The Double Semion State in Infinite Volume

Abstract

We describe in a simple setting how to extract a braided tensor category from a collection of superselection sectors of a two-dimensional quantum spin system, corresponding to abelian anyons. We extract from this category its fusion ring as well as its F and R-symbols. We then construct the double semion state in infinite volume and extract the braided tensor category describing its semion, anti-semion, and bound state excitations. We verify that this category is equivalent to the representation category of the twisted quantum double $\mathcal {D}^{\phi }(\mathbb {Z}_2)$.

1 Introduction

Gapped ground states of two-dimensional quantum lattice systems may support anyonic excitations. Paradigmatic examples include Kitaev’s quantum double models [15] and, more generally, the string-net models of Levin and Wen [16]. It is widely believed that the types of anyons supported by a given ground state are organised in a unitary braided fusion category, whose simple 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, 9,10,11] to the setting of microscopic lattice systems [3, 19, 21]. This body of work manages to associate with a large class of gapped ground states a strict braided $C^*$-tensor category whose objects are localised and transportable endomorphisms of the observable algebra, and shows that this category is a robust invariant for the gapped phase [1] to which the ground state belongs.

One can extract from any braided tensor category its fusion ring and F-symbols, which encode the tensor structure of the category, as well as R-symbols, which encode the braided structure of the category. This yields in particular a microscopic definition of fusion rules, F-symbols, and R-symbols, which are commonly employed in the physics literature to describe anyon theories. We note that a microscopic definition of F and R-symbols has been given before in [14]. Extracting the F and R-symbols from the DHR-type analysis has the advantage that it is clear that these data yield a robust invariant of gapped phases [3, 21].

In Sect. 2 of the paper, we review the construction of a braided 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 the set of anyon types is finite, and that each anyon has an anti-particle, we show that corresponding fusion ring is group-like and abelian and is equipped with 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 [16] in infinite volume. This is the simplest state that supports abelian anyons whose braided tensor category has non-trivial F-symbols. We show that the double semion ground state satisfies the assumptions of Sect. 2 with anyon content consisting of the vacuum, the semion, the anti-semion, and a semion–anti-semion bound state. We extract the fusion ring, F-symbols, and R-symbols from the corresponding braided tensor category. Finally, we use a result from [12] to conclude that the braided tensor category describing these anyons is equivalent to the category $\textrm{Rep}_f(\mathcal {D}^{\phi }(\mathbb {Z}_2))$ of finite- dimensional representations of the twisted quantum double algebra of $\mathbb {Z}_2$. Since the double semion model is the string-net model defined by input fusion category $\mathcal {F}= \textrm{Vec}_{\mathbb {Z}_2}^{\phi }$ with $\phi$ a non-trivial 3-cocycle of $\mathbb {Z}_2$, this partially verifies the conjecture that the resulting anyon theory is described by the Drinfeld centre $\mathcal {Z}(\textrm{Vec}_{\mathbb {Z}_2}^{\phi }) = \textrm{Rep}_f(\mathcal {D}^{\phi }(\mathbb {Z}_2))$, see [17].

In appendix, we show various properties of the double semion state that are used in Sect. 3. In particular, we introduce a new technique based on results from [13] to show that the double semion state we construct is pure. This technique is applicable to the construction and proof of purity of ground states and excited states of a large class of lattice spin models including all Levin–Wen string-net models.

2 Braided Tensor Category for Abelian Anyons

2.1 Setup

2.1.1 Algebra of Observables and State

Consider a countable set $\Gamma \subset \mathbb {R}^2$ of sites in the plane. To each site $x \in \Gamma$, we associate an algebra $\mathcal {A}_x \simeq \textrm{End}(\mathbb {C}^d)$ for some fixed $d \ge 2$. For any finite $X \subset \Gamma$, we set $\mathcal {A}_X = \bigotimes _{x \in X} \mathcal {A}_x$. If $X \subset Y$ are finite subsets of $\Gamma$, then there is a natural norm-preserving inclusion by tensoring with the identity.

For any infinite subset $Y \subset \Gamma$, we then have a local net of algebras $\mathcal {A}_X$ for finite $X \subset Y$, whose direct limit is $\mathcal {A}_{Y, \textrm{loc}}$, the algebra of local observables supported in Y. Its norm completion is $\mathcal {A}_Y:= \overline{\mathcal {A}_{Y, \textrm{loc}}}^{ \Vert \cdot \Vert }$, the algebra of quasi-local observables supported in Y, and we get inclusions also for infinite $X \subset Y$. We write $\mathcal {A}_{\textrm{loc}} = \mathcal {A}_{\Gamma , \textrm{loc}}$ and $\mathcal {A}= \mathcal {A}_{\Gamma }$ for the algebra of all local and all quasi-local observables, respectively. The support of an observable $O \in \mathcal {A}$ is the smallest set $X \subset \Gamma$ such that $O \in \mathcal {A}_X$.

Similarly, the support of an automorphism w of $\mathcal {A}$ is the smallest set Y such that $w|_{\mathcal {A}_{Y^c}}=\text {id}_{\mathcal {A}_{Y^c}}$. Any subset $Z \subset \mathbb {R}^2$ of the plane determines a subset ${\overline{Z}} = Z \cap \Gamma$ of $\Gamma$, and we denote $\mathcal {A}_{Z}:= \mathcal {A}_{{\overline{Z}}}$.

A major role is played by cones. The cone with apex at $a \in \mathbb {R}^2$, axis ${\hat{v}} \in \mathbb {R}^2$ of unit length, and opening angle $\theta \in (0, 2\pi )$ is

$$\begin{aligned} \Lambda _{a, {\hat{v}}, \theta }:= \{ x \in \mathbb {R}^2 \, | \, (x - a) \cdot {\hat{v}} > \Vert x-a\Vert \cos (\theta /2) \}. \end{aligned}$$

(1)

We will assume that ${\overline{\Lambda }}$ is infinite for any cone $\Lambda$. In particular, this holds if $\Gamma$ is a lattice.

We will consider a pure state $\omega$ on $\mathcal {A}$ with GNS representation $(\pi _1, \mathcal {H}, \Omega )$. For any $X \subset \Gamma$, we put $\mathcal {R}(X):= (\pi _1(\mathcal {A}_X))''$, the von Neumann algebra associated with X. For $Z \subset \mathbb {R}^2$, we also write $\mathcal {R}(Z) = \mathcal {R}({\overline{Z}})$. We note that if $\Lambda$ is a cone, then $\mathcal {R}(\Lambda )$ is an infinite factor (Theorem 5.2 of [19]).

2.1.2 Superselection Sectors

Definition 2.1

An irreducible representation $(\pi , \mathcal {H})$ of $\mathcal {A}$ is said to satisfy the superselection criterion with respect to $\pi _1$ if for any cone $\Lambda$ there is a unitary $U \in \mathcal {B}(\mathcal {H})$ such that

$$\begin{aligned} U \pi (O) U^* = \pi _1(O) \quad \text {for all} \,\, O \in \mathcal {A}_{\Lambda ^c} \end{aligned}$$

(2)

If two representations $\pi , \pi '$ are unitarily equivalent, then we write $\pi \simeq \pi '$.

We assume that we have a finite set of irreducible representations $\mathcal {O}= \{ \pi _a \, | \, a \in I \}$ of $\mathcal {A}$ indexed by a labelling set I. We assume moreover that $\pi _a \simeq \pi _b$ if and only if $a = b$, so all sectors in $\mathcal {O}$ are truly distinct. Moreover, $1 \in I$ so that $\pi _1 \in \mathcal {O}$. We call $\pi _1$ the vacuum sector.

We will now make some additional Assumptions (1–4) on these sectors which will in particular imply that they satisfy the superselection criterion with respect to $\pi _1$. These assumptions are not generically satisfied by gapped ground states, but Assumptions (1–3) and the first part of Assumption 4 are verified for the Toric code model in [19] and for all abelian quantum double models in [8]. The second part of Assumption 4 can be shown for these models using similar methods. Below, we will verify these assumptions for the double semion model. More generally, the authors expect that these assumptions hold for all abelian string-net models [16].

Assumption 1

For any cone $\Lambda$ and any $a \in I$, there is an automorphism $w_{a, \Lambda }$ supported on $\Lambda$ such that

$$\begin{aligned} \pi _a \simeq \pi _1 \circ w_{a, \Lambda }. \end{aligned}$$

(3)

In particular, we take $w_{1, \Lambda } = \textrm{id}$ for all cones $\Lambda$.

The following assumption says that the anyons we study are abelian.

Assumption 2

For any $a, b \in I$ there is a unique $c \in I$ such that for any two cones $\Lambda _1, \Lambda _2$ we have $\pi _c \simeq \pi _1 \circ w_{a, \Lambda _1} \circ w_{b, \Lambda _2}$. We write $c = a \times b$.

The following assumption says that each anyon has an antiparticle.

Assumption 3

For each $a \in I$, there is an $a^* \in I$ such that $a \times a^* = 1$.

The final assumption is of a technical nature it plays an important role in constructing a tensor category in Sect. 2.2.

Assumption 4

Assumption 1 implies that if $\Lambda _1, \Lambda _2 \subset \Lambda$ are cones, then $\pi _1 \circ w_{a, \Lambda _1} \simeq \pi _1 \circ w_{a, \Lambda _2}$. We assume that any unitary $V \in \mathcal {B}(\mathcal {H})$ implementing this equivalence belongs to the von Neumann algebra $\mathcal {R}(\Lambda )$.

Similarly, it follows from Assumption 2 that $\pi _1 \circ w_{a, \Lambda } \circ w_{b, \Lambda } \simeq \pi _1 \circ w_{a \times b, \Lambda }$. We assume that any unitary $V \in \mathcal {B}(\mathcal {H})$ implementing this equivalence belongs to the von Neumann algebra $\mathcal {R}(\Lambda )$.

This assumption is implied by Haag duality (cf. the discussion at the end of Sect. 2.2.4) and can also often be proven directly if the automorphisms $w_{a, \Lambda }$ are known explicitly, for example, for the abelian quantum double models [8, 19] and for the double semion model treated below.

Assumptions 1–3 have a few elementary but important consequences.

Lemma 2.2

The representations $\pi _a$, $a \in I$ satisfy the superselection criterion w.r.t. $\pi _1$.

Proof

Fix a cone $\Lambda$. By Assumption 1, there is an automorphism $w_{a, \Lambda }$ supported in $\Lambda$ such that $\pi _1 \circ w_{a, \Lambda } \simeq \pi _a$. i.e. there is a unitary $U \in \mathcal {B}(\mathcal {H})$ such that

$$\begin{aligned} U \pi _a(O) U^* = \pi _1( w_{a, \Lambda }(O) ) \end{aligned}$$

(4)

for all $O \in \mathcal {A}$. Since $w_{a, \Lambda }$ is supported in $\Lambda$, we have $w(O) = O$ for $O \in \mathcal {A}_{\Lambda ^c}$ and therefore

$$\begin{aligned} U \pi _a(O) U^* = \pi _1(O) \quad \text {for all} \,\, O \in \mathcal {A}_{\Lambda ^c}. \end{aligned}$$

(5)

$\square$

Lemma 2.3

The binary operation $\times : I \times I \rightarrow I$ makes I into an abelian group with unit 1 and inverse $a^{-1} = a^*$.

Proof

We first show that $\times$ is abelian. Take $a, b \in I$. Assumption 2 says that for any two cones $\Lambda _1, \Lambda _2$ there are automorphisms $w_{a, \Lambda _1}$, $w_{b, \Lambda _2}$ such that $\pi _{a \times b} \simeq \pi _1 \circ w_{a, \Lambda _1} \circ w_{b, \Lambda _2}$. Exchanging the roles of a and b and of $\Lambda _1$ and $\Lambda _2$, we have $\pi _{b \times a} \simeq \pi _1 \circ w_{b, \Lambda _2} \circ w_{a, \Lambda _1}$. If we now take $\Lambda _1$ and $\Lambda _2$ to be disjoint, then certainly $w_{a, \Lambda _1} \circ w_{b, \Lambda _2} = w_{b, \Lambda _2} \circ w_{a, \Lambda _1}$ and therefore $\pi _{a \times b} \simeq \pi _{b \times a}$. But we assumed that two representations in $\mathcal {O}$ are unitarily equivalent only if they are the same, so $a \times b = b \times a$.

We now show that 1 is the identity for the product $\times$. Fix cones $\Lambda _1$ and $\Lambda _2$. By Assumptions 1 and 2, there are automorphisms $w_{1, \Lambda _1} = \textrm{id}$ and $w_{a, \Lambda _2}$ such that $\pi _{1 \times a} \simeq \pi _1 \circ \textrm{id}\circ w_{a, \Lambda _2} = \pi _1 \circ w_{a, \Lambda _2} \simeq \pi _{a}$, hence $1 \times a = a$. We already know that $\times$ is abelian, so also $a \times 1 = a$.

Finally, Assumption 3 states that $a^*$ is the inverse of a. $\square$

We will often write $ab = a \times b$ for the product of elements $a, b \in I$.

2.2 Braided Tensor Category

It is well understood how to associate a braided tensor category with a pure state on a quantum spin system [3, 19, 21]. In this section, we recap this construction and use Assumptions 1–4 to identify a braided fusion subcategory with abelian fusion rules.

2.2.1 Tensor Category of Localised and Transportable Endomorphisms

Fix a unit vector ${\hat{f}} \in \mathbb {R}^2$, representing a ‘forbidden direction’. We say a cone $\Lambda _{a,{\hat{v}},\theta }$ with axis ${\hat{v}}$ and opening angle $\theta$ is forbidden if it contains the forbidden direction ${\hat{f}}$, i.e. if ${\hat{v}} \cdot {\hat{f}} > \cos (\theta /2)$. A cone that is not forbidden is said to be allowed. We define an allowed algebra by

$$\begin{aligned} \mathcal {B}:= \overline{ \bigcup _{\text {allowed} \, \Lambda } \mathcal {R}(\Lambda ) }^{ \Vert \cdot \Vert } \subset \mathcal {B}(\mathcal {H}). \end{aligned}$$

(6)

This is a $C^*$-algebra that contains $\pi _1(\mathcal {A})$ since any local observable is supported in some allowed cone.

