The double semion state in infinite volume

We describe in a simple setting how to extract a unitary braided fusion category from a collection of superselection sectors of a two-dimensional quantum spin system, corresponding to abelian anyons. The structure of the unitary braided fusion category is given by F and R-symbols, which describe fusion and braiding of the anyons. We then construct the double semion state in infinite volume and extract the unitary braided fusion category describing its semion, anti-semion, and bound state excitations. We verify that this category corresponds to the representation category of the twisted quantum double of Z_2.


Introduction
Gapped ground states of two dimensional quantum lattice systems may support anyonic exitations.Paradigmatic examples include Kitaev's quantum double models [12] and, more generally, the string net models of Levin and Wen [13].It is widely believed that the types of anyons supported by a given ground state are organized in a unitary braided fusion category, whose objects are the anyon types, and whose structure is described by F and R-matrices.
Recent years have seen a lot of progress towards justifying this belief, by adapting the DHR analysis of superselection sectors in algebraic quantum field theory [5,6,7,8,9] to the setting of microscopic lattice systems [14,3,15].This body of work manages to associate to a large class of gapped ground states a strict braided C * -tensor category whose objects are localized and transportable endomorphisms of the observable algebra, and shows that this category is a robust label for the gapped phase [1] to which the ground state belongs.
To obtain a unitary braided fusion category described in terms of F and R-symbols, one may skeletonise the braided C * -tensor category refered to above, yielding in particular a microscopic definition of the F and R-symbols.We note that a microscopic definition of F and R-symbols has been given before in [11].Extracting the F and R-symbols from the DHR-type analysis has the advantage that it is clear that this data yields a robust label of gapped phases [3,15].
In section 2 of the paper we review the construction of a braided C * -tensor category from a given ground state under some simplifying assumptions.We assume in particular that the anyons we consider are abelian.By moreover assuming that our set of anyon types is finite and that each anyon has an anti-particle, we obtain by skeletonization a unitary braided fusion category with the anyons types as objects, and structure given by F and R-symbols that satisfy the pentagon and hexagon equations.
Section 3 is devoted to the construction and analysis of the double semion state [13] in infinite volume.This is the simplest state that supports abelian anyons whose unitary braided fusion category has non-trivial F-symbols.
In the appendix we show various properties of the double semion state that are used in Section 3. In particular, the appendix contains a proof that the double semion state is pure.

Braided fusion category for abelian anyons
2.1.Setup. 1 California Institute of Technology, Pasadena, CA 91125, USA 2 QMATH, Department of Mathematical Sciences, University of Copenhagen, Univer-2.1.1.Algebra of observables and state.Consider a countable set Γ ⊂ R 2 of sites in the plane.To each site x ∈ Γ we associate an algebra A x ≃ End(C d ) for some fixed d.For any finite X ⊂ Γ we set A X = x∈X A x .If X ⊂ Y are finite subset of Γ then there is a natural norm-preserving inclusion A X − → A Y by tensoring with the identity.For any infinite subset Y ⊂ Γ we have a local net of algebras A X with X ⊂ Y , whose union is A Y,loc , the algebra of local observables supported in Y .Its completion is A Y := A Y,loc ∥•∥ , the algebra of quasi-local observables supported in Y , and we get inclusions A X − → A Y also for infinite X ⊂ Y .We write A loc = A Γ,loc and A = A Γ for the algebra of all local and all quasi-local observables respectively.The support of an observable O ∈ A is the smallest set X ⊂ Γ such that O ∈ A X .
Similarly, the support of an automorphism w of A is the smallest set Y such that w| A Y c = id A Y c .Any subset S ⊂ R 2 of the plane determines a subset S = S ∩ Γ of Γ, and we denote A S := A S .
A major role is played by cones.The cone with apex at a ∈ R 2 , axis v ∈ R 2 of unit length, and opening angle θ ∈ (0, 2π) is (1) We will consider a pure state ω on A with GNS representation (π 1 , H, Ω).For any X ⊂ Γ we put R(X) := (π 1 (A X )) ′′ , the von Neumann algebra associated to X.For S ⊂ R 2 we also write R(S) = R(S).We note that if Λ is a cone, then R(Λ) is a factor (Theorem 5.2 of [14]).

