ∗ α ( V ) . This unitary intertwines α ⊗ β and β ⊗ α . One can show that it has all theproperties a “braiding” should have (compare for example with [22]).Note that in the previous section we have constructed a sequence V n con-verging to V in the weak operator topology. Since α can be extended to aweakly continuous map on the auxiliary algebra A Λ a , we can calculate α ( V ) =w-lim n →∞ α ( V n ). We are interested here in the calculation of the modular ma-trix S , whose entries are in the present case given by S α,β = ε α,β ◦ ε β,α . Notethat because of irreducibility of α ⊗ β this is an element of C I and hence canbe identified with a scalar. It only depends on the equivalence classes of α and β , so that we can choose representatives in a convenient way. We do this as inFig. 7: we choose two non-intersecting ribbons ρ i that can be localised in thesame cone Λ. For the transported automorphisms we choose b ρ = ρ and for b ρ a ribbon to the left of the cone Λ, such that it is inside a cone b Λ that is35isjoint from Λ. A sequence V n of charge transporters can then be constructedas in Section 6.2, and it remains to calculate α ( V n ).This amounts to a straightforward application of the definitions. For conve-nience we can choose the ribbons connecting the n -th triangle of ρ and b ρ insuch a way that they cross the ribbon ρ exactly once. Now note that for each n >
0, there is an integer N ( n ) such that α ( V n ) = F χ ,c ( ρ ) N ( n ) V n ( F χ ,c ( ρ ) N ( n ) ) ∗ , by Proposition 6.1. Note that by construction V n is a product of two ribbonoperators. Since the ribbon on which V n is defined and the ribbon ρ crossexactly once, we can commute V n with the ribbon operator on the left of it inthe expression above, at the expense of a phase according to equation (2.11). Itfollows that α ( V ) = χ ( d ) χ ( c ) V and hence ε α,β = χ ( d ) χ ( c ) I .The operator ε β,α can be found in the same way: we move the charge α tothe left cone (and back). Since in this case the ribbons used in the constructionof the appropriate intertwiner W do not cross, it follows that ε β,α = W ∗ β ( W ) = W ∗ W = I . This gives us Verlinde’s matrix S [41], whose entries are S α,β = ε α,β ◦ ε β,α in the special case that each sector is abelian (as is the case here).A more thorough discussion of S in the context of the theory of superselectionsectors can be found in [35, 36]. In the end we obtain S ( χ ,c ) , ( χ ,d ) = χ ( d ) χ ( c ) . This is (up to a factor due to a different choice of normalization), preciselythe matrix obtained in [2, Thm. 3.2.1] for the representation category of thequantum double D ( G ).This is of course no coincidence. There is a correspondence between thesuperselection sectors constructed here and the finite dimensional representa-tions of D ( G ), seen as a Hopf algebra. It is well known that the irreduciblerepresentations of D ( G ) are in one-one correspondence with pairs consisting ofan equivalence class of G and irreducible representations of the centraliser of arepresentative of this equivalence class (see for example [2, 4]). In the presentcase of abelian groups this reduces to the pairs ( χ, c ). The fusion rules estab-lished above are precisely those obtained from the representation theory. Witha little bit of work one can in fact show that the sector theory is completelydetermined by the representation theory of D ( G ), where the D ( G )-linear mapsbetween finite dimensional representations correspond to intertwiners betweenthe sectors constructed here. In the language of category theory, this can bephrased as stating that the category of localised endomorphisms and the cat-egory of finite dimensional representations of D ( G ) are equivalent as braidedfusion categories. With the help of the results above, the arguments are verysimilar to the toric code case [29], and hence we will not repeat them here. Inany case, the upshot is that understanding the sector theory is the same asunderstanding the representation theory of D ( G ) (a well studied subject), andthat all physical properties of the excitations can be obtained by representationtheory.There is still a point that has not been answered, however. In principle, itmay be the case that we have not constructed all sectors. That is, there may beirreducible representations that satisfy the selection criterion (6.1), but are notunitarily equivalent to one of the charged representations constructed above.36he question if such additional charges exist can be answered by computingthe Jones-Kosaki-Longo index for pairs of cones [32]. In essence one has toconsider two disjoint cones Λ ∪ Λ and the von Neumann algebra generatedby the local observables in these cones, and in addition the algebra generatedby the commutant of everything in the complement of the two cones. Thisalgebra contains the von Neumann algebra generated by the observables insidethe cones, but also charge transporters that move charges from one cone