Definition 2.4

We say an endomorphism ${\bar{\rho }}$ of $\mathcal {B}$ is localised on $\Lambda$ if ${\bar{\rho }}( \pi _1(O) ) = \pi _1(O)$ for all $O \in \mathcal {A}_{\Lambda ^c}$, and that ${\bar{\rho }}$ is localised if it is localised on some allowed cone $\Lambda$. We say that an endomorphism ${\bar{\rho }}$ of $\mathcal {B}$ that is localised on an allowed cone $\Lambda$ is transportable if for any allowed cone $\Lambda '$ there is an endomorphism ${\bar{\rho }}'$ of $\mathcal {B}$, localised on $\Lambda '$, and a unitary $U \in \mathcal {B}$ such that $U \, {\bar{\rho }}( \pi _1(O) ) = {\bar{\rho }}'( \pi _1(O) ) \, U$ for all $O \in \mathcal {A}$ and such that $U \in \mathcal {R}(\widetilde{\Lambda })$ for any allowed cone $\widetilde{\Lambda }$ that contains the cones $\Lambda$ and $\Lambda '$. We denote by $\Delta$ the set of all localised and transportable endomorphisms.

The localised and transportable endomorphisms of $\mathcal {B}$ are the objects of a $\mathbb {C}$-linear category with morphisms

$$\begin{aligned} ({\bar{\rho }}, {\bar{\sigma }}):= \{ R \in \mathcal {B}\,: \, R {\bar{\rho }}( \pi _1(O) ) = {\bar{\sigma }}( \pi _1(O) ) R \,\,\, \text {for all } \,\,\, O \in \mathcal {A}\} \end{aligned}$$

(7)

for any ${\bar{\rho }}, {\bar{\sigma }} \in \Delta$. Morphisms are referred to as intertwiners.

Direct sums of objects can be constructed as in Lemma 6.1 of [19]. Indeed, for any cone $\Lambda$, Corollary 5.3 of [19] shows that there are isometries $V_1, V_2 \in \mathcal {R}(\Lambda )$ such that $V_i^* V_j = \delta _{i, j} {{\,\mathrm{\mathbb {1}}\,}}$ and $V_1 V_1^* + V_2 V_2^* = {{\,\mathrm{\mathbb {1}}\,}}$. If ${{\overline{\rho }}}, {{\overline{\sigma }}}$ are localised on $\Lambda$, then $\textrm{Ad}[V_1] \circ {{\overline{\rho }}} + \textrm{Ad}[V_2] \circ {{\overline{\sigma }}}$ is a direct sum of ${{\overline{\rho }}}$ and ${{\overline{\sigma }}}$ which is still localised on $\Lambda$. Moreover, if ${{\overline{\rho }}}$ and ${{\overline{\sigma }}}$ are transportable, then for any allowed cone $\Lambda '$ there are endomorphisms ${{\overline{\rho }}}', {{\overline{\sigma }}}'$ localised on $\Lambda '$ and unitary morphisms $U_1 \in ({{\overline{\rho }}}, {{\overline{\rho }}}')$ and $U_2 \in ({{\overline{\sigma }}}, {{\overline{\sigma }}}')$ such that $U_1, U_2 \in \mathcal {R}(\widetilde{\Lambda })$ for any allowed cone $\widetilde{\Lambda }\supseteq \Lambda \cup \Lambda '$. Then take isometries $V'_1, V'_2 \in \mathcal {R}(\Lambda ')$ as above and consider the direct sum $\textrm{Ad}[V'_1] \circ {{\overline{\rho }}}' + \textrm{Ad}[V_2'] \circ {{\overline{\sigma }}}'$ of ${{\overline{\rho }}}'$ and ${{\overline{\sigma }}}'$, which is localised on $\Lambda '$. The unitary $W = V'_1 \, U_1 \, V_1^* + V'_2 \, U_2 \, V_2^* \in \mathcal {R}(\widetilde{\Lambda })$ then intertwines the two direct sums, showing that direct sums of localised and transportable endomorphisms are again localised and transportable.

The category of localised and transportable endomorphisms of $\mathcal {B}$ can be equipped with a monoidal structure. For any ${\bar{\rho }}, {\bar{\sigma }} \in \Delta$, we define their tensor product by

$$\begin{aligned} {\bar{\rho }} \otimes {\bar{\sigma }}:= {\bar{\rho }} \circ {\bar{\sigma }}, \end{aligned}$$

(8)

and for any intertwiners $R \in ({\bar{\rho }}, {\bar{\rho }}')$ and $S \in ({\bar{\sigma }}, {\bar{\sigma }}')$ we define the tensor product by

$$\begin{aligned} R \otimes S:= R {\bar{\rho }}(S) \in ( {\bar{\rho }} \otimes {\bar{\sigma }}, {\bar{\rho }}' \otimes {\bar{\sigma }}' ). \end{aligned}$$

(9)

2.2.2 Subcategory of Abelian Anyons

We use Assumptions 1–4 to obtain a subcategory of $\Delta$ whose simple objects correspond to the anyon types $a \in I$.

The following lemma shows that the automorphisms $w_{a, \Lambda }$ of Assumption 1 can be extended to localised and transportable endomorphisms of the allowed algebra $\mathcal {B}$, i.e. they yield objects in the category $\Delta$.

Lemma 2.5

(Proposition 4.6 [19]). For each allowed cone $\Lambda$ and each $a \in I$, the automorphism $w_{a, \Lambda }$ of Assumption 1 has a unique extension to an endomorphism ${\overline{w}}_{a, \Lambda }$ of $\mathcal {B}$ that is weakly continuous on $\mathcal {R}(\Lambda ')$ for any allowed cone $\Lambda '$. If $\Lambda ' \supset \Lambda$, then ${\overline{w}}_{a, \Lambda }( \mathcal {R}(\Lambda ') ) = \mathcal {R}(\Lambda ')$. Moreover, ${\overline{w}}_{a, \Lambda }$ is localised on $\Lambda$, and is transportable. In particular, ${\overline{w}}_{a, \Lambda } \in \Delta$.

Proof

Let $\Lambda ' \supset \Lambda$ be any allowed cone that contains $\Lambda$. From Assumption 1, we have for any forbidden cone $\Lambda ''$ with $\Lambda ' \cap \Lambda '' = \emptyset$ that $\pi _1 \circ w_{a, \Lambda } \simeq \pi _1 \circ w_{a, \Lambda ''}$. Let $V \in \mathcal {B}(\mathcal {H})$ be a unitary implementing this equivalence. We have for any $O \in \mathcal {A}_{\Lambda '}$ that

$$\begin{aligned} \pi _1 \big (w_{a, \Lambda }(O) \big ) = V \pi _1 \big ( w_{a, \Lambda ''}(O) \big ) V^* = V \pi _1(O) V^* \end{aligned}$$

(10)

where we used that $w_{a, \Lambda ''}$ is supported on $\Lambda ''$, which is disjoint form $\Lambda '$. We define the action of ${\overline{w}}_{a, \Lambda }$ on $\mathcal {R}(\Lambda ')$ by $\textrm{Ad}(V)$, which is weakly continuous, and is uniquely determined by the action of $w_{a, \Lambda }$ on $\mathcal {A}_{\Lambda '}$. Clearly this action on $\mathcal {R}(\Lambda ')$ does not depend on the choice of $\Lambda ''$. It follows that the extensions to $\mathcal {R}(\Lambda ')$ for different $\Lambda '$ are consistent with each other. Together with the weak continuity, this shows that the extension ${\overline{w}}_{a, \Lambda }$ is well defined on all of $\mathcal {B}$. Moreover, we have $w_{a, \Lambda }(\mathcal {A}_{\Lambda '}) = \mathcal {A}_{\Lambda '}$ and weak continuity then implies ${\overline{w}}_{a, \Lambda }(\mathcal {R}(\Lambda ')) = (w_{a, \Lambda }(\mathcal {A}_{\Lambda '}))'' = \mathcal {R}(\Lambda ')$ as required. This implies the inclusion ${\overline{w}}_{a, \Lambda }(\mathcal {B}) \subseteq \mathcal {B}$, so ${\overline{w}}_{a, \Lambda }$ is indeed an endomorphism of $\mathcal {B}$.

The endomorphism ${\overline{w}}_{a, \Lambda }$ is localised on $\Lambda$ by construction. To see that it is transportable, let $\Lambda '$ be any allowed cone and let ${\overline{w}}_{a, \Lambda '}$ be the extension of the automorphism $w_{a, \Lambda '}$ to the allowed algebra $\mathcal {B}$. By Assumption 4, there is a unitary $U \in \mathcal {B}(\mathcal {H})$ such that $U \in \mathcal {R}(\Lambda '')$ for any allowed cone $\Lambda '' \supset \Lambda \cup \Lambda '$ and

$$\begin{aligned} U \, \pi _1 \big ( w_{a, \Lambda }(O) \big ) = \pi _1( w_{a, \Lambda '}(O) ) \, U \end{aligned}$$

(11)

for all $O \in \mathcal {A}$. It follows from this and the construction of ${\overline{w}}_{a, \Lambda }$ and ${\overline{w}}_{a, \Lambda '}$ that $U \in ({\overline{w}}_{a, \Lambda }, {\overline{w}}_{a, \Lambda '})$, showing that ${\overline{w}}_{a, \Lambda }$ is transportable. $\square$

Lemma 2.6

The endomorphisms ${\overline{w}}_{a, \Lambda }$ are simple objects of $\Delta$, and two such objects ${\overline{w}}_{a, \Lambda }$ and ${\overline{w}}_{b, \Lambda '}$ are isomorphic if and only if $a = b$. That is,

$$\begin{aligned} ({\overline{w}}_{a, \Lambda }, {\overline{w}}_{b, \Lambda '}) \simeq {\left\{ \begin{array}{ll} \mathbb {C}{{\,\mathrm{\mathbb {1}}\,}}\quad \text {if} \,\,\, a = b \\ \{0\} \quad \text {otherwise.}\end{array}\right. } \end{aligned}$$

(12)

Proof

Suppose $R \in ({\overline{w}}_{a, \Lambda }, {\overline{w}}_{b, \Lambda '})$, i.e. the operator $R \in \mathcal {B}$ satisfies

$$\begin{aligned} R \, {\overline{w}}_{a, \Lambda }( \pi _1(O) ) = {\overline{w}}_{b, \Lambda '}( \pi _1(O) ) R \end{aligned}$$

(13)

for all $O \in \mathcal {A}$. By construction of ${\overline{w}}_{a, \Lambda }$, this implies

$$\begin{aligned} R \, (\pi _1 \circ w_{a, \Lambda })(O) = (\pi _1 \circ w_{b, \Lambda '})(O) \, R \end{aligned}$$

(14)

for all $O \in \mathcal {A}$. By Assumption 1, we have $\pi _1 \circ w_{a, \Lambda } \simeq \pi _a$ and $\pi _1 \circ w_{b, \Lambda } \simeq \pi _b$, so there are unitaries $U_a, U_b \in \mathcal {B}(\mathcal {H})$ such that

$$\begin{aligned} R U_a \, \pi _a(O) \, U_a^* = U_b \, \pi _b(O) \, U_b^* R \end{aligned}$$

(15)

for all $O \in \mathcal {A}$. We see that $U_b^* R U_a$ intertwines the irreducible representations $\pi _a$ and $\pi _b$. By assumption, we have $\pi _a \simeq \pi _b$ if and only if $a = b$. So if $a = b$ then we must have $U_b^* R U_a \in \mathbb {C}{{\,\mathrm{\mathbb {1}}\,}}$ which holds if and only if $R \in \mathbb {C}I$, and if $a \ne b$ then we must have $U_b^* R U_a = 0$ hence $R = 0$. $\square$

Recall from Assumption 2 and Lemma 2.3 that the set of anyon types I is equipped with an abelian product $\times$.

Lemma 2.7

If $\Lambda$ and $\Lambda '$ are allowed cones and $a, b \in I$, then ${\overline{w}}_{a, \Lambda } \otimes {\overline{w}}_{b, \Lambda '} \simeq {\overline{w}}_{a \times b, \Lambda ''}$ for any allowed cone $\Lambda ''$. If $\widetilde{\Lambda }\supseteq \Lambda \cup \Lambda ' \cup \Lambda ''$, then the intertwiners that realise this isomorphism are elements of $\mathcal {R}(\widetilde{\Lambda })$.

Proof

By Assumption 4, we have $\pi _1 \circ w_{a, \Lambda } \simeq \pi _1 \circ w_{a, \Lambda ''}$ implemented by unitaries $V \in \mathcal {R}(\widetilde{\Lambda })$. It follows from this and Lemma 2.6 that $({\overline{w}}_{a, \Lambda }, {\overline{w}}_{a, \Lambda ''})$ is spanned by a unitary $U_a \in \mathcal {R}(\widetilde{\Lambda })$. Similarly, $({\overline{w}}_{b, \Lambda '}, {\overline{w}}_{b, \Lambda ''})$ is spanned by a unitary $U_b \in \mathcal {R}(\widetilde{\Lambda })$.

Again by Assumption 4, we have $\pi _1 \circ w_{a, \Lambda ''} \circ w_{b, \Lambda ''} \simeq \pi _1 \circ w_{a \times b, \Lambda ''}$ with unitary intertwiners all belonging to $\mathcal {R}(\Lambda '')$. It follows that ${\overline{w}}_{a, \Lambda ''} \otimes {\overline{w}}_{b, \Lambda ''}$ is isomorphic to ${\overline{w}}_{a \times b, \Lambda ''}$ and is therefore simple. Moreover, $( {\overline{w}}_{a, \Lambda ''} \otimes {\overline{w}}_{b, \Lambda ''}, {\overline{w}}_{a \times b, \Lambda ''} )$ is spanned by a unitary $V_{a\times b} \in \mathcal {R}(\Lambda '') \subset \mathcal {R}(\widetilde{\Lambda })$.