Superselection sectors.
Definition 2.1.An irreducible representation (π, H) of A is said to satisfy the superselection criterion w.r.t.π 1 if for any cone Λ there is a unitary U ∈ B(H) such that If two representations π, π ′ are unitarily equivalent then we write π ≃ π ′ .We assume that we have a finite set of irreducible representations O = {π a | a ∈ I} of A indexed by a labeling.We assume moreover that π a ≃ π b if and only if a = b, so all sectors in O are truly distinct.Moreover, 1 ∈ I so that π 1 ∈ O.We call π 1 the vacuum sector.
We will now make some additional assumptions on these sectors which will in particular imply that they satify the superselection criterion w.r.t.π 1 .These assumptions are not generically satisfied by gapped ground states, but they are satisfied for a large class of models, for example all abelian string net models [13] including the double semion model studied below.Assumption 1.For any cone Λ and any a ∈ I there is an automorphism w a,Λ supported on Λ such that π a ≃ π 1 • w a,Λ .
The following assumption says that the anyons we study are abelian.Assumption 2. For any a, b ∈ I there is a unique c ∈ I such that for any two cones The following assumption says that each anyon has an antiparticle.Assumption 3.For each a ∈ I there is an a * ∈ I such that a × a * = 1.
The final assumption is of a technical nature, it plays an important role in constructing a tensor category in Section 2.2.
We assume that any unitary V ∈ B(H) implementing this equivalence belongs to the von Neumann algebra R(Λ).
Similarly, it follows from assumption 2 that π 1 • w a,Λ • w b,Λ ≃ π 1 • w a×b,Λ .We assume that any unitary V ∈ B(H) implementing this equivalence belongs to the von Neumann algebra R(Λ).
This assumption is implied by Haag duality, and can also be proven directly if the automorphisms w a,Λ have a lot of structure, for example for the Toric code [14] and for the double semion model treated below.
Assumptions 1-3 have a few elementary but important consequences.
Proof : Fix a cone Λ.By assumption 1 there is an automorphism w a,Λ supported in Λ such that π 1 • w a,Λ ≃ π a .i.e. there is a unitary U ∈ B(H) such that Proof : We first show that × is abelian.Take a, b ∈ I. Assumption 2 says that for any two cones Λ 1 , Λ 2 there are automorphisms Exchanging the roles of a and b and of Λ 1 and Λ 2 we have If we now take Λ 1 and Λ 2 to be disjoint then certainly w a,Λ 1 • w b,Λ 2 = w b,Λ 2 • w a,Λ 1 and therefore π a×b ≃ π b×a .But we assumed that two representations in O are unitarily equivalent only if they are the same, so a × b = b × a.
We now show that 1 is the identity for the product ×.Fix cones Λ 1 and Λ 2 .By assumptions 1 and 2 there are automorphisms w 1,Λ 1 = id and w a,Λ 2 such that We already know that × is abelian, so also a × 1 = a.
Finally, assumption 3 states that a * is the inverse of a. □ We will often write ab = a × b for the product of elements a, b ∈ I.

Braided tensor category.
It is well understood how to associate a braided tensor category to a pure state on a quantum spin system [14,3,15].In this section we recap this constuction in the very simple setting of assumptions 1-4.
2.2.1.Category of automorphisms.Fix a unit vector f ∈ R 2 , representing a 'forbidden direction'.We say a cone Λ a,v,θ with axis v and opening angle θ is forbidden if it contains the forbidden direction f , i.e. if v • f > cos(θ/2).A cone that is not forbidden is said to be allowed.
Let ∆ be the group of automorphisms generated by w a,Λ and their inverses for a ∈ I and Λ allowed.
Since π 1 is a faithful representation of A we will often identify A with its image π 1 (A) and simply write ρ instead of π 1 • ρ for automorphisms ρ on A.
The automorphisms in ∆ are the objects of a C * -category with morphisms Morphisms are referred to as intertwiners.(Direct sums of objects can be constructed as in Lemma 6.1 of [14]).
Lemma 2.4.Each ρ ∈ ∆ is supported on some allowed cone, and there is a unique a ∈ I such that π 1 • ρ ≃ π a .
Proof : Since ∆ is generated by the w a,Λ and their inverses the first claim follows by noting that w −1 a,Λ is supported on Λ, and for any two allowed cones Λ 1 , Λ 2 there is an allowed cone Λ such that Λ ⊃ Λ 1 , Λ 2 , so compositions of automorphisms supported on allowed cones are also supported on allowed cones.For the second claim, note first that where we used assumption 2. Thus, ρ is a finite composition of automorphisms The hom-sets (ρ, σ) are either zero-or one-dimensional.
Proof : By Lemma 2.4 there are unique a, b ∈ I such that π a ≃ π 1 • ρ and π b ≃ π 1 • σ.Each morphism in (ρ, σ) then yields a distinct intertwiner from π a to π b .In particular, the space of intertwiners from π a to π b has at least the dimension of (ρ, σ).Since π a and π b are irreducible representations, this dimension is at most one.□ It follows from the above reasoning that every non-trivial hom-set contains a unitary intertwiner.The need for a forbidden direction will become clear in the next section, where we equip ∆ with a tensor product structure.
Proof : By Lemma 2.4 the automorphism ρ is supported on an allowed cone Λ, and π 1 •ρ ≃ π a for some anyon type a ∈ I.It then follows from assumption 1 that there is a forbidden cone Λ We define the action of ρ on R(Λ) by Ad(V ), which is weakly continuous, and is uniquely determined by the action of ρ on A Λ .Clearly this action on R(Λ) does not depend on the choice of Λ ′ .It follows that the extensions to R(Λ) for different Λ are consistent with each other.Thus the extension ρ is well-defined on all of B. Finally, ρ(A Λ ) = A Λ for any allowed cone Λ ⊃ Λ and weak continuity then implies ρ(R( Λ)) = (ρ(A Λ )) ′′ = R( Λ) for all such cones.□ Lemma 2.7.Let Λ be an allowed cone and denote by ∆ Λ the subgroup of ∆ generated by w a, Λ with Λ ⊂ Λ and their inverses.If ρ, σ ∈ ∆ Λ then (ρ, σ) ⊂ R(Λ).
Since the assumptions on ρ and σ hold for all generators of ∆ Λ , it follows by induction that for all ρ ∈ ∆ Λ the intertwiners (ρ, w a,Λ ) belong to R(Λ).Now take U ∈ (ρ, w a,Λ ) and V ∈ (σ, w a,Λ ), then V * U ∈ R(Λ) is an intertwiner from ρ to σ, and since the (ρ, σ) are at most one-dimensional, this proves the claim.□ We equip the C * -category ∆ with a tensor product structure as follows.For ρ, σ ∈ ∆ we put and for R ∈ (ρ, ρ ′ ) and S ∈ (σ, σ ′ ) we define 2.2.3.Braiding.Consider two automorphisms ρ, σ ∈ ∆ Λ 0 both supported in an allowed cone Λ 0 .Pick allowed cones Λ L and Λ R as in Figure 1.i.e. the disjoint allowed cones Λ R , Λ 0 and Λ L are arranged in a counterclockwise order from the forbidden direction, and there are allowed We sey Λ L is to the left of Λ 0 , and Λ R is to the right of Λ 0 .By Assumption 1 there are automorphisms Definition 2.8.The braiding intertwiner ϵ(ρ, σ) ∈ (ρ ⊗ σ, σ ⊗ ρ) is given by To get the last equality, we use σ R (U ) = U .One verifies that ϵ(ρ, σ) is indeed an intertwiner by the fact that ρ L , σ R commute.
10.The braiding intertwiners satisfy the braid equations where Proof : Let us prove the first equation, the second is shown in the same way.Choose ρ L , σ L supported in Λ L and morphisms  12), defining the F-symbols F (a, b, c).Each node represents a fusion operator.The diagrams represents two different compositions of fusion operators both yielding intertwiners from 3. Braided fusion category.We construct a category whose objects are labeled by the anyon types a ∈ I.This category is obtained by a 'skeletonization' of the category ∆.