to theother. The Jones-Kosaki-Longo attaches a number to the relative size of thesetwo algebras, and one can show that it is related to the quantum dimensionof the charges. We expect that the proof given in the toric code case [32] canbe extended to quantum double models for abelian G , by using the methodsdeveloped in this paper. The techniques used are very close to the proof ofHaag duality given in Section 4, and therefore we do not attempt to give a fullproof here. In any case, we expect that this index is equal to | G | , and that wehave therefore found all charges in the model. Many of the results so far are only proven for abelian groups G . A natural ques-tion is if the same results hold for non-abelian groups. Such models are partic-ularly interesting because they have non-abelian anyons, for which the braidingoperators are not just a phase, but in general give rise to higher dimensionalrepresentations of the braid group. In Kitaev’s model this can be exploitedto implement unitary operations ( gates ) from which quantum circuits can bebuild. Such quantum circuits perform quantum computation tasks. Under suit-able conditions on the group G Kitaev’s model is in fact universal , meaning thatin principle any quantum computation algorithm could be implemented on topof the model [27, 28].We believe that these non-abelian models can be studied along the samelines as the abelian ones. From a technical point of view, however, the anal-ysis is much more involved. The difficulties mainly stem from the fact thatfor non-abelian G the quantum double D ( G ) has higher dimensional irreduciblerepresentations. This has a few consequences. First of all, rather than a single ribbon operator being associated with a certain irreducible representation (suchas the F χ,cρ we used above), one has to deal with “multiplets” of ribbon oper-ators, see for example equation (B.66) of [6]. When acting with the operatorsin such a multiplet on the ground state Ω, one can span a finite dimensionalvector space. The star and plaquette operators at one of the endpoints of theribbon give a natural action of the quantum double on this finite dimensionalvector space, which then transforms as an irreducible representation under thisaction.The second point is related to the tensor product of two irreducible represen-tations. In the abelian case such a tensor product was again irreducible and ofthe same form. This is no longer true in general in the non-abelian case. Indeed,there are tensor products of representations that are the direct sum of more thanone irreducible representation. On the level of the ribbon operators this has, forexample, the consequence that when we multiply two ribbon operators actingalong the same ribbon, in general it is not of the form of a single ribbon operatorany more. This naturally makes the analysis more complicated. In addition the37nterchange of two ribbon operators is more complicated than just introducinga phase. Nevertheless, the representation theory of the quantum double is wellunderstood, so we expect that the main ideas in our proof can be transferredto the non-abelian case. In particular, one should be able to use this knowledgeof the representation theory to study the commutation properties of the ribbonoperators, which are essential in the proof of Haag duality.The non-abelianness also makes it more difficult to explicitly construct rep-resentatives of the charged sectors: instead of automorphisms one has to dealwith endomorphisms and we cannot just conjugate with the ribbon operatorsto define them. Instead, one way would be to use amplimorphisms , which arenothing but morphisms ρ : A → M n ( A ), the n -by- n matrices with entries in A .Such methods have been employed to describe localised (in intervals) charges inquantum spin systems on the line [33, 39]. Unlike in the case of finitely localisedexcitations, in the case of conelike localisation we expect to be able to obtainproper endomorphisms again. One way to do this is to note that the cone al-gebras are infinite factors. This allows us to find isometries V i ( i = 1 , . . . , n ) inthe cone algebra whose ranges sum up to the identity projection. In this way wecan identify H with ⊕ ni =1 H , and obtain an identification of an amplimorphism ρ as above with an endomorphism of the cone algebra. This should make itpossible to carry over the well-known structure of the amplimorphisms to en-domorphisms, and build up representatives of each sector and find the braidingoperators. We hope to return to this issue in the future. Acknowledgements:
LF is supported by the European Research Coun-cil (ERC) through the Discrete Quantum Simulator (DQSIM) project. PN issupported by the Dutch Organisation for Scientific Research (NWO) through aRubicon grant and partly through the EU project QFTCMPS and the clusterof excellence EXC 201 Quantum Engineering and Space-Time Research.
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