We now find that $V_{a \times b} (U_a \otimes U_b) \in \mathcal {R}(\widetilde{\Lambda })$ is a unitary intertwiner in $({\overline{w}}_{a, \Lambda } \otimes {\overline{w}}_{b, \Lambda '}, {\overline{w}}_{a \times b, \Lambda ''})$. Since ${\overline{w}}_{a \times b, \Lambda ''}$ is simple, so is ${\overline{w}}_{a, \Lambda } \otimes {\overline{w}}_{b, \Lambda '}$, and it follows that $({\overline{w}}_{a, \Lambda } \otimes {\overline{w}}_{b, \Lambda '}, {\overline{w}}_{a \times b, \Lambda ''})$ is actually spanned by $V_{a \times b} (U_a \otimes U_b)$. This proves the claim. $\square$

We now identify a full subcategory of $\Delta$ whose isomorphism classes correspond to sums of anyons types $a \in I$.

For any allowed cone $\Lambda$, we fix isometries $V_1, V_2 \in \mathcal {R}(\Lambda )$ such that $V_i^* V_j = \delta _{i, j} {{\,\mathrm{\mathbb {1}}\,}}$ and $V_1 V_1^* + V_2 V_2^*={{\,\mathrm{\mathbb {1}}\,}}$ (existence follows from Corollary 5.3 of [19]) and define for any ${\bar{\rho }}, {\bar{\sigma }} \in \Delta$ the concrete direct sum ${\bar{\rho }} \, \oplus _{\Lambda } {\bar{\sigma }}:= \textrm{Ad}[V_1] \circ {\bar{\rho }} + \textrm{Ad}[V_2] \circ {\bar{\sigma }}$.

Definition 2.8

We let $\Delta ^I_{\Lambda }$ be the full tensor subcategory of $\Delta$ generated by the simple objects $\{ {\overline{w}}_{a, \Lambda } \}_{a \in I}$ using the tensor product $\otimes$ and the direct sum $\oplus _{\Lambda }$.

By construction, all objects of $\Delta ^I_{\Lambda }$ are localised on $\Lambda$. Moreover, the category $\Delta _{\Lambda }^I$ is semisimple as show in the following lemma.

Lemma 2.9

Every object of $\Delta _{\Lambda }^I$ is isomorphic to an object of the form

$$\begin{aligned} {{\overline{\rho }}}_1 \oplus _{\Lambda } {{\overline{\rho }}}_2 \, \oplus _{\Lambda } \cdots \oplus _{\Lambda } {{\overline{\rho }}}_n \end{aligned}$$

(16)

where the ${{\overline{\rho }}}_i$ are tensor products of the simple objects ${\overline{w}}_{a, \Lambda }$ and the isomorphism is given by a unitary in $\mathcal {R}(\Lambda )$.

(Note that the direct sum $\oplus _{\Lambda }$ is not associative, so the expression (16) is to be interpreted as being defined by some choice of bracketing. Different bracketings are isomorphic through a unitary in $\mathcal {R}(\Lambda )$).

Proof

For the duration of this proof, we write $\oplus = \oplus _{\Lambda }$. Denote by $V_1, V_2 \in \mathcal {R}(\Lambda )$ the isometries used to construct the direct sum $\oplus _{\Lambda }$, and for any ${{\overline{\rho }}} \in \Delta$ we write $\oplus _{{{\overline{\rho }}}}$ for the direct sum constructed using the isometries ${{\overline{\rho }}}(V_1), {{\overline{\rho }}}(V_2)$.

Suppose the claim is true for two objects ${{\overline{\rho }}}$ and ${{\overline{\sigma }}}$ of $\Delta _{\Lambda }^I$, i.e. ${{\overline{\rho }}} \simeq {{\overline{\rho }}}_1 \oplus \cdots \oplus {{\overline{\rho }}}_n$ and ${{\overline{\sigma }}} \simeq {{\overline{\sigma }}}_1 \oplus \cdots \oplus {{\overline{\sigma }}}_m$ with isomorphisms implemented by unitaries in $\mathcal {R}(\Lambda )$, and where the ${{\overline{\rho }}}_i$ and ${{\overline{\sigma }}}_j$ are finite tensor products of the simple objects ${\overline{w}}_{a, \Lambda }$. The claim then holds trivially for ${{\overline{\rho }}} \oplus {{\overline{\sigma }}}$. Let us now show that the claim holds for the tensor product ${{\overline{\rho }}} \otimes {{\overline{\sigma }}}$. We have

$$\begin{aligned} {{\overline{\rho }}} \otimes {{\overline{\sigma }}}&\simeq ( {{\overline{\rho }}}_1 \oplus \cdots \oplus {{\overline{\rho }}}_n ) \otimes ({{\overline{\sigma }}}_1 \oplus \cdots \oplus {{\overline{\sigma }}}_m) \\&= \bigoplus _{\kappa = 1}^n \, ( {{\overline{\rho }}}_{\kappa } \otimes {{\overline{\sigma }}}_1 ) \oplus _{{{\overline{\rho }}}_{\kappa }} \cdots \oplus _{{{\overline{\rho }}}_{\kappa }} ({{\overline{\rho }}}_{\kappa } \otimes {{\overline{\sigma }}}_m) \end{aligned}$$

where the isomorphism is implemented by a unitary in $\mathcal {R}(\Lambda )$. Noting that the direct sums $\oplus _{{{\overline{\rho }}}_{\kappa }}$ are isomorphic to $\oplus$ through a unitary in $\mathcal {R}(\Lambda )$ we obtain the required equivalence of ${{\overline{\rho }}} \otimes {{\overline{\sigma }}}$ to an object of the form (16). We have shown that if ${{\overline{\rho }}}$ and ${{\overline{\sigma }}}$ both satisfy the claim, then so do ${{\overline{\rho }}} \oplus {{\overline{\sigma }}}$ and ${{\overline{\rho }}} \otimes {{\overline{\sigma }}}$. Since the claim holds trivially for the simple objects $\{ {\overline{w}}_{a, \Lambda } \}_{a \in I}$ and the category $\Delta _{\Lambda }^I$ is by definition generated by these simple objects using $\otimes$ and $\oplus$, we conclude that every object of $\Delta _{\Lambda }^I$ is isomorphic to an object of the form (16) through a unitary in $\mathcal {R}(\Lambda )$, as we wanted to show. $\square$

Lemma 2.10

Let $\Lambda$ be an allowed cone and $\Lambda _1, \Lambda _2 \subset \Lambda$ two allowed subcones. For any two objects ${{\overline{\rho }}} \in \Delta _{\Lambda _1}^I$ and ${{\overline{\sigma }}} \in \Delta _{\Lambda _2}^I$, we have $({{\overline{\rho }}}, {{\overline{\sigma }}}) \subset \mathcal {R}(\Lambda )$.

Proof

Let ${{\overline{\rho }}}_1, {{\overline{\rho }}}_2 \in \Delta ^I_{\Lambda _1}$ and ${{\overline{\sigma }}}_1, {{\overline{\sigma }}}_2 \in \Delta ^I_{\Lambda _2}$ and suppose $({\overline{\rho }}_k,{\overline{\sigma }}_l)\subset \mathcal {R}(\Lambda )$ for $k,l\in \{1,2\}$. We will show that $({{\overline{\rho }}}_1 \oplus _{\Lambda _1} {{\overline{\rho }}}_2, {{\overline{\sigma }}}_1 \oplus _{\Lambda _2} {{\overline{\sigma }}}_2) \subset \mathcal {R}(\Lambda )$. According to Lemma 2.9, the result then follows by induction on the number of summands in (16), since it holds for the simple objects $\{ {\overline{w}}_{a, \Lambda _1} \}_{a \in I}$ and $\{ {\overline{w}}_{a, \Lambda _2} \}_{a \in I}$ by Lemma 2.6 and for finite tensor products of these by repeated application of Lemma 2.7.

For $i\in \{1,2\}$, let $V_1^{(i)}, V_2^{(i)}\in \mathcal {R}(\Lambda _i)$ be the isometries used to define $\oplus _{\Lambda _i}$, and let $p_k^{(i)} = V_k^{(i)} (V_k^{(i)})^*$ for $k=1,2$. Then $(V_k^{(i)})^* p^{(i)}_l = \delta _{k, l} (V_k^{(i)})^*$ and $p^{(i)}_k V^{(i)}_l = \delta _{k, l} V_k^{(i)}$. Suppose $R \in ({{\overline{\rho }}}_1 \oplus _{\Lambda _1} {{\overline{\rho }}}_2, {{\overline{\sigma }}}_1 \oplus _{\Lambda _2} {{\overline{\sigma }}}_2)$, then

$$\begin{aligned}{} & {} R \big ( \textrm{Ad}[V_1^{(1)}] \circ {{\overline{\rho }}}_1(O) + \textrm{Ad}[V_2^{(1)}] \circ {{\overline{\rho }}}_2(O) \big )\nonumber \\{} & {} \quad = \big ( \textrm{Ad}[V_1^{(2)}] \circ {{\overline{\sigma }}}_1(O) + \textrm{Ad}[V_2^{(2)}] \circ {{\overline{\sigma }}}_2(O) \big ) R \end{aligned}$$

(17)

for all $O \in \mathcal {A}$. Multiplying from the left with $(V_k^{(2)})^*$ and from the right with $V_l^{(1)}$ yields

$$\begin{aligned} (V_k^{(2)})^* R \, V_l^{(1)} \, {{\overline{\rho }}}_l(O) = {{\overline{\sigma }}}_k(O) \, (V_k^{(2)})^* R \, V_l^{(1)} \end{aligned}$$

(18)

for all $O \in \mathcal {A}$, so $(V_k^{(2)})^* R \, V_l^{(1)} \in ({{\overline{\rho }}}_l, {{\overline{\sigma }}}_k)$. So then by hypothesis, $(V_k^{(2)})^* R \, V_l^{(1)} \in \mathcal {R}(\Lambda )$ for all $k, l = 1, 2$, and also $p_k^{(2)} R \, p_l^{(1)} \in \mathcal {R}(\Lambda )$. Therefore, $R = \sum _{k, l} p_k^{(2)} R \, p_l^{(1)} \in \mathcal {R}(\Lambda )$ as required. $\square$

Fig. 1

Cones $\Lambda _0$, $\Lambda _L$ and $\Lambda _R$ used in the definition of the braiding intertwiners $\epsilon (\rho , \sigma )$

2.2.3 Braided Structure

Fix an allowed cone $\Lambda _0$ and consider two endomorphisms ${\bar{\rho }}, {\bar{\sigma }} \in \Delta ^I_{\Lambda _0}$. Pick allowed cones $\Lambda _L$ and $\Lambda _R$ as in Fig. 1. i.e. the disjoint allowed cones $\Lambda _R, \Lambda _0$ and $\Lambda _L$ are arranged in a counterclockwise order from the forbidden direction, and there are allowed cones $\widetilde{\Lambda }_L \supset \Lambda _L \cup \Lambda _0$ and $\widetilde{\Lambda }_R \supset \Lambda _R \cup \Lambda _0$ such that $\widetilde{\Lambda }_L \cap \Lambda _R = \widetilde{\Lambda }_R \cap \Lambda _L = \emptyset$. We say $\Lambda _L$ is to the left of $\Lambda _0$, and $\Lambda _R$ is to the right of $\Lambda _0$. By transportability, there are endomorphisms ${\bar{\rho }}_L \in \Delta ^I_{\Lambda _L}$ and ${\bar{\sigma }}_R \in \Delta ^I_{\Lambda _R}$, and unitary intertwiners $U \in ({\bar{\rho }}, {\bar{\rho }}_L)$ and $V \in ({\bar{\sigma }}, {\bar{\sigma }}_R)$ such that $U \in \mathcal {R}(\widetilde{\Lambda }_L)$ and $V \in \mathcal {R}(\widetilde{\Lambda }_R)$.

Definition 2.11

The braiding intertwiner $\epsilon ({\bar{\rho }}, {\bar{\sigma }}) \in ({\bar{\rho }} \otimes {\bar{\sigma }}, {\bar{\sigma }} \otimes {\bar{\rho }})$ is given by

$$\begin{aligned} \epsilon ({\bar{\rho }}, {\bar{\sigma }}):= (V^* \otimes U^*)(U \otimes V) = V^* {\bar{\rho }}(V). \end{aligned}$$

(19)

To get the last equality, we use ${{\overline{\sigma }}}_R(U) = U$ which holds because ${\bar{\sigma }}_R$ is localised on $\Lambda _R$ while $U \in \mathcal {R}( \widetilde{\Lambda }_L )$. Using ${\bar{\rho }}_L \otimes {\bar{\sigma }}_R = {\bar{\sigma }}_R \otimes {\bar{\rho }}_L$, one easily verifies that $\epsilon ({\bar{\rho }}, {\bar{\sigma }})$ is indeed an intertwiner from ${\bar{\rho }} \otimes {\bar{\sigma }}$ to ${\bar{\sigma }} \otimes {\bar{\rho }}$.

Lemma 2.12

The braiding $\epsilon ({\bar{\rho }}, {\bar{\sigma }})$ is independent of the choice of cones $\Lambda _L, \Lambda _R$, the choice of objects ${\bar{\rho }}_L \in \Delta _{\Lambda _L}^I$ and ${\bar{\sigma }}_R \in \Delta _{\Lambda _R}^I$ and the choice of intertwiners $U \in ({\bar{\rho }}, {\bar{\rho }}_L)$ and $V \in ({\bar{\sigma }}, {\bar{\sigma }}_R)$.