2.3.1.
Fusion and F-symbols.Fix an allowed cone Λ 0 and write w a := w a,Λ 0 .Pick unitary intertwiners Ω(a, b) ∈ (w a ⊗ w b , w a×b ) ⊂ R(Λ 0 ) called fusion operators.The Ω(a, b) are unique up to phase.Note that as an automorphism on A, we have Ad(Ω(a, b) These F (a, b, c) are the F -symbols. Figure 2 gives a graphical representation of Eq. ( 12).The F-symbols satisfy a pentagon equation, which in our setting of abelian anyons takes the form of a cocycle relation.
Proposition 2.11.The F-symbols satisfy Proof : A graphical proof is shown in Figure 3.In equations, we have  but also And the desired equality follows.□ The phases R(a, b) are the R-symbols.Figure 4 gives a graphical representation of Eq. ( 14).Proposition 2.12.We have and  Proof : By the braid equations (Proposition 2.10) and the fact that Ad(Ω(a, b)) ∈ ∆, we have Proposition 2.13.The F and R-symbols satisfy the hexagon equations and Proof : The left diagram in Figure 6 suggests the following two equalities of morphisms: and where we used the Yang-Baxter equation to obtain the second line.The coefficients of the right hand sides must be equal, yielding the first hexagon equation.
The second hexagon equation is obtained in exactly the same way, following the right diagram in Figure 6.□ 2.3.5.Braided fusion category.We now consider the category with objects a ∈ I thought of as one-dimensional vector spaces over C. The monoidal structure is given by the fusion rules: The unit is 1 ∈ I and the left and right unitors are simply the identity map.The associators ) are given by multiplication with F (a, b, c).Proposition 2.11 shows that these associators satisfy the pentagon identity, and since F (a, 1, c) = 1 for all a, c, also the triangle identity is satisfied.The braiding ϵ a,b : a⊗b → b⊗a is given by multiplication with R(a, b).Proposition 2.13 shows that this braiding and the associators given by the F-symbols satisfy the hexagon identities.
All this extends uniquely to direct sums.Finally, each anyon a ∈ I has dual a * .
2.3.6.Dependence of F and R-symbols on the choice of Λ 0 and the phases of Ω(a, b).Suppose we chose different phases for the intertwiners Ω(a, b), i.e. we consider for phases χ(a, b).This yields new F-symbols by which are related to the original F-symbols by i.e.F ′ is related to F by the coboundary dχ.It follows that only the cohomology class ) is well-defined.The R-symbols are also affected by the different choice of phases.Indeed, the new R-symbols defined by It follows that the self-statistcs R(a, a) and the double braidings R(a, b)R(b, a) are invariants.Next, we investigate the dependence of the F and R-symbols on the choice of allowed cone Λ 0 .We will find no additional ambiguity beyond the one just discussed.
Let Λ ′ 0 be another allowed cone.Then there is an allowed cone Λ 0 containing Λ 0 ∪Λ ′ 0 .Denote w ′ a = w a,Λ ′ 0 .Then there are unitaries These unitaries are unique up to phase.This leads to new intertwiners Ω ′ (a, b), uniquely defined up to phase by A short computation shows that the right hand side becomes It follows that we can choose phases for the Ω ′ (a, b) such that (Note that we already understood the dependence of the F and R-symbols on the choice of phases of the Ω ′ (a, b), so in this discussion we can choose these phases as it suits us.) The new F-symbols are determined by Using Eq. 28 we compute and Recall that the braiding intertwiners ϵ(a, b) are defined in terms of automorphisms w a supported in Λ 0 , w L a supported in Λ L and w R a supported in Λ R as follows.Pick intertwiners (unique up to phase) U a ∈ (w a , w L a ) and It was shown in Lemma 2.9 that this braiding intertwiner is independent of the choice of the automorphisms w L a and w R a and therefore of the precise choice of cones Λ L and Λ R .Moreover, ϵ(a, b) is independent of the choice of phase for the intertwiners U a and V a .
Let's choose the left and right cones Λ L and Λ R such that they are to the left and right of Λ 0 respectively.
With the new choice of automorphisms w ′ a related to the old w a by Eq. ( 25) we get new intertwiners (33) A short computation relates this to the braiding ϵ(a, b) as The new R-symbol is determined by Using Eqs. ( 28) and (34) the left hand side becomes and We conclude that Eqs. ( 22) and ( 24) are the only ambiguities in the F and R-symbols.
The braided fusion category with the old F and R-symbols is braided monoidally equivalent to the braided fusion category with the new F and R-symbols through an equivalence F which