Proof

Choose different cones $\Lambda _L'$ and $\Lambda _R'$ to the left and to the right of $\Lambda _0$. Then there are allowed cones $\Lambda ''_L \supseteq \Lambda _L \cup \Lambda '_L$ and $\Lambda ''_R \supseteq \Lambda _R \cup \Lambda '_R$ that are also to the left and to the right of $\Lambda _0$, respectively. Choose objects ${\bar{\rho }}'_L \in \Delta _{\Lambda '_L}^I$ and ${\bar{\sigma }}'_R \in \Delta _{\Lambda '_R}^I$, as well as morphisms $U' \in ({\bar{\rho }}, {\bar{\rho }}'_L)$ and $V' \in ({\bar{\sigma }}, {\bar{\sigma }}'_R)$. Then $V'V^* \in ({\bar{\sigma }}_R, {\bar{\sigma }}'_R) \subset \mathcal {R}(\Lambda ''_R)$ by Lemma 2.10. The new choice ${\bar{\rho }}'_L, {\bar{\sigma }}'_R$ leads to a braiding intertwiner

$$\begin{aligned} \epsilon '({\bar{\rho }}, {\bar{\sigma }})&= (V'^* \otimes U'^*)(U' \otimes V') = V'^* {\bar{\rho }} (V') \\&= V^* (V' V^*)^* {\bar{\rho }} \big ( (V' V^*) V \big ) = V^* {\bar{\rho }}( V ) \\&= \epsilon ({\bar{\rho }}, {\bar{\sigma }}) \end{aligned}$$

where we used ${\bar{\rho }}( V'V^* ) = V'V^*$ since $V'V^* \in \mathcal {R}(\Lambda ''_R)$ and ${\bar{\rho }}$ is supported in $\Lambda _0$, which is disjoint from $\Lambda ''_R$. $\square$

Lemma 2.13

The braiding intertwiners satisfy the braid equations

$$\begin{aligned} \epsilon (\rho \otimes \sigma , \tau )&= \big ( \epsilon (\rho , \tau ) \otimes 1_{\sigma } \big ) \big ( 1_{\rho } \otimes \epsilon (\sigma , \tau ) \big ) \\ \epsilon (\rho , \sigma \otimes \tau )&= \big ( 1_{\sigma } \otimes \epsilon (\rho , \tau ) \big ) \big ( \epsilon (\rho , \sigma ) \otimes 1_{\tau } \big ) \end{aligned}$$

where $1_{\rho } = {{\,\mathrm{\mathbb {1}}\,}}\in (\rho , \rho )$.

Proof

Let us prove the first equation, the second is shown in the same way. Choose $\rho _L, \sigma _L$ supported in $\Lambda _L$ and morphisms $U_{\rho } \in (\rho , \rho _L)$, $U_{\sigma } \in (\sigma , \sigma _L)$. Choose $\tau _R$ supported in $\Lambda _R$ and a morphism $V_{\tau } \in (\tau , \tau _R)$. Then

$$\begin{aligned} \epsilon (\rho \otimes \sigma , \tau )&= \big ( V_{\tau }^* \otimes (U_{\rho } \otimes U_{\sigma })^* \big ) \big ( (U_{\rho } \otimes U_{\sigma } ) \otimes V_{\tau } \big ) \\&= V_{\tau }^* \overline{\rho \otimes \sigma }( V_{\tau } ) = V_{\tau }^* {{\overline{\rho }}} \big ( {{\overline{\sigma }}} \big ( V_{\tau } \big ) \big ) \\&= V_{\tau }^* {{\overline{\rho }}}(V_{\tau }) {{\overline{\rho }}}(V_{\tau }^*) {{\overline{\rho }}} \big ( {{\overline{\sigma }}}( V_{\tau }) \big ) \\&= \big ( \epsilon (\rho , \tau ) \otimes 1_{\sigma } \big ) \big ( {{\overline{\rho }}}( V_{\tau }^* {{\overline{\sigma }}}(V_{\tau }) ) \big ) \\&= \big ( \epsilon (\rho , \tau ) \otimes 1_{\sigma } \big )\big ( 1_{\rho } \otimes \epsilon (\sigma , \tau ) \big ). \end{aligned}$$

$\square$

2.2.4 Relation to Previous Work

The state $\omega$ is said to satisfy Haag duality for cones if $\mathcal {R}(\Lambda ) = \mathcal {R}(\Lambda ^c)'$ for all cones $\Lambda$. Haag duality for cones has been verified for abelian quantum double models in [8, 20]. We believe that the double semion state introduced below can also be shown to satisfy Haag duality for cones using similar methods.

Under the assumption that the pure state $\omega$ satisfies (approximate) Haag duality for cones, it is shown in [21] that the category of localised and transportable endomorphisms $\Delta$ is a braided C$^*$-tensor category with isomorphism classes of simple objects in one-to-one correspondence with (equivalence classes of) irreducible representations of the observable algebra that satisfy the superselection criterion (Definition 2.1).

The braided tensor categories $\Delta _{\Lambda }^I$ constructed above are full subcategories of $\Delta$. If $\omega$ satisfies Haag duality for cones, then $\Delta$ is equipped with a braiding given by Definition 4.11 of [21]. The restriction of this braiding to $\Delta _{\Lambda }^I$ agrees with the braiding defined in 2.11. It follows that the category $\Delta _{\Lambda }^I$ completely describes the closed system of abelian anyons corresponding to the irreducible representations $\{ \pi _a \}_{a \in I}$ in a way that is consistent with the theory of [21]. These abelian anyons form a subset of all anyon types present in the model (anyon types correspond to irreducible representations of the observables algebra that satisfy the superselection criterion). In contrast, the theory of [21] captures all anyon types, in particular also all non-abelian anyons.

The reason that (approximate) Haag duality allows a description of all anyon types is twofold. Most importantly, Haag duality allows one to construct for any representation $\pi$ that satisfies the superselection criterion with respect to $\pi _1$, and for any cone $\Lambda$, a transportable endomorphism ${{\overline{\rho }}}_{\pi , \Lambda }$ localised on $\Lambda$ such that $\pi \simeq {{\overline{\rho }}}_{\pi , \Lambda } \circ \pi _1$ (Definition 2.13 and Lemma 2.14 of [21]). This ensures that $\Delta$ contains objects corresponding to any superselection sector. In our setting, we obtain localised and transportable endomorphisms for the superselection sectors $\{ \pi _a \}_{a \in I}$ using Assumptions 1 and 4, see Lemma 2.5.

Haag duality also allows a braiding to be defined for the entire tensor category $\Delta$. Note that for the braiding of definition 2.11 to be well defined, we must show that it is independent of the choice of intertwiners U and V used in the definition. This is done in Lemma 2.12 using the fact that if ${\bar{\rho }} \in \Delta _{\Lambda _1}^I$ and ${\bar{\sigma }} \in \Delta _{\Lambda _2}^I$, then all morphisms in $({\bar{\rho }}, {\bar{\sigma }})$ are elements of $\mathcal {R}(\Lambda )$ for any allowed cone $\Lambda \supseteq \Lambda _1 \cup \Lambda _2$. In our setting, this follows from Assumption 4 (Lemma 2.10). With Haag duality for cones, this locality property of morphisms follows immediately. Indeed, suppose ${{\overline{\rho }}}$ and ${{\overline{\sigma }}}$ are both localised on an allowed cone $\Lambda$, and suppose $R \in ({{\overline{\rho }}}, {{\overline{\sigma }}})$. Then for any $O \in \mathcal {A}_{\Lambda ^c}$ we have

$$\begin{aligned} {{\overline{\rho }}}( \pi _1(O) ) = {{\overline{\sigma }}}( \pi _1(O) ) = \pi _1(O), \end{aligned}$$

(20)

and hence

$$\begin{aligned} R \pi _1(O) = R {{\overline{\rho }}}( \pi _1(O) ) = {{\overline{\sigma }}}( \pi _1(O) ) R = \pi _1(O) R. \end{aligned}$$

(21)

We see that $R \in \pi _1(\mathcal {A}_{\Lambda ^c})' = \mathcal {R}(\Lambda ^c)' = \mathcal {R}(\Lambda )$ by Haag duality.

Using the same argument, one can use Haag duality for cones to prove Assumption 4, as mentioned above.

2.3 Fusion Ring, F-Symbols, and R-Symbols

2.3.1 Fusion Ring of $\Delta _{\Lambda }^I$

To any semisimple tensor category, one may associate its fusion ring, see Section 4.5 of [7] for details. For the category $\Delta _{\Lambda }^I$, the construction goes as follows. First, note that the isomorphism classes of simple objects of $\Delta _{\Lambda }^I$ are labelled by the elements of I, they are precisely the classes $[ {\overline{w}}_{a, \Lambda } ]$ for $a \in I$. Since $\Delta _{\Lambda }^I$ is semisimple, any object ${\bar{\rho }}$ of $\Delta _{\Lambda }^I$ is isomorphic to a direct sum of simple objects ${\overline{w}}_{a, \Lambda }$. The number of times that ${\overline{w}}_{a, \Lambda }$ appears in such a direct sum decomposition is independent on the particular choice of direct sum decomposition and is called the multiplicity of a in ${\bar{\rho }}$, and denoted by $[{\bar{\rho }}: a]$. The fusion ring of $\Delta _{\Lambda }^{I}$ is the free abelian group generated by the isomorphism classes of simple objects $\{ [{\overline{w}}_{a, \Lambda }] \}_{a \in I}$, which we can identify with the elements of I. Addition in this group corresponds to the direct sum in the category. The isomorphism class of a general object ${\bar{\rho }} \in \Delta _{\Lambda }^I$ corresponds to an element of fusion ring given by

$$\begin{aligned}{}[{\bar{\rho }}]:= \sum _{a \in I} [ {\bar{\rho }}: a ] \, a. \end{aligned}$$

(22)

The multiplication of the fusion ring is given by the tensor product of the category. It is sufficient to define the multiplication on the generators by

$$\begin{aligned} a \times b:= [ {\overline{w}}_{a, \Lambda } \otimes {\overline{w}}_{b, \Lambda } ] = [ {\overline{w}}_{a \times b, \Lambda } ] = ab, \end{aligned}$$

(23)

which corresponds precisely to the multiplication on I introduced in Assumption 2 and Lemma 2.3. We see that the fusion ring of $\Delta _{\Lambda }^I$ is given by $\mathbb {Z}[I]$, the ring of polynomials over the abelian group I with integer coefficients.

Two tensor categories that are monoidally equivalent have isomorphic fusion rings. The converse certainly does not hold; there are inequivalent tensor categories that have isomorphic fusion rings (for example, all categories $\textrm{Vec}_G^{\omega }$ for 3-cocycle $\omega : G^{\times 3} \rightarrow \mathbb {C}^{\times }$ have fusion ring $\mathbb {Z}[G]$). It turns out that the fusion ring together with certain cohomological data is sufficient information to characterise a tensor category up to monoidal equivalence, see, for example, Proposition 1.1 of [23]. Below, we will describe these cohomological data for the category $\Delta _{\Lambda }^I$ in terms of ‘F-symbols’, a nomenclature common in the physics literature. Since we are interested in braided tensor categories, the question arises to what extent the fusion ring together with the F-symbols characterises a braided tensor category. The braiding induces more structure on the fusion ring in the form of ‘R-symbols’, which will be described for our $\Delta _{\Lambda }^I$ below. To the best of the authors’ knowledge, it is not known if a fusion ring together with F and R-symbols is enough information to characterise a braided tensor category completely. In the special case where the fusion ring takes the form $\mathbb {Z}[G]$ for a finite abelian group G however, this does turn out to be the case, see Proposition 7.5.2 of [12].

2.3.2 Fusion and F-Symbols

Fix an allowed cone $\Lambda _0$ and write ${\overline{w}}_{a}:= {\overline{w}}_{a, \Lambda _0}$. Pick unitary intertwiners $\Omega (a, b) \in ({\overline{w}}_a \otimes {\overline{w}}_b, {\overline{w}}_{a \times b}) \subset \mathcal {R}(\Lambda _0)$ called fusion operators. Note that the $\Omega (a, b)$ are unique up to phase. The unitaries

$$\begin{aligned} \Omega (ab, c) (\Omega (a, b) \otimes 1_c)&= \Omega (ab, c) \Omega (a, b) \\ \Omega (a, bc) (1_a \otimes \Omega (b, c))&= \Omega (a, bc) {\overline{w}}_a( \Omega (b, c) ) \end{aligned}$$

are both intertwiners from ${\overline{w}}_{a} \otimes {\overline{w}}_b \otimes {\overline{w}}_c$ to ${\overline{w}}_{abc}$. Since $( {\overline{w}}_a \otimes {\overline{w}}_b \otimes {\overline{w}}_c, {\overline{w}}_{abc})$ is one-dimensional, there are phases $F(a, b, c) \in U(1)$ such that

$$\begin{aligned} \Omega (ab, c) \Omega (a, b) = F(a, b, c) \times \Omega (a, bc) {\overline{w}}_a(\Omega (b, c)). \end{aligned}$$

(24)

These F(a, b, c) are the F-symbols. Figure 2 gives a graphical representation of Eq. (24).

Fig. 2

Graphical representation of Eq. (24), defining the F-symbols F(a, b, c). Each node represents a fusion operator. The diagrams represent two different compositions of fusion operators both yielding intertwiners from $w_a \otimes w_b \otimes w_c$ to $w_{abc}$

The F-symbols satisfy a pentagon equation, which in our setting of abelian anyons takes the form of a cocycle relation.

Proposition 2.14

The F-symbols satisfy

$$\begin{aligned} (\textrm{d}F)(a, b, c, d):= \frac{F(a, b, c) F(a, bc, d) F(b, c, d)}{F(ab, c, d) F(a, b, cd)} = 1. \end{aligned}$$

(25)

Proof

A graphical proof is shown in Fig. 3. In equations, we have

$$\begin{aligned}&\Omega (abc, d) \Omega (ab, c) \Omega (a, b)\\&\quad = F(ab, c, d) \times \Omega (ab, cd) {\overline{w}}_{ab} \big ( \Omega (c, d) \big ) \Omega (a, b) \\&\quad = F(ab, c, d) \times \Omega (ab, cd) \Omega (a, b) {\overline{w}}_a \big ( {\overline{w}}_b \big ( \Omega (c, d) \big ) \big ) \\&\quad = F(ab, c, d) F(a, b, cd) \times \Omega (a, bcd) {\overline{w}}_{a}( \Omega (b, cd)) {\overline{w}}_a( {\overline{w}}_b( \Omega (c, d)) ) \\&\quad = F(ab, c, d) F(a, b, cd) \times \Omega (a, bcd) {\overline{w}}_a \big ( \Omega (b, cd) {\overline{w}}_b( \Omega (c, d) ) \big ) \end{aligned}$$

but also

$$\begin{aligned}&\Omega (abc, d) \Omega (ab, c) \Omega (a, b)\\&\quad = F(a, b, c) \times \Omega (abc, d) \Omega (a, bc) {\overline{w}}_a( \Omega (b, c) ) \\&\quad = F(a, b, c) F(a, bc, d) \times \Omega (a, bcd) {\overline{w}}_{a}( \Omega (bc, d) ) {\overline{w}}_a( \Omega (b, c) ) \\&\quad = F(a, b, c) F(a, bc, d) \times \Omega (a, bcd) {\overline{w}}_a \big ( \Omega (bc, d) \Omega (b, c) \big ) \\&\quad = F(a, b, c) F(a, bc, d) F(b, c, d) \times \Omega (a, bcd) {\overline{w}}_a \big ( \Omega (b, cd) {\overline{w}}_b( \Omega (c, d) ) \big ). \end{aligned}$$

And the desired equality follows. $\square$

Fig. 3

A graphical proof of the Pentagon equation

2.3.3 Braiding and R-Symbols

We simply set $\epsilon (a, b):= \epsilon ({\overline{w}}_a, {\overline{w}}_b)$ for any $a, b \in I$. The unitaries $\Omega (b, a) \epsilon (a, b)$ and $\Omega (a, b)$ are both intertwiners from ${\overline{w}}_a \otimes {\overline{w}}_b$ to ${\overline{w}}_{ab}$. Since $({\overline{w}}_a \otimes {\overline{w}}_b, {\overline{w}}_{ab})$ is one-dimensional, there exist phases $R(a, b) \in U(1)$ such that

$$\begin{aligned} \Omega (b, a) \epsilon (a, b) = R(a, b) \times \Omega (a, b). \end{aligned}$$

(26)

The phases R(a, b) are the R-symbols. Figure 4 gives a graphical representation of Eq. (26).

Fig. 4

Graphical representation of Eq. (26), defining the R-symbols R(a, b). The point where the a-line passes under the b-line represents the braiding intertwiner $\epsilon (a, b)$

2.3.4 Yang–Baxter Equation

The braidings $\epsilon (a, b)$ and fusions $\Omega (a, b)$ satisfy the Yang–Baxter equations, see Fig. 5.

Proposition 2.15

We have

$$\begin{aligned} {\overline{w}}_c \big (\Omega (a, b) \big ) \epsilon (a, c) {\overline{w}}_a \big ( \epsilon (b, c) \big ) = \epsilon (ab, c) \Omega (a, b) \end{aligned}$$

(27)

and

$$\begin{aligned} \Omega (b, c) {\overline{w}}_b \big ( \epsilon (a, c) \big ) \epsilon (a, b) = \epsilon (a, bc) {\overline{w}}_a \big ( \Omega (b, c) \big ). \end{aligned}$$

(28)

Proof

We prove Eq. (27); the proof of Eq. (28) is similar. Choose a cone $\Lambda _R$, an endomorphism ${\overline{w}}_c^R \in \Delta _{\Lambda _R}^I$, and a unitary $V \in ({\overline{w}}_c, {\overline{w}}_c^R)$ as in Definition 2.11. Then

$$\begin{aligned} \epsilon (ab, c) \Omega (a, b)&= \epsilon ( {\overline{w}}_{ab}, {\overline{w}}_c ) \Omega (a, b) \\&= V^* \, {\overline{w}}_{ab} \big ( V \big ) \Omega (a, b) \\&= V^* \Omega (a, b) {\overline{w}}_a \big ( {\overline{w}}_b (V) \big ) \\&= V^* {\overline{w}}_c^R \big ( \Omega (a, b) \big ) {\overline{w}}_a( V ) \, {\overline{w}}_a \big ( V^* {\overline{w}}_b(V) \big ) \\&= {\overline{w}}_c \big ( \Omega (a, b) \big ) \, V^* {\overline{w}}_a( V ) \, {\overline{w}}_a \big ( V^* {\overline{w}}_b(V) \big ) \\&= {\overline{w}}_c \big ( \Omega (a, b) \big ) \, \epsilon (a, c) \, {\overline{w}}_a \big ( \epsilon (b, c) \big ) \end{aligned}$$

where we used $\Omega (a, b) \in ({\overline{w}}_{a} \otimes {\overline{w}}_b, {\overline{w}}_{ab})$ to obtain the third line. We used the fact that $\Omega (a, b) \in \mathcal {R}(\Lambda _0)$ so ${\overline{w}}_c^R( \Omega (a, b) ) = \Omega (a, b)$ to obtain the fourth line. The fifth line follows from $V \in ({\overline{w}}_c, {\overline{w}}_c^R)$ and the fact that V is unitary, and the final line follows by the definition of the braidings $\epsilon (a, c)$ and $\epsilon (b, c)$. $\square$

Fig. 5

Graphical representation of the Yang–Baxter equations

2.3.5 Hexagon Equation

Fig. 6

Graphical representations of the first and second hexagon equations

Using the Yang–Baxter equation, we obtain the Hexagon equation, see Fig. 6.

Proposition 2.16

The F and R-symbols satisfy the hexagon equations

$$\begin{aligned} \frac{F(a, b, c) F(c, a, b)}{F(a, c, b)} = \frac{R(a, c) R(b, c)}{R(ab, c)} \end{aligned}$$

(29)

and

$$\begin{aligned} \frac{F(a, b, c) F(b, c, a)}{F(b, a, c)} = \frac{R(a, bc)}{R(a, b) R(a, c)}. \end{aligned}$$

(30)

Proof

The left diagram in Fig. 6 suggests the following two equalities of morphisms:

$$\begin{aligned}&\Omega (ca, b) \Omega (c, a) \epsilon (a, c) {\overline{w}}_a(\epsilon (b, c))\\&\quad = R(a, c) \times \Omega (ca, b) \Omega (a, c) {\overline{w}}_a(\epsilon (b, c)) \\&\quad = R(a, c) F(a, c, b) \times \Omega (a, cb) {\overline{w}}_a (\Omega (c, b)) {\overline{w}}_a( \epsilon (b, c)) \\&\quad = R(a, c) F(a, c, b) \times \Omega (a, cb) {\overline{w}}_a \big ( \Omega (c, b) \epsilon (b, c) \big ) \\&\quad = R(a, c) F(a, c, b) R(b, c) \times \Omega (a, cb) {\overline{w}}_a( \Omega (b, c) ) \end{aligned}$$

and

$$\begin{aligned}&\Omega (ca, b) \Omega (c, a) \epsilon (a, c) {\overline{w}}_a(\epsilon (b, c))\\&\quad = F(c, a, b) \times \Omega (c, ab) {\overline{w}}_c( \Omega (a, b) ) \epsilon (a, c) {\overline{w}}_a(\epsilon (b, c)) \\&\quad = F(c, a, b) \times \Omega (c, ab) \epsilon (ab, c) \Omega (a, b) \\&\quad = F(c, a, b) R(ab, c) \times \Omega (ab, c) \Omega (a, b) \\&\quad = F(c, a, b) R(ab, c) F(a, b, c) \times \Omega (a, bc) {\overline{w}}_a(\Omega (b, c)) \end{aligned}$$

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 Fig. 6. $\square$

2.3.6 Dependence of F and R-Symbols on the Choice of $\Lambda _0$ and the Phases of $\Omega (a, b)$.

Suppose we chose different phases for the intertwiners $\Omega (a, b)$, i.e. we consider

$$\begin{aligned} \Omega '(a, b) = \chi (a, b) \Omega (a, b) \end{aligned}$$

(31)

for phases $\chi (a, b)$. This yields new F-symbols by

$$\begin{aligned} \Omega '(ab, c) \Omega '(a, b) = F'(a, b, c) \times \Omega '(a, bc) {\overline{w}}_a (\Omega '(b, c)) \end{aligned}$$

(32)

which are related to the original F-symbols by

$$\begin{aligned} F'(a, b, c) = (\textrm{d}\chi )(a, b, c) F(a, b, c) = \frac{\chi (b, c) \chi (a, bc)}{\chi (ab, c) \chi (a, b)} F(a, b, c). \end{aligned}$$

(33)

i.e. $F'$ is related to F by the coboundary $\textrm{d}\chi$. It follows that only the cohomology class $[F] \in H^3(I, U(1))$ is well defined.

The R-symbols are also affected by the different choice of phases. Indeed, the new R-symbols defined by

$$\begin{aligned} \Omega '(b, a) \epsilon (a, b) = R'(a, b) \times \Omega '(a, b) \end{aligned}$$

(34)

are related to the old by

$$\begin{aligned} R'(a, b) = \frac{\chi (b, a)}{\chi (a, b)} R(a, b). \end{aligned}$$

(35)

It follows that the self-statistics 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 $\Lambda _0$. We will find no additional ambiguity beyond the one just discussed.

Let $\Lambda _0'$ be another allowed cone. Then there is an allowed cone $\widetilde{\Lambda }_0$ containing $\Lambda _0 \cup \Lambda _0'$. Denote ${\overline{w}}'_a = {\overline{w}}_{a, \Lambda _0'}$. Then there are unitaries $W_a \in \mathcal {R}(\widetilde{\Lambda }_0)$ such that

$$\begin{aligned} W_a \in ( {\overline{w}}_a, {\overline{w}}'_a). \end{aligned}$$

(36)

These unitaries are unique up to phase.

This leads to new fusion intertwiners $\Omega '(a, b) \in ( {\overline{w}}'_a \otimes {\overline{w}}'_b, {\overline{w}}'_{ab} )$ given by

$$\begin{aligned} \Omega '(a, b) = W_{ab} \Omega (a, b) ( W_a \otimes W_b )^*. \end{aligned}$$

(37)

The new F-symbols are determined by

$$\begin{aligned} \Omega '(ab, c) \Omega '(a, b) = F'(a, b, c) \times \Omega '(a, bc) {\bar{w}}'_a( \Omega '(b, c) ). \end{aligned}$$

(38)

Using Eq. 37, we compute

$$\begin{aligned} \Omega '(ab, c) \Omega '(a, b) = W_{abc} \Omega (ab, c) \Omega (a, b) {\bar{w}}_a \big ( {\bar{w}}_b( W_c^* ) W_b^* \big ) W_a^* \end{aligned}$$

(39)

and

$$\begin{aligned} \Omega '(a, bc) {\bar{w}}'_a( \Omega '(b, c) ) = W_{abc} \Omega (a, bc) {\bar{w}}_a( \Omega (b, c) ) {\bar{w}}_a \big ( {\bar{w}}_b( W_c^* ) W_b^* \big ) W_a^*. \end{aligned}$$

(40)

It follows that $F'(a, b, c) = F(a, b, c)$ for all $a, b, c \in I$.

Recall that the braiding intertwiners $\epsilon (a, b)$ are defined in terms of endomorphisms ${\overline{w}}_a$ localised on $\Lambda _0$, ${\overline{w}}_a^L$ localised on $\Lambda _L$ and ${\overline{w}}_a^R$ localised on $\Lambda _R$ as follows. Pick intertwiners (unique up to phase) $U_a \in ({\overline{w}}_a, {\overline{w}}_a^L)$ and $V_a \in ({\overline{w}}_a, {\overline{w}}_a^R)$, then

$$\begin{aligned} \epsilon (a, b) = (V_b^* \otimes U_a^*)(U_a \otimes V_b) = V_b^* \overline{w}_a( V_b ). \end{aligned}$$

(41)

It is shown in Lemma 2.12 that this braiding intertwiner is independent of the choice of cones $\Lambda _L, \Lambda _R$. Moreover, $\epsilon (a, b)$ is independent of the choice of phase for the intertwiners $U_a$ and $V_a$.

In order to make the comparison with the braiding on $\Delta _{\widetilde{\Lambda }_0}^I$, let us choose the left and right cones $\Lambda _L$ and $\Lambda _R$ such that they are to the left and right of both $\Lambda _0$ and $\widetilde{\Lambda }_0$.

With the new endomorphisms ${\overline{w}}'_a$ related to the old ${\overline{w}}_a$ by Eq. (36), we get new intertwiners $U'_a = U_a W^*_a \in ({\overline{w}}'_a, {\overline{w}}_a^L)$ and $V'_a = V_a W^*_a \in ({\overline{w}}'_a, {\overline{w}}_a^R)$ and therefore new braiding intertwiners

$$\begin{aligned} \epsilon '(a, b) = ((V'_b)^* \otimes (U'_a)^*)(U'_a \otimes V'_b) = (V'_b)^* {\bar{w}}'_a \big ( V'_b \big ). \end{aligned}$$

(42)

A short computation relates this to the braiding $\epsilon (a, b)$ as

$$\begin{aligned} \epsilon '(a, b) = W_b V_b^* W_a V_b \epsilon (a, b) {\overline{w}}_a( W^*_b ) W^*_a. \end{aligned}$$

(43)

The new R-symbol is determined by

$$\begin{aligned} \Omega '(b, a) \epsilon '(a, b) = R'(a, b) \times \Omega '(a, b). \end{aligned}$$

(44)

Using Eqs. (37) and (43), the left-hand side becomes

$$\begin{aligned} \Omega '(b, a) \epsilon '(a, b) = W_{ba} \Omega (b, a) \epsilon (a, b) \overline{w}_a( W^*_b ) W^*_a \end{aligned}$$

(45)

and

$$\begin{aligned} \Omega '(a, b) = W_{ab} \Omega (a, b) \overline{w}_a( W^*_b ) W^*_a \end{aligned}$$

(46)

so, noting that $ab = ba$, we find $R'(a, b) = R(a, b)$.