The double semion state
We construct an infinite volume version of the ground state of the double semion model, first introduced in [13].We identify superselection sectors corresponding to semion, anti-semion, and bound state anyons and find that the braided fusion category describing these anyons corresponds to the representation category of a twisted quantum double algebra D ϕ (Z 2 ).
3.1.Construction of the double semion state.We take Γ to be the (midpoints of the) edges of the hexagonal lattice (Figure 7) and We interpret σ Z x = −1 as the edge x being occupied by a string, while σ Z x = 1 means that the edge is unoccupied.Let ω 0 be the pure product state without any strings, i.e. ω 0 (σ Z x ) = 1 for all x ∈ Γ.
for any hexagon p let and for any finite set Π of hexagons, let Note that A Π produces a string around the region Π when it acts on ω 0 , see Figure 8.Let (π 0 , H 0 , Ω 0 ) be the GNS triple for ω 0 .Let Π n be an increasing sequence of sets of hexagons as in Figure 9 and let where ♯Π is the number of connected components of Π. i.e.Ω n is a superposition of closed string configurations supported in Π n , with phases determined by the parity of the number of components of the string configuration.
The vectors Ω n determine a sequence of pure states ω n on A. The following theorem is proven in the appendix (Appendix A) Theorem 3.1.The sequence ω n converges in the weak-* topology to a pure state ω.
This pure state ω is the double semion state.Let (π 1 , H, Ω) be the GNS triple for ω.

String operators.
Let P be an oriented edge-self-avoiding path in Γ.Following [13], we define three types of non-trivial string operators.The semion string is given by where and j, k are the edges of P that go in and out of the vertex v respectively.The terms 'R-legs' and 'L-vertices' are explained in Figure 10.
The anti-semion string is given by and the bound-state string is given by We further define string operators for the vacuum sector W 1 [P ] = 1, all equal to the identity.
We set I = {1, S, S, B} and we denote by    For any cone Λ whose opening angle is less than π, the edges of ∂Π Λ whose center lies a distance further than n > 2 from the apex of Λ form two half-infinite paths P Let Λ be a cone, and let Λ (L) and Λ (R) be its left and right half-cones, see Figure 12.Take n > 2 sufficiently large such that w a,Λ := w a [P Let i and f be the initial and final edges of the path P (if they exist) and let v[P ] be the automorphism given by conjugation with Lemma 3.3.We have ω • v[P ] = ω for any path P .
Proof : We first take P finite and show that v[P ] leaves any ω n invariant.Recall that ω n is the expectation value of the GNS vector We have (48) Indeed, A Π is a product of σ X j for all j ∈ ∂Π, a closed path.Now, any such closed path supports an even number of factors σ Z j of the unitary V [P ].Indeed, if ∂Π travels along P , then the two edges of P along an R-leg carry no σ Z , while the two edges along an L-vertex both have a σ Z .The closed path ∂Π must enter/leave the path P an even number of times.If it enters through an R-leg, it picks up a σ Z from the R-leg.If it enters through an L-vertex, then it picks up exactly one of the σ Z 's of the two edges of P next to the L-vertex.Finally, if ∂Π enters P through an endpoint of P , then the factors σ Z i , σ Z f at the initial/final edges ensure that a factor σ Z is picked up.In all, we see that V [P ]A Π = A Π V [P ], because the computation involves an even number of commutations of a σ X with a σ Z .Obviously V [P ]Ω 0 = Ω 0 so V [P ]A Π Ω 0 = A Π Ω 0 and V [P ]Ω n = Ω n .It follows that ω n • v[P ] = ω n for any n and any finite P .
It follows immediately that ω • v[P ] = ω for any finite P .If P is infinite, then for any strictly local observalbe O we can find a finite P ′ such that v Since the strictly local observables are dense in A, this proves the claim.