We conclude that Eqs. (33) and (35) are the only ambiguities in the F and R-symbols.

We call two sets of F and R-symbols (F, R) and $(F', R')$ on the fusion ring $\mathbb {Z}(I)$ gauge equivalent if they are related by (33) and (35) for some phases $\chi (a, b)$.

3 The Double Semion State

We construct an infinite volume version of the ground state of the double semion model, first introduced in [16]. 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 $\mathcal {D}^{\phi }(\mathbb {Z}_2)$.

3.1 Construction of the Double Semion State

Fig. 7

Degrees of freedom of the double semion state live on the edges of a hexagonal lattice

Fig. 8

$\Pi$ is the set of edges of the hexagons shaded blue. Acting with $A_{\Pi }$ on the state $\omega _0$ yields a string configuration with two connected components (color figure online)

Fig. 9

Increasing sequence of balls $\Pi _n$ depicted in the primal hexagonal lattice

Let $\Gamma ^V \subset \mathbb {R}^2$ be the vertices of the hexagonal lattice. We take $\Gamma = \Gamma ^E$ to be the (midpoints of the) edges of the hexagonal lattice (Fig. 7) and to each edge $e \in \Gamma$ we associate a degree of freedom $\mathcal {A}_e \simeq \textrm{End}(\mathbb {C}^2)$. We fix Pauli matrices $\sigma _e^X, \sigma _e^Y, \sigma _e^Z$ in each $\mathcal {A}_e$. We denote by $\Gamma ^F = (\Gamma ^V)^*$ be the set of faces of the hexagonal lattice.

We say an edge $e \in \Gamma$ belongs to a hexagon $p \in \Gamma ^F$ and write $e \in p$ if e is one of the six boundary edges of p. We write $\partial p$ for the set of six edges that belong to p. For any subset $\Pi \subset \Gamma ^F$, we write $\Pi ^E = \cup _{p \in \Pi } \partial p$ for all edges that belong to some hexagon in $\Pi$ and by $\partial \Pi = \Pi ^E \cap (\Pi ^c)^E$ the collection of edges that belong to exactly one hexagon in $\Pi$. That is, $\partial \Pi$ is the boundary of $\Pi$.

We interpret $\sigma _e^Z = -1$ as the edge e being occupied by a string, while $\sigma _e^Z=1$ means that the edge is unoccupied.

For any hexagon $p \in \Gamma ^F$, let

$$\begin{aligned} A_p = \prod _{e \in \partial p} \sigma _e^X \end{aligned}$$

(47)

and for any finite set $\Pi$ of hexagons, let

$$\begin{aligned} A_{\Pi } = \prod _{p \in \Pi } A_p =\prod _{e \in \partial \Pi } \sigma _e^X. \end{aligned}$$

(48)

Note that $A_{\Pi }$ produces a string around the region $\Pi$ when it acts on $\omega _0$, see Fig. 8.

Let us fix a hexagon $p_0 \in \Pi ^F$ as an origin and define $\Pi _n = \{ p \in \Pi ^F \,: \, \textrm{dist}(p, p_0) \le n \}$, where $\textrm{dist}(\cdot , \cdot )$ is the graph distance for $\Pi ^F$, see Fig. 9.

Let $\omega _0$ be the pure product state without any strings, i.e. $\omega _0(\sigma _e^Z) = 1$ for all $e \in \Gamma$. Let $(\pi _0, \mathcal {H}_0, \Omega _0 )$ be the GNS triple for $\omega _0$, and let

$$\begin{aligned} \Omega _n:= \sqrt{\frac{1}{2^{\left| \Pi _n\right| }}} \, \sum _{\Pi \subset \Pi _n} \, (-1)^{\sharp \Pi } \, A_{\Pi } \Omega _0 \end{aligned}$$

(49)

where $\sharp \Pi$ is the number of connected components of $\Pi$. i.e. $\Omega _n$ is a superposition of closed string configurations supported in $\Pi ^E_n$, with phases determined by the parity of the number of components of the string configuration.

The vectors $\Omega _n$ determine a sequence of pure states $\omega _n$ on $\mathcal {A}$. The following theorem is proved in Appendix A.

Theorem 3.1

The sequence $\omega _n$ converges in the weak-* topology to a pure state $\omega$.

We call this pure state $\omega$ the double semion state and denote its GNS triple by $(\pi _1, \mathcal {H}, \Omega )$.

3.2 String Operators

An oriented edge is a pair $(v_0, v_1)$ of neighbouring vertices of $\Gamma ^V$. We say $v_0$ ($v_1$) is the initial (final) vertex of $(v_0, v_1)$. A path P is a collection of oriented edges such that there is a sequence of vertices $( \cdots , v_{i-1}, v_i, v_{i+1} \cdots )$ such that each oriented edge in P is of the form $e = (v_i, v_{i+1})$ for some i. Such a sequence is called a vertex sequence for P. The set of vertices appearing in any vertex sequence for P is uniquely determined by P and denoted by $P^V$. We call $P^V$ the vertex set of P; it is the set of vertices that are the initial or final vertex of some edge in P. We, moreover, require paths to be self-avoiding in the sense that any vertex sequence for P consists of distinct vertices, except for possibly the initial vertex, which may be equal to the final vertex of the sequence, if these exist. In the latter case, we say the path P is closed. Similarly, if a vertex sequence for P is bi-infinite, we also say P is closed. If P has finite vertex sequence with all its vertices being distinct, then the vertex sequence is uniquely determined by P and we denote by $\partial _0 P$ the first vertex of the vertex sequence and by $\partial _1 P$ the final vertex of the vertex sequence. We say an edge e belongs to P if P contains an oriented edge corresponding to e. With slight abuse of notation, we write $e \in P$ if e belongs to P.

Let P be a path. An edge e is said to be a leg of a path P at $v \in P^V$ if e is the unique edge with endpoint v such that e does not belong to P. If e is a leg of P, then e either lies to the left or to the right of P w.r.t. the orientation of P and the standard orientation of the plane. A leg that lies to the left is called an L-leg of P, and a leg of P that lies to the right is called an R-leg of P. Let $P^V$ be the vertex set of P. If $v \in P^V$ is the endpoint of an L−leg of P, then we say v is an L−vertex of P. Similarly, if $v \in P^V$ is the endpoint of an R-leg of P, then we say v is an R-vertex of P, see Fig. 10.

Following [16], we define three types of non-trivial string operators.

The semion string is given by

$$\begin{aligned} W_S[P]:= \left( \prod _{e \in P} \sigma _e^X \right) \left( \prod _{\text {R-legs} \, e} \textrm{i}^{\frac{1 - \sigma _e^Z}{2}} \right) \left( \prod _{\text {L-vertices} \, v} (-1)^{s_v} \right) \end{aligned}$$

(50)

where $s_v = \frac{1}{4}(1 - \sigma _e^Z)(1 + \sigma _{e'}^Z)$ and $e, e'$ are the edges of P that go in and out of the vertex v, respectively.

The anti-semion string is given by

$$\begin{aligned} W_{{\bar{S}}}[P]:= \left( \prod _{e \in P} \sigma _e^X \right) \left( \prod _{\text {R-legs} \, e} (-\textrm{i})^{\frac{1 - \sigma _e^Z}{2}} \right) \left( \prod _{\text {L-vertices} \, v} (-1)^{s_v} \right) \end{aligned}$$

(51)

and the bound-state string is given by

$$\begin{aligned} W_{B}[P]:= \left( \prod _{\text {R-legs} \, e} \sigma _e^Z \right) . \end{aligned}$$

(52)

Fig. 10

An oriented path P in solid blue with its L-vertices fattened and its R-legs marked with dotted lines (color figure online)

We further define string operators for the vacuum sector $W_1[P] = {{\,\mathrm{\mathbb {1}}\,}}$. We set $I = \{1, S,{\bar{S}}, B\}$, and we denote by $w_a[P]$ the automorphism defined by conjugation with the (possibly formal) unitary $W_a[P]$. We say $W_a[P]$ or $w_a[P]$ is a closed string operator whenever P is a closed path.

One easily checks that $W_{{\bar{S}}}[P]=W_{S}[P]W_{B}[P]=W_{B}[P]W_{S}[P]$, and $W_B[P]^2=\mathbb {1}$. We will see later that this implies that the anyons $1, S, {\bar{S}}$ and B satisfy fusion rules $S \times B = {\bar{S}}$, ${\bar{S}} \times B = S$ and $B \times B = 1$. The fusion rules $S \times S = 1$ and ${\bar{S}} \times {\bar{S}} = 1$ do not follow so simply. For example, $W_S[P]^2 \ne {{\,\mathrm{\mathbb {1}}\,}}$, see Eq. (54). This failure of the string operators to form a strict representation of the fusion rules is the origin of the non-trivial F-symbols of the double semion model, see Sect. 3.3.

These string operators have the following important property:

Proposition 3.2

Closed string operators leave the ground state invariant. i.e. if P is a closed string, then

$$\begin{aligned} \omega \circ w_a[P] = \omega \end{aligned}$$

(53)

for all $a \in I$.

The proof is in Appendix B.

3.2.1 Definition of $w_{a, \Lambda }$

For any set $Z \subset \mathbb {R}^2$, let $\Pi _{Z}$ be the set of hexagons (regarded as open subsets of $\mathbb {R}^2$) that have some overlap with the set Z. For a cone $\Lambda$, we interpret the boundary $\partial \Pi _{\Lambda }$ as an infinite closed path oriented counterclockwise around $\Lambda$. Assuming the opening angle of $\Lambda$ is less than $\pi$, the edges of $\partial \Pi _{\Lambda }$ whose centre lies a distance further than $n > 2$ from the apex of $\Lambda$ form two half-infinite oriented paths $P^{(n)}_{\Lambda , +}$ and $P^{(n)}_{\Lambda , -}$, as shown in Fig. 11.

Fig. 11

A cone $\Lambda$ with the oriented path $\partial \Pi _{\Lambda }$ going around it in a counterclockwise direction. The paths $P^{(n)}_{\Lambda , +}, P^{(n)}_{\Lambda , -}$ straddle the left and right side of $\Lambda$, respectively

Let $\Lambda$ be a cone, and let $\Lambda ^{(L)}$ and $\Lambda ^{(R)}$ be its left- and right half-cones, see Fig. 12. Take $n > 2$ sufficiently large such that $w_{a, \Lambda }:= w_a[P^{(n)}_{\Lambda ^{(R)}, +}]$ and $v_{a, \Lambda }: = w_a[ P^{(n)}_{\Lambda ^{(L)}, -} ]$ are supported in $\Lambda$ for all $a \in I$. Denote $P_{\Lambda }:= P^{(n)}_{\Lambda ^{(R)}, +}$ and ${\overline{P}}_{\Lambda }:= P^{(n)}_{\Lambda ^{(L)}, -}$.

Fig. 12

A cone $\Lambda$ divided into its left and right cones $\Lambda ^{(L)}$ and $\Lambda ^{(R)}$. The path $P_{\Lambda }$ is the largest part of $\partial \Pi _{\Lambda ^{(R)}}$ such that $w_a[P_{\Lambda }]$ is supported in $\Lambda$

3.2.2 Fusion Rules

In this section, we show the fusion rules for the string operators $w_{a, \Lambda }$. In particular, we will show that $w_{a, \Lambda } \circ w_{b, \Lambda }$ is unitarily equivalent to $w_{ab, \Lambda }$.

Let us begin with semion-semion fusion. For any oriented path P, the automorphism $w_S[P] \circ w_S[P]$ is given by conjugation with the formal unitary

$$\begin{aligned} W_S[P]^2 = \Omega _{S, S}[P]:= \left( \prod _{\text {R-legs} \, e} \sigma _e^Z \right) \left( \prod _{\begin{array}{c} \text {L-vertices} \\ v = (e, e') \end{array}} \sigma _e^Z \sigma _{e'}^Z \right) . \end{aligned}$$

(54)

Let $e_i$ and $e_f$ be the initial and final edges of the path P (if they exist) and let u[P] be the automorphism given by conjugation with

$$\begin{aligned} U[P]:= \Omega _{S, S}[P] \times \sigma _{e_i}^Z \sigma _{e_f}^Z. \end{aligned}$$

(55)

Lemma 3.3

We have $\omega \circ u[P] = \omega$ for any path P.

Proof

We first take P finite and show that u[P] leaves any $\omega _n$ invariant.

Recall that $\omega _n$ is a vector state in the GNS representation of $\omega _0$, given by

$$\begin{aligned} \Omega _n = \sqrt{\frac{1}{2^{\left| \Pi _n\right| }}}\, \sum _{\Pi \subset \Pi _n} (-1)^{\sharp \Pi } \, A_{\Pi } \Omega _0. \end{aligned}$$

(56)

We have

$$\begin{aligned} U[P] A_{\Pi } \Omega _0 = A_{\Pi } \Omega _0. \end{aligned}$$

(57)

Indeed, $A_{\Pi }$ is a product of $\sigma _e^X$ for all $e \in \partial \Pi$, a finite closed path. Now, any such closed path supports an even number of factors $\sigma _e^Z$ of the unitary U[P]. Indeed, if $\partial \Pi$ travels along P, then the two edges of P along an R-leg carry no $\sigma ^Z$, while the two edges along an L-vertex both have a $\sigma ^Z$. The closed path $\partial \Pi$ must enter/leave the path P an even number of times. If it enters through an R-leg, it picks up a $\sigma ^Z$ from the R-leg. If it enters through an L-vertex, then it picks up exactly one of the $\sigma ^Z$’s of the two edges of P next to the L-vertex. Finally, if $\partial \Pi$ enters P through an endpoint of P, then the factors $\sigma _i^Z$, $\sigma _f^Z$ at the initial/final edges ensure that a factor $\sigma ^Z$ is picked up. In all, we see that $U[P] A_{\Pi } = A_{\Pi } U[P]$, because the computation involves an even number of commutations of a $\sigma ^X$ with a $\sigma ^Z$. Obviously $U[P] \Omega _0 = \Omega _0$ so $U[P] A_{\Pi } \Omega _0 = A_{\Pi } \Omega _0$ and $U[P] \Omega _n = \Omega _n$. It follows that $\omega _n \circ u[P] = \omega _n$ for any n and hence $\omega \circ u[P] = \omega$ for any finite P.

If P is infinite, then for any strictly local observable O we can find a finite $P'$ such that $u[P](O) = u[P'](O)$ so $(\omega \circ u[P])(O) = (\omega \circ u[P'])(O) = \omega (O)$. Since the strictly local observables are dense in $\mathcal {A}$, this proves the claim. $\square$

Lemma 3.4

If u[P] is supported in a cone $\Lambda$, then $\pi _1 \circ u[P] \simeq \pi _1$, and the unitary implementing this equivalence belongs to the von Neumann algebra $\mathcal {R}(\Lambda )$.

Proof

The unitary equivalence $\pi _1 \circ u[P] \simeq \pi _1$ follows immediately from Lemma 3.3. Let U be the unitary implementing this equivalence, i.e. 

$$\begin{aligned} U O U^* = u[P](O) \end{aligned}$$

(58)

for all $O \in \mathcal {A}$, and $U \Omega = \Omega$. (We identify $\mathcal {A}$ with its image under the faithful representation $\pi _1$.)

If P is finite, then actually $U \in \mathcal {A}_{\Lambda } \subset \mathcal {R}(\Lambda )$. If P is infinite, let $P_n$ be the path consisting of edges of P whose midpoints lie in $\Pi _n$. Then $U[P_n] \in \mathcal {A}_{\Lambda }$ has $U[P_n] \Omega = \Omega$ for all n, and for any strictly local observables $O, O'$ we have

$$\begin{aligned} \langle O \Omega , U O' \Omega \rangle= & {} \langle O \Omega , u[P](O') U \Omega \rangle = \lim _{n \uparrow \infty } \langle O \Omega , u[P_n](O') \Omega \rangle \nonumber \\= & {} \lim _{n \uparrow \infty } \langle O \Omega , U[P_n] O' \Omega \rangle . \end{aligned}$$

(59)

Since the vectors $O \Omega$, $O' \Omega$ for $O, O'$ strictly local observables are dense in $\mathcal {H}$, this shows that the sequence $U[P_n]$ converges weakly to U. Since $U[P_n] \in \mathcal {A}_{\Lambda }$ for all n, it follows that $U \in \mathcal {R}(\Lambda )$. $\square$

Lemma 3.5

For any cone $\Lambda$, we have that $\pi _1 \circ w_{S, \Lambda } \circ w_{S, \Lambda } \simeq \pi _1$, and the unitary $U_{\Lambda }$ implementing this equivalence belongs to the von Neumann algebra $\mathcal {R}(\Lambda )$.

Proof

By definition, $w_{S, \Lambda } = w_S[ P_{\Lambda }]$ so $w_{S, \Lambda } \circ w_{S, \Lambda } = \textrm{Ad}(\sigma _f^Z) \circ u[P_{\Lambda }]$ where $e_f$ is the final edge of the half-infinite path $P_{\Lambda }$ (cf. Eqs. (54) and (55)). From Lemma 3.4, we find that there exists a unitary $U_{\Lambda } \in \mathcal {R}(\Lambda )$ such that $u[P_{\Lambda }] = \textrm{Ad}(U_{\Lambda })$, hence

$$\begin{aligned} \pi _1 \circ w_{S, \Lambda } \circ w_{S, \Lambda } = \pi _1 \circ \textrm{Ad}(\sigma _{e_f}^Z U_{\Lambda }), \end{aligned}$$

(60)

proving the claim. $\square$

We can now easily show

Proposition 3.6

For each cone $\Lambda$, there are unitaries $\Omega (a, b) \in \mathcal {R}(\Lambda )$ such that

$$\begin{aligned} \textrm{Ad}(\Omega (a, b)) \circ w_{a, \Lambda } \circ w_{b, \Lambda } = w_{a \times b, \Lambda } \end{aligned}$$

(61)

for all $a, b \in I = \{1, S, {\bar{S}}, B\}$ and where $\times$ is an abelian product on I given by

$$\begin{aligned} \begin{array}{ c|c c c c } \times &{} 1\quad &{} S \quad &{} \bar{S} \quad &{} B\quad \\ \hline 1 &{} 1\quad &{} S\quad &{} \bar{S}\quad &{} B\quad \\ S &{} S\quad &{} 1 \quad &{} B\quad &{} \bar{S} \\ \bar{S} &{} \bar{S} \quad &{} B\quad &{} 1\quad &{} S \\ B &{} B\quad &{} \bar{S}\quad &{} S\quad &{} 1 \\ \end{array} \end{aligned}$$

Proof

Lemma 3.5 shows that the claim holds for $S \times S = 1$. The rest of the claim follows from this case and

$$\begin{aligned} w_{{\bar{S}}, \Lambda } = w_{S, \Lambda } \circ w_{B,\Lambda } = w_{B, \Lambda } \circ w_{S, \Lambda }, \quad w_{B, \Lambda } \circ w_{B, \Lambda } = \textrm{id}. \end{aligned}$$

(62)

$\square$

3.2.3 Transportability

Lemma 3.7

If $\Lambda _1$ and $\Lambda _2$ are cones with axes ${\hat{w}}_1$ and ${\hat{w}}_2$, both contained in a cone $\Lambda$ and such that ${\hat{w}}_1$ points to the right of ${\hat{w}}_2$ relative to $\Lambda$ (see Fig. 13). Then $\pi _1 \circ w_{a, \Lambda _1} \simeq \pi _1 \circ v_{a, \Lambda _2}^{-1}$ and the unitary implementing this equivalence belongs to the von Neumann algebra $\mathcal {R}(\Lambda )$.

Fig. 13

Axis of $\Lambda _1$ points to the right of the axis of $\Lambda _2$ relative to the cone $\Lambda$. The region Z has $w_a[\partial \Pi _Z]$ supported in $\Lambda$. Moreover, the path $\partial \Pi _{Z}$ differs from the union of the paths $P_{\Lambda _1}$, ${\overline{P}}_{\Lambda _2}$ by a finite number of edges

Proof

Fix points $x_1, x_2$ on the central axes of the cones $\Lambda _1, \Lambda _2$ such that the region Z 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 $w_{a, \partial \Pi _{Z}}$ supported in $\Lambda$, see Fig. 13.

By construction, $w_{a, \partial \Pi _{Z}}$ differs from $w_{a, \Lambda _1} \circ v_{a, \Lambda _2}$ by the action of a local unitary W supported on $\Lambda$. Since $\partial \Pi _{Z}$ is a closed path, it follows from Proposition 3.2 that there exists a unitary $V \in \mathcal {B}(\mathcal {H})$ such that $\pi _1 \circ w_{a, \partial \Pi _{Z}} = \textrm{Ad}(V) \circ \pi _1$ and $V \Omega = \Omega$; hence,

$$\begin{aligned} \pi _1 \circ w_{a, \Lambda _1} = \textrm{Ad}(VW) \circ \pi _1 \circ v_{a, \Lambda _2}^{-1}. \end{aligned}$$

(63)

This shows the required unitary equivalence. It remains to show that $V \in \mathcal {R}(\Lambda )$.

Let $Z_n = Z \cap B_n$ where $B_n$ is the disk of radius n centred at the origin of $\mathbb {R}^2$. Then $\partial \Pi _{Z_n}$ are closed paths and the automorphisms $w_{\partial \Pi _{Z_n}} = \textrm{Ad}( W_a[\partial \Pi _{Z_n}] )$ leave the ground state invariant, and are supported in $\Lambda$. In particular, there exist phases $\phi _n$ such that $V_n:= \phi _n W_a[\partial \Pi _{Z_n}]$ satisfies $V_n \Omega = \Omega$. For any strictly local observables $O, O' \in \pi _1(\mathcal {A})$ we have

$$\begin{aligned} \langle O \Omega , V O' \Omega \rangle= & {} \langle O \Omega , w_{a, \partial \Pi _Z}(O') V \Omega \rangle = \lim _{n \uparrow \infty } \langle O \Omega , w_{a, \partial \Pi _{Z_n}}(O') V_n \Omega \rangle \nonumber \\= & {} \lim _{n \uparrow \infty } \langle O \Omega , V_n O' \Omega \rangle . \end{aligned}$$

(64)

Since the vectors $O \Omega , O' \Omega$ are dense in $\mathcal {H}$, this shows that $V_n$ converges weakly to V. Since each $V_n$ is in $\mathcal {A}_{\Lambda }$, we conclude that $V \in \mathcal {R}(\Lambda )$. $\square$

Proposition 3.8

If $\Lambda _1$ and $\Lambda _2$ are cones both contained in a cone $\Lambda$, then $\pi _1 \circ w_{a, \Lambda _1} \simeq \pi _1 \circ w_{a, \Lambda _2}$ and the unitary implementing this equivalence belongs to the von Neumann algebra $\mathcal {R}(\Lambda )$.

Proof

Let ${\hat{w}}_1, {\hat{w}}_2$ be the axes of the cones $\Lambda _1, \Lambda _2$ and take a cone $\Lambda _3 \subset \Lambda$ such that its axis ${\hat{w}}_3$ points to the right of both ${\hat{w}}_1$ and ${\hat{w}}_2$ relative to $\Lambda$. Then, Lemma 3.7 implies that there are unitaries $V_1, V_2 \in \mathcal {R}(\Lambda )$ such that

$$\begin{aligned} \pi _1 \circ w_{a, \Lambda _1} = \textrm{Ad}(V_1) \circ \pi _1 \circ v_{a, \Lambda _3}^{-1}, \quad \pi _1 \circ w_{a, \Lambda _2} = \textrm{Ad}(V_2) \circ \pi _1 \circ v_{a, \Lambda _3}^{-1}, \end{aligned}$$

(65)

hence

$$\begin{aligned} \pi _1 \circ w_{a, \Lambda _1} = \textrm{Ad}(V_2^* V_1) \circ \pi _1 \circ w_{a, \Lambda _2}. \end{aligned}$$

(66)

Since $V_2^* V_1 \in \mathcal {R}(\Lambda )$, this proves the claim. $\square$

3.2.4 Distinct Sectors

Fix a cone $\Lambda _0$ with axis (0, 1) and let $\pi _a:= \pi _1 \circ w_{a, \Lambda _0}$ for $a \in I$.

Proposition 3.9

For all $a, b \in I$, we have $\pi _a \simeq \pi _b$ if and only if $a = b$.

Proof

For any n large enough such that the endpoint of $P_{\Lambda _0}$ is contained in $\Pi _{n-2}$, consider the S-matrix

$$\begin{aligned} S_{ab}:= \frac{1}{2} (\omega \circ w_{a, \Lambda _0})( W_b[\partial \Pi _n] ). \end{aligned}$$

(67)

An easy calculation shows that these quantities are independent of n and given by

$$\begin{aligned} S = \frac{1}{2} \begin{bmatrix} 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 \\ 1 &{}\quad -1 &{}\quad 1 &{}\quad -1 \\ 1 &{}\quad 1 &{}\quad -1 &{}\quad -1 \\ 1 &{}\quad -1 &{}\quad -1 &{}\quad 1 \end{bmatrix}. \end{aligned}$$

(68)

It follows that for any $a \ne b$ there is a c such that $(\omega \circ w_{a, \Lambda _0})(W_c[\partial \Pi _n]) = - (\omega \circ w_{b, \Lambda _0})(W_c[\partial \Pi _n])$ for all n sufficiently large. Corollary 2.6.11 of [2] then implies that $\pi _a$ and $\pi _b$ are disjoint. $\square$

3.2.5 Verification of Assumptions

The four faithful irreducible representations $\pi _1, \pi _S, \pi _{{\bar{S}}}, \pi _B$ defined by $\pi _a = \pi _1 \circ w_{a, \Lambda _0}$ for $a \in \{1, S, {\bar{S}}, B\} = I$ are pairwise disjoint by Proposition 3.9.

For any cone $\Lambda$ and any $a \in I$, we defined an automorphism $w_{a, \Lambda }$ supported in $\Lambda$. 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 3.6 and 3.8.

3.3 Computation of F-Symbols

Having fixed the cone $\Lambda _0$ with axis (0, 1), we use for all $a \in I$ the shorthand notations $w_a:= w_{a, \Lambda _0}$ and ${\overline{w}}_a:= {\overline{w}}_{a, \Lambda _0}$, where the latter are the extensions of $w_a$ to the allowed algebra $\mathcal {B}$ constructed in Lemma 2.5.

Let $e_f$ be the final edge of the path $P_{\Lambda _0}$ and let $U \in \mathcal {R}(\Lambda _0)$ be the unitary such that $\pi _1 \circ u[P_{\Lambda _0}] = \textrm{Ad}(U) \circ \pi _1$ and $U \Omega = \Omega$ provided by Lemma 3.4. The proof of Lemma 3.5 shows that

$$\begin{aligned} \textrm{Ad}(\Omega (S, S) ) \circ ({\overline{w}}_S \circ {\overline{w}}_S) = {\overline{w}}_1 \end{aligned}$$

(69)

with $\Omega (S, S) = \sigma _{e_f}^Z U$.