and the unitary implementing this equivalence belongs to the von Neumann algebra R(Λ).
Proof : The unitary equivalence π 1 • v[P ] ≃ π 1 follows immediately from the previous Lemma.Let V be the unitary implementing this equivalence, i.e.
for all O ∈ A, and V Ω = Ω.(We identify A with its image under the faithful representation π 1 ) If P where finite, then actually V ∈ A Λ ⊂ R(Λ).If P is infinite, let P n be the path consisting of edges of P whose midpoints lie in Π n .Then V [P n ] ∈ A Λ has V [P n ]Ω = Ω for all n, and for any strictly local obervables O, O ′ we have Since the vectors OΩ, O ′ Ω for O, O ′ strictly local observables are dense in H, this shows that the sequence V [P n ] converges weakly to V .Since V [P n ] ∈ A Λ for all n, it follows that V ∈ R(Λ).□ Lemma 3.5.For any cone Λ we have that π 1 •w S,Λ •w S,Λ ≃ π 1 , and the unitary V Λ implementing this equivalence belongs to the von Neumann algebra R(Λ).
Proof : By definition, w S,Λ = w S [P Λ ] so w S,Λ • w S,Λ = Ad(σ Z f ) • v[P Λ ] where f is the final edge of the half-infinite path P Λ .From Lemma 3.4, we find that there exists a unitary proving the claim.□ We can now easily show Proposition 3.6.For each cone Λ there are unitaries Ω(a, b) ∈ R(Λ) such that for all a, b ∈ I = {1, S, S, B} and where × is an abelian product on I given by Proof : Lemma 3.5 shows that the claim holds for S × S = 1.The rest of the claim follows from this case and Lemma 3.7.If Λ 1 and Λ 2 are cones with axes ŵ1 and ŵ2 , both contained in a cone Λ and such that ŵ1 points to the right of ŵ2 relative to Λ (see Figure 13).Then Proof : Fix points x 1 , x 2 on the central axes of the cones Λ 1 , Λ 2 such that the region S bounded by the half-infinite parts of these axes starting at x 1 , x 2 , and the line between x 1 and x 2 is convex and has w a,∂Π S supported in Λ, see Figure 13.
By construction, w a,∂Π S differs from w a,Λ 1 •v a,Λ 2 by the action of a local unitary W supported on Λ.Since ∂Π S is a closed path, it follows from Proposition 3.2 that there exists a unitary V ∈ B(H) such that π 1 • w a,∂Π S = Ad(V ) • π 1 and BΩ = Ω, hence This shows the required unitary equivalence.It remains to show that V ∈ R(Λ).
Let S n = S ∩ B b where B n is the disk of radius n centered at the origin of R 2 .Then ∂Π Sn are closed paths and the automorphisms w ∂Π Sn = Ad(W a [∂Π Sn ]) leave the ground state invariant, and are supported in Λ.In particular, there exist phases ϕ n such that Since the vectors OΩ, O ′ Ω are dense in H, this shows that V n converges weakly to V .since each V n is in A Λ , we conclude that V ∈ R(Λ).□ Proposition 3.8.If Λ 1 and Λ 2 are cones both contained in a cone Λ, then π 1 •w a,Λ 1 ≃ π 1 •w a,Λ 2 and the unitary implementing this equivalence belongs to the von Neumann algebra R(Λ).Proof : Let ŵ1 , ŵ2 be the axes of the cones Λ 1 , Λ 2 and take a cone Λ 3 ⊂ Λ such that its axis ŵ3 points to the right of both ŵ1 amd ŵ2 relative to Λ. Then Lemma 3.7 implies that there unitaries hence Proof : For any n large enough such that the endpoint of P Λ 0 is contained in Π n−2 , consider the S-matrix An easy calculation shows that these quantities are independent of n, and given by for all n sufficiently large.Corollary 2.6.11 of [2] then implies that π a and π b are not unitarily equivalent.□ 3.2.5.Verification of assumptions.The four faithful irreducible representations π 1 , π S , π S , π B defined by π a = π 1 • w a,Λ 0 for a ∈ {1, S, S, B} = I are pairwise not unitarily equivalent by Proposition 3.9.
For any cone Λ and any a ∈ I we defined an automorphism w a,Λ supported in Λ.This collection of automorphisms satisfies assumption 1 by Proposition 3.8.Assumptions 2 and 3 are verified by Proposition 3.6.Finally, assumption 4 holds by Propositions 3.8 and 3.6.
Table 1.The fusion intertwiners Ω(a, b) for the double semion state.
Let f be the endpoint of the path P Λ 0 and let V ∈ R(Λ 0 ) be the unitary such that π 1 •v[P Λ 0 ] = Ad(V ) • π 1 and V Ω = Ω provided by Lemma 3.3.The proof of Lemma 3.5 shows that Ad(Ω(S, S)) ) for all a, b ∈ I with fusion interwiners Ω(a, b) given in Table 1.
In order to compute the F-symbols, we first show Lemma 3.10.
Proof : Since f is the last edge of the path P Λ 0 , we have w V .Since V is the weak limit of the sequence V n = V [P n ] where P n is the path consisting of edges of P Λ 0 whose midpoints lie in Π n (cf. the proof of Lemma 3.4), it is sufficient to show This follows similarly to the argument in the proof of Lemma 3.4.Since V n is a product of σ s Z we have that w B (V n ) = V n , and By design, the unitary V n has an even number of σ Z 's on the path P n .Indeed, there are two factors of σ Z for every L-vertex, zero for every R-leg, and another two for the enpoints.We conclude that w We can now start computing the F -symbols.If in Eq. ( 12) we take a = 1 then  Finally, we consider the case where a, b, c ∈ {S, S}.Then since ab, bc ∈ {1, B} we have Using Lemma 3.10 we conclude that F (a, b, c) = −1 for a, b, c ∈ {S, S}.
To compute the braiding intertwiners ϵ(a, b (Recall that the automorphisms v a,Λ were defined in section 3.2.1).
It follows from Lemma 3.7 and Proposition 3.8 that there are unitaries U a ∈ (w a , w L a ) and V a ∈ (w a , w R a ) which are unique up to phase.By definition 2.8, In order to compute w a (V b ), let us realise V b as the weak limit of a sequence of strictly local unitaries.
Let K be the cone whose legs coincide with the central axes of Λ 0 and Λ R , see Figure 15.Then the path ∂Π K contains P Λ 0 and P Λ R and the path does not depend on n (for n large enough, and possible redefining the phases of the V (n) b ), and is supported near the path Q.Since ∂Π K is a closed path, the automorphism w b [∂Π K ] leaves the grounds state invariant by Proposition 3.2, so there is a unique unitary V ∈ B(H) such that V Ω = Ω and w b [∂Π K ] = Ad( V ) (as automorphisms on π 1 (A)).Now, for strictly local operators O, O ′ we have  showing that the sequence V (n) converges weakly to V .It follows that the sequence We can now compute the braiding intertwiners. Obviously for any a ∈ {1, S, S, B}, so ϵ(B, a) = 1 for all a, while because the path P Λ 0 contains a single R-leg of the path P n .So ϵ(S, B) = ϵ( S, B) = − 1.
Let us now compute ϵ(S, S).Note that the path P n enters the path P Λ 0 at an L-vertex of P Λ 0 .Let (i, j) be the edges of P Λ 0 before and after this L-vertex, see Figure 16.We find Table 2.The braiding intertwiners ϵ(a, b) for the double semion state.
Table 3.The R-symbols R(a, b) for the double semion state.
We now use the braid equations (Lemma 2.10) (where I ρ denotes the identity intertwiner from ρ to itself) to compute Thus we have computed all braiding intertwiners,see Table 2 for a summary.The R-symbols are defined (14) by Since Ω(a, b) = Ω(b, a) for all a, b, we find that the R-symbols are as in table 3.One can verify that the F and R-symbols indeed satisfy the pentagon and hexagon equations.Finally, each object is its own dual a * = a with evaluation maps ev a : a ⊗ a * = 1 → 1 given by multiplication with −1 if a ∈ {S, S} and multiplication by 1 if a ∈ {1, b}, and coevaluation maps i a : 1 → a ⊗ a * = 1 given by multiplication by 1.
Since the F and R-symbols satisfy the pentagon and hexagon equations, this data describes a braided fusion category.In the rest of this section, we show that this is precisely the category of representations of the quasi quantum double D ϕ (Z 2 ) where ϕ is a representative of the nontrivial class of H 3 (Z 2 , U (1)).
Let ϕ : (Z 2 ) 3 → U (1) be the normalized representative of the non-trivial class in H 3 (Z 2 , U (1)): For each f , the map c f : The quasi quantum double D ϕ (Z 2 ) is an algebra spanned by {P x f } x,f ∈Z 2 with multiplication The unit for this multiplication is x∈Z 2 (P x 1).The quasi quantum double is moreover equipped with a coproduct ∆ : Associativity and coassociativity follow readily from Eq. (77).That ∆ is an algebra morphism follows from the identity There is a counit ϵ : D ϕ (Z 2 ) → C and an antipode S : D ϕ (Z 2 ) → D ϕ (Z 2 ): These give D ϕ (Z 2 ) the structure of a quasi Hopf algebra.This quasi Hopf algebra is moreover quasitriangular with universal R-matrix R = x,y (P x 1) ⊗ (P y x). (82) with where [x], [f ] are the additive representation of x and f .i.e. ε − (−) = i and all other components are equal to one.ε x is a cocycle and Since we have a coproduct, we can define the tensor product of representations: One easily verifies that Π (x,χ) ⊗ Π (y,σ) = Π (xy,χσ) .
The representation Π (1,1) is an identity for this tensor product (with trivial left and right unitors).We will show that ω m | n is a mixed state which is an equal-weight convex combination of pure states η n (b) where b is a boundary condition, namely an assignment of up-or down to each out edge of the region Π n such that an even number of edges are up, see Figure 17 The state η n (b) is then given by a uniform superposition of all loop soups that satisfy the boundary condition b, weighed by ±1 depending on whether a fixed 'closure' of the boundary condition has an even or an odd number of closed loops.n can be 'closed up' by connecting neighbouring marked edges in two ways, see figure 17.Pick one such 'pairing' of marked boundary edges and let ♯α be the number of loops of α closed up with the chosen pairing.Then we have normalized vectors We have ⟨η Here, 2 6n−1 is the number of boundary conditions b.Indeed, there are 6n outer edges where the boundary condition either forces or does not force a string to pass, and the number of edges where a string is forced to end must be even.There are as many even boundary conditions as there are odd boundary conditions.Indeed, flipping a fixed edge gives a bijection.
Proof : By Lemma A.1 it is sufficient to consider m = n + 1.The state ω n+1 on Π n+1 is a uniform superpostition of closed loop soups in Π n+1 .Any such loop soup α defines a boundary condition b(α) by the outer edges of Π n that are occupied by strings of α.We can therefore organize the α according to which boundary condition they induce: The factor of 2 appearing in the third line is the number of choices of completing a loop soup α ′ in Π n with boundary condition b to a closed loop soup α in Π n+1 .The phase (−) ♯α ′ +♯β ′ does not depend on which (common) completion is chosen.Indeed, changing the completion changes both ♯α ′ and ♯β ′ by an odd amount if the number of marked edges is a multiple of 4, and both by an even amount otherwise (Lemma A.4).To get the fourth line we used that n , denote by ♯ 1 α and ♯ 2 α the number of loops in the two possible completions.then ♯ 1 α − ♯ 2 α is odd if the number of marked points for b is a multiple of 4, and even otherwise.
Proof : Assume first that α has no closed loops.Let the number of marked points be 2n.The following construction is illustrated in Figure 18.Abstract the region Π n to a disk with the marked points sitting on the boundary.Then the two completions correspond to two sets of n intervals that 'interlace' along the boundary of the disk.Choose one of them.The loop soup α connects these n intervals into groups.The number of groups g is the number of closed loops in this completion, say ♯ 1 α = g.Put a vertex on each interval for this completion, and add a vertex in each group.Connect this vertex by edges to the vertices of the intervals in the group.Finally, connect the vertices on the intervals by edges along the boundary of the disk.This gives a connected graph with V = n + g vertices and E = 2n edges.By the Euler formula, this graph has F = 1 − V + E = 1 + n − g internal faces.The number of internal faces corresponds precisely to ♯ 2 α, and we find which is odd if n is even and vice versa.Any closed loops of α remain connected components of both completions, so internal loops do not contribute to ♯ 1 α = ♯ 2 α. □

We further show
□ A.2. Purity of the limit state.We will now show that ω is a pure state by making use of the following Lemma, which is a special case of Lemma 2.1.of [10].
Lemma A.6 (Lemma 2.1 of [10]).A state ω on a UHF algebra realized as the inductive limit of a sequence of finite matrix algebras {M m } is pure if the following holds: For each n there exists m > n such that if ρ is a linear functional on M m that satifies then ρ| Mn = λω| Mn (98) for some λ ∈ R.
Let us also introduce terms imposing boundary conditions: This is also a sum of orthogonal projections, and they all commute with each other and with the B p and A v appearing in H Πn .We now consider the commuting projection Hamiltonians Let Π n be the collection of edges that have an endpoint in Π n .Then H n ∈ A Πn .Moreover, the state ω n restricts to A Πn as a pure state.Let us continue to denote this restriction by ω n .We have Lemma B.1.The state ω n on A Πn is the unique ground state of H n .
Proof : The state ω n is defined by the expectation in the vector state where Ω 0 has all σ Z i = 1.The state Ω n is a superposition of closed string configurations in Π n .Each such closed string configuration satisfies for all v ∈ Π n , all i ∈ ∂Π n and all Π ⊂ Π n .
To see that Ω n is a ground state of H n it remains to show that it is in the kernel of all B p for p ∈ Π n .One can check that To see that Ω n is the unique ground state, observe that any ground state must be in the kernel of all the 1 − A v for v ∈ Π n and all the 1 2 (1 − σ Z i ) for i ∈ ∂Π n .The space of states that are simultaneously in the kernels of all these projections is spanned by the closed string states We must find in this space a state that is in the kernel of all the B p , equivalently a -1 eigenstate of all the W S [p] for p ∈ Π n .Consider a general state for all Π ′ ⊂ Π n .Indeed, and Π can be related to any Π ′ by a sequence of symmetric differences with elementary plaquettes p.This shows that Ψ ≃ Ω n , so Ω n is indeed the unique ground state of H n on A Πn .□ Lemma B.2.If P is a closed path entirely contained in Π n , then W a [P ] commutes with H n .
Proof : This is shown for the string operators of any Levin-Wen model using a graphical representation of the string operators in [13].In our case of the double semion model, we can also show it by brute force.That W S [P ] commutes with the star operators A v and with the boundary terms in H ∂Πn is obvious.Let us show that W S [P ] commutes with B p for p ∈ Π n .
To this end, note simply that if Q is the path, possibly consisting of multiple components, made up of edges of P that are also edges or R-legs of p, oriented with the same orientation as P , then

□ Lemma 2 . 3 .
The binary operation × : I × I → I makes I into an abelian group with unit 1 and inverse a −1 = a * .

Figure 2 .
Figure 2. Graphical representation of Eq. (12), defining the F-symbols F (a, b, c).Each node represents a fusion operator.The diagrams represents two different compositions of fusion operators both yielding intertwiners from w a ⊗ w b ⊗ w c to w abc .

Figure 3 .
Figure 3.A graphical proof of the Pentagon equation.

Figure 4 .
Figure 4. Graphical representation of Eq. (14), defining the R-symbols R(a, b).The point where the a-line passes under the b-line represents the braiding intertwiner ϵ(a, b).

2. 3 . 2 .
Braiding and R-symbols.We simply set ϵ(a, b) := ϵ(w a , w b ) for any a, b ∈ I.The unitaries Ω(b, a)ϵ(a, b) and Ω(a, b) are both intertwiners from w a ⊗ w b to w ab

Figure 5 .
Figure 5. Graphical representation of the Yang-Baxter equations.

Figure 6 .
Figure 6.Graphical representations of the first and second hexagon equations.

Figure 7 .
Figure 7.The degrees of freedom of the double semion state live on the edges of a hexagon lattice.

Figure 8 .
Figure 8. Acting with A Π on the state ω 0 yields a string configuration with two connected components.

3 . 2 .
w a [P ] the automorphism defined by conjugation with the (possibly formal) unitary W a [P ].These string operators have the following important property Proposition Closed string operators leave the ground state invariant.i.e. if P is a closed string then ω • w a [P ] = ω (44) for all a ∈ I.The proof is in appendix B.

Figure 10 .
Figure 10.An oriented path P in solid blue with its L-vertices fattened and its R-legs marked with dotted lines.

Figure 11 .
Figure 11.A cone Λ with the oriented path ∂Π Λ going around it in a counter clockwise direction.

Figure 12 .
Figure 12.A cone Λ divided into its left and right cones Λ (L) and Λ (R) .The path P Λ is the largest part of ∂Π Λ (R) such that w a [P Λ ] is supported in Λ.
following the left and right legs of Λ, respectively.

2
and the unitary implementing this equivalence belongs to the von Neumann algebra R(Λ).

Figure 13 .
Figure 13.The axis of Λ 1 points to the right of the axis of Λ 2 relative to the cone Λ.The region S has w a [∂Π S ] supported in Λ.Moreover, the path ∂Π S differs from the union of the paths P Λ 1 , P Λ 2 by a finite number of edges.

Lemma 3 . 11 .
The sequence V (n) b converges weakly to V b .Proof : Consider first the sequence of closed paths P n = ∂Π Kn and corresponding string operators W b [ P n ].By Proposition 3.2, the unitaries W b [ P n ] leave the ground state invariant up to a phase, so there are phases ϕ

Figure 15 .
Figure 15.Sets K n and their boundary paths P n used in the construction of the sequence of unitaries V (n) b that converge weakly to the intertwiner V b .

Figure 16 .
Figure 16.Edges i and j playing a role in the computation of ϵ(S, S).

3. 5 . 3 . 5 . 1 .
Equivalence to Rep f D ϕ (Z 2 ).The braided fusion category of anyons.Let C be the category whose objects are the four anyons types {1, s, s, b} seen as one-dimensional vector spaces over C. The monoidal structure is determined by the fusion rules:a ⊗ b = a × b(74)which has unit object 1, and the associators are given by the F -symbols.i.e. α a,b,c : (a⊗b)⊗c → a ⊗ (b ⊗ c) is given by multiplication with F (a, b, c).The braiding intertwiners are given by the R-symbols, i.e. ϵ a,b : a ⊗ b → b ⊗ a is given by multiplication with R(a, b).

Figure 17 .
Figure 17.A loop soup α ∈ P (b) 4 with boundary condition b corresponding to the red edges.The dotted red paths indicate one of two ways of pairing neighbouring red edges, resulting in a closed loop soup.

Lemma A. 2 .
There are 2 |Πn| such loop soups for each boundary condition b.Proof : For the boundary condition with all spins up this is obvious, because then the loop soups are precisely closed loop soups in Π n .To obtain loop soups for an arbitrary boundary condition b, act on any closed loop soup with A p on the plaquettes between pairs of boundary edges where b forces a loop to end (choose one of two possible pairings).This yields a loop soup that satisfies the boundary condition, and two different closed loop soups give two different loop soups satisfying the boundary condition.Conversely, evey loop soup satisfying the boundary condition arises in this way, because acting on loop soups satisfying b with A v 's on the vertices between pairs of boundary edges where b forces a loop to end yields a closed loop soup.□ Write P (b) n for the loop soups in Π n that satisfy the boundary condition b.For a given boundary condition b any α ∈ P (b) n ⟩ = δ b,b ′ , i.e. these vectors form an orthonormal set.Denote by η (b) n the pure state on M n corresponding to the vector |η (b) n ⟩.Proposition A.3.For m > n ≥ 1,ω m | n = 1 2 6n+1 b η (b) n .(93)Since the |η (b) n ⟩ form an orthonormal set, this is a Schmidt decomposition of ω m | n .

)
The states |α⟩ are orthonormal product states.If O is supported on Π n then the matrix elements ⟨β, Oα⟩ only depend on the configuration of α and β on Π n .This information still allows us to deduce the boundary conditions b(α) and b(β).Moreover, the matrix element vanishes if b

Figure 18 .
Figure 18.The red dotted paths completing α to a closed loop soup are marked with vertices (black), and so are the closed regions (green) resulting from this completion.The white regions correspond one-to-one to faces of the black graph.Each such white region corresponds to a loop of the alternative completion of α to a closed loop soup.

Lemma A. 5 .
For any boundary condition b and any O supported on Π n−1 we have η (b) n (O) = ω(O).Proof : From Lemma A.1, it is sufficient to show that η (b) n (O) = ω n (O).(I write ω n for the restriction of this state to Π n ).Note that ω n = η ∅ n , where ∅ stand for the trivial boundary condition.For any other boundary condition b, let A b be the product of A p operators over plaquettes between pairs of marked edges of b.Clearly, A b is supported outside Π n−1 , so A * b OA b = O and since A b bijectively maps loop soups satisfying b to closed loop soups, we find η (b) n (O) = η (b) n (A * b OA b ) = η ∅ n (O) = ω n (O) = ω(O).