Using

$$\begin{aligned} w_{{\bar{S}}} = w_B \circ w_S = w_S \circ w_{B}, \quad w_B \circ w_B = \textrm{id}, \end{aligned}$$

(70)

we find that

$$\begin{aligned} \textrm{Ad}(\Omega (a, b)) \circ (w_a \circ w_b) = w_{a \times b} \end{aligned}$$

(71)

for all $a, b \in I$ with fusion intertwiners $\Omega (a, b)$ given in Table 1. It follows that $\Omega (a, b) \in ({\overline{w}}_a \otimes {\overline{w}}_b, {\overline{w}}_{a \times b})$ for all $a, b \in I$.

Table 1 Fusion intertwiners $\Omega (a, b)$ for the double semion state

In order to compute the F-symbols, we first show

Lemma 3.10

$$\begin{aligned} {\overline{w}}_S( U \sigma _{e_f}^Z ) = - U \sigma _{e_f}^Z, \quad {\overline{w}}_B(U \sigma _{e_f}^Z) = U \sigma _{e_f}^Z, \quad {\overline{w}}_{{\bar{S}}}(U \sigma _{e_f}^Z) = - U \sigma _{e_f}^Z. \end{aligned}$$

(72)

Proof

Since $e_f$ is the final edge of the path $P_{\Lambda _0}$, we have $w_S(\sigma _{e_f}^Z) = w_{{\bar{S}}}(\sigma _{e_f}^Z) = - \sigma _{e_f}^Z$ and $w_B(\sigma _{e_f}^Z) = \sigma _{e_f}^Z$. It remains to show that ${\overline{w}}_{S}(U) = {\overline{w}}_{{\bar{S}}}(U) = {\overline{w}}_B(U) = U$.

Since U is the weak limit of the sequence $U_n = U[P_n]$ where $P_n$ is the path consisting of edges of $P_{\Lambda _0}$ whose midpoints lie in $\Pi _n$ (cf. the proof of Lemma 3.4), it is sufficient to show $w_{S}(U_n) = w_{{\bar{S}}}(U_n) = w_B(U_n) = U_n$. This follows similar to the argument in the proof of Lemma 3.4. Since $U_n$ is a product of $\sigma ^Z$’s, we have that $w_B(U_n) = U_n$, and

$$\begin{aligned} w_S(U_n) = w_{{\bar{S}}}(U_n) = \left( \prod _{e \in P_n} \sigma _e^X \right) U_n \left( \prod _{e \in P_n} \sigma _e^X \right) . \end{aligned}$$

(73)

By design, the unitary $U_n$ has an even number of $\sigma ^Z$’s on the path $P_n$. Indeed, there are two factors of $\sigma ^Z$ for every L-vertex, zero for every R-leg, and another two for the endpoints. We conclude that $w_S(U_n) = W_{{\bar{S}}}(U_n) = U_n$ for all n.

$\square$

We can now start computing the F-symbols. If in Eq. (24) we take $a = 1$, then

$$\begin{aligned} \Omega (b, c) \Omega (1, b) = F(1, b, c)\Omega (1, bc) \Omega (B, C). \end{aligned}$$

(74)

Since $\Omega (1, b) = \Omega (1, bc) = {{\,\mathrm{\mathbb {1}}\,}}$, we find that $F(1, b, c) = 1$ for all b, c.

Similarly we find $F(a, 1, c) = F(a, b, 1) = 1$ for all a, b, c.

Let us now consider F-symbols that involve the bound state B, for example,

$$\begin{aligned} \Omega (Bb, c) \Omega (B, b) = F(B, b, c) \Omega (B, bc) {\overline{w}}_B( \Omega (b, c) ). \end{aligned}$$

(75)

Since $\Omega (B, b) = \Omega (B, bc) = {{\,\mathrm{\mathbb {1}}\,}}$, this reduces to

$$\begin{aligned} \Omega (Bb, c) = F(B, b, c) {\overline{w}}_B( \Omega (b, c) ). \end{aligned}$$

(76)

If $b = B$ or $c = B$, then then $\Omega (Bb, b) = \Omega (b, c) = {{\,\mathrm{\mathbb {1}}\,}}$ so $F(B, b, c) = 1$. If $b, c \in \{S, {\bar{S}}\}$, then $\Omega (Bb, c) = \Omega (b, c) = U \sigma _{e_f}^Z$, so using Lemma 3.10 we find again $F(B, b, c) = 1$ for all b, c.

Similar considerations show that $F(B, b, c) = F(a, B, c) = F(a, b, B) = 1$ for all a, b, c.

Finally, we consider the case where $a, b, c \in \{ S, {\bar{S}} \}$. Then since $ab, bc \in \{ 1, B \}$ we have $\Omega (ab, c) = \Omega (a, bc) = {{\,\mathrm{\mathbb {1}}\,}}$ so

$$\begin{aligned} U \sigma _{e_f}^Z = F(a, b, c) {\overline{w}}_{a}( U \sigma _{e_f}^Z ). \end{aligned}$$

(77)

Using Lemma 3.10, we conclude that $F(a, b, c) = -1$ for $a, b, c \in \{S, {\bar{S}}\}$.

3.4 Computation of R-Symbols

Choose cones $\Lambda _L$ with axis $(-1, 0)$ and $\Lambda _R$ with axis (1, 0), both disjoint from $\Lambda _0$ as in Fig. 14. Let $\widetilde{\Lambda }_L \supseteq \Lambda _0 \cup \Lambda _L$ and $\widetilde{\Lambda }_R \supseteq \Lambda _0 \cup \Lambda _R$ be allowed cones such that $\widetilde{\Lambda }_L \cap \Lambda _R = \widetilde{\Lambda }_R \cap \Lambda _L = \emptyset$.

To compute the braiding intertwiners $\epsilon (a, b) = \epsilon ({\overline{w}}_a, {\overline{w}}_b)$, set ${\overline{v}}_a^L = {\overline{v}}_a[\Lambda _L]^{-1}$, ${\overline{v}}_a^R = {\overline{v}}_a[\Lambda _R]^{-1}$ as well as ${\overline{w}}_a^L = {\overline{w}}_{a, \Lambda _L}$ and ${\overline{w}}_a^R = {\overline{w}}_{a, \Lambda _R}$ for all $a = 1, S, {\bar{S}}, B$. (Recall that the automorphisms $v_{a, \Lambda }$ are defined in Sect. 3.2.1.)

It follows from Lemma 3.7 that there are unitaries $U_a \in ({\overline{w}}_a, {\overline{v}}^L_a)$ and $V_b \in ({\overline{w}}_b, {\overline{v}}^R_b)$ such that $U_a \in \mathcal {R}(\widetilde{\Lambda }_L)$ and $V_b \in \mathcal {R}(\widetilde{\Lambda }_R)$. By the same lemma, there are unitaries $U'_a \in ({\overline{v}}^L_a, {\overline{w}}^L_a)$ and $V'_b \in ({\overline{v}}^R_b, {\overline{w}}^R_{b})$ such that $U'_a \in \mathcal {R}(\Lambda _L)$ and $V'_b \in \mathcal {R}(\Lambda _R)$. We therefore have unitary morphisms $U'_a U_a \in ({\overline{w}}_a, {\overline{w}}^L_a)$ and $V'_b V_b \in ({\overline{w}}_b, {\overline{w}}^R_b)$ with $U'_a U_a \in \mathcal {R}(\widetilde{\Lambda }_L)$ and $V'_b V_b \in \mathcal {R}(\widetilde{\Lambda }_R)$. By definition 2.11 and the fact that ${\overline{w}}_a( V'_b ) = V'_b$, we have

$$\begin{aligned} \epsilon (a, b) = V_b^* \, {\overline{w}}_a \big ( V_b \big ). \end{aligned}$$

(78)

Fig. 14

Cones $\Lambda _0$, $\Lambda _L$ and $\Lambda _R$ used to define the braiding intertwiners

In order to compute ${\overline{w}}_a(V_b)$, 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 $\Lambda _0$ and $\Lambda _R$, see Fig. 15. Then the path $\partial \Pi _{K}$ contains $P_{\Lambda _0}$ and ${\overline{P}}_{\Lambda _R}$ and the path $Q = \partial \Pi _{K} {\setminus } ( P_{\Lambda _0} \cup P_{\Lambda _R} )$ is finite. For each n, let $K_n = K \cap B_n$ where $B_n \subset \mathbb {R}^2$ is the disk of radius n centred at the origin of $\mathbb {R}^2$. Consider the sequence of paths $P_n = \partial \Pi _{K_n} \setminus Q$ and set $V_b^{(n)}:= W_b[P_n]$.

Fig. 15

Sets $K_n$ and the paths $P_n$ and Q used in the construction of the sequence of unitaries $V_b^{(n)}$ that converge weakly to the intertwiner $V_b$

Lemma 3.11

There are phases $\phi _n$ such that the sequence $\phi _n V_b^{(n)}$ converges weakly to $V_b$.

Proof

Consider first the sequence of finite closed paths $\partial \Pi _{K_n} = P_n \cup Q$ and corresponding string operators $W_b[\partial \Pi _{K_n}]$. By Proposition 3.2, the unitaries $W_b[\partial \Pi _{K_n}]$ leave the ground state invariant up to a phase, so there are phases $\phi _n$ such that ${\widetilde{W}}^{(n)}:= \phi _n W_b[\partial \Pi _{K_n}]$ satisfy ${\widetilde{W}}^{(n)} \Omega = \Omega$.

Since $\partial \Pi _K$ is a closed path, the automorphism $w_b[\partial \Pi _{K}]$ leaves the grounds state invariant by Proposition 3.2. It follows that there is a unique unitary ${\widetilde{W}} \in \mathcal {B}(\mathcal {H})$ such that ${\widetilde{W}} \Omega = \Omega$ and $w_b[\partial \Pi _K] = \textrm{Ad}({\widetilde{W}})$ (as automorphisms on $\pi _1(\mathcal {A})$).

Now, for any strictly local observable O we have $w_b[\partial \Pi _K](O)$ = $w_b[\partial \Pi _{K_n}](O)$ for all n large enough. Thus, for any strictly local operators $O, O'$ we have

$$\begin{aligned} \langle O \Omega , {\widetilde{W}} \, O' \Omega \rangle&= \langle O \Omega , w_b[\partial \Pi _K]( O' ) \, {\widetilde{W}} \Omega \rangle \\&= \lim _{n \uparrow \infty } \langle O \Omega , w_b[\partial \Pi _{K_n}](O') \, \Omega \rangle \\&= \lim _{n \uparrow \infty } \langle O \Omega , {\widetilde{W}}^{(n)} O' \Omega \rangle , \end{aligned}$$

showing that the sequence ${\widetilde{W}}^{(n)}$ converges weakly to ${\widetilde{W}}$.

Now note that the paths $\partial \Pi _{K_n}$ and $P_n = \partial \Pi _{K_n} {\setminus } Q$ differ by the same path $Q = \partial \Pi _{K_n} {\setminus } P_n$ for all n. It follows that the corresponding string operators $W_b[\partial \Pi _{K_n}]$ and $V^{(n)}_b = W_b[P_n]$ satisfy $V_b^{(n)}(W_b[{\widetilde{P}}_n])^* = W$ for a unitary W that is independent of n and is supported on the path Q and edges adjacent to Q. (In fact, W is equal to $W_b[Q]^*$ up local operators supported near the endpoints of the path Q.) Therefore, $V_b^{(n)} = W W_b[\partial \Pi _{K_n}]$ and $\phi _n V_b^{(n)} = W {\widetilde{W}}^{(n)}$.

Since ${\widetilde{W}}^{(n)}$ converges weakly to ${\widetilde{W}}$, it follows that the sequence $\phi _n V_b^{(n)} = W {\widetilde{W}}^{(n)}$ converges weakly to $W {\widetilde{W}}$. By construction, $\textrm{Ad}(W {\widetilde{W}}) = w_b \circ (w_b^R)^{-1} = \textrm{Ad}(V_b)$ so $W {\widetilde{W}} = \mu V_b$ for some phase $\mu$. We then find that the sequence $\mu ^* \phi _n V_n^{(n)}$ converges weakly to $V_b$. This proves Lemma. $\square$

We can now compute the braiding intertwiners.

Obviously ${\overline{w}}_1 = \textrm{id}$ and $V_1 = I$ so $\epsilon (1, a) = \epsilon (a, 1) = {{\,\mathrm{\mathbb {1}}\,}}$ for all $a \in I$. It is also easy to see that

$$\begin{aligned} w_{B}(V_a^{(n)}) = V_a^{(n)} \end{aligned}$$

(79)

for any $a \in \{1, S, {\bar{S}}, B\}$, so $\epsilon (B, a) = {{\,\mathrm{\mathbb {1}}\,}}$ for all a, while

$$\begin{aligned} w_S(V_B^{(n)}) = w_{{\bar{S}}}(V_B^{(n)}) = -V_B^{(n)}, \end{aligned}$$

(80)

because the path $P_{\Lambda _0}$ contains a single R-leg of the path $P_n$. So $\epsilon (S, B) = \epsilon ({\bar{S}}, B) = -{{\,\mathrm{\mathbb {1}}\,}}$.

Let us now compute $\epsilon (S, S)$. Note that the path $P_n$ enters the path $P_{\Lambda _0}$ at an L-vertex of $P_{\Lambda _0}$. Let $(e, e')$ be the edges of $P_{\Lambda _0}$ before and after this L-vertex, see Fig. 16. We find

$$\begin{aligned}&(V_S^{(n)})^* \, w_S( V_S^{(n)} )\\&\quad = \left( \sigma _{e'}^X \left( \textrm{i}^{\frac{1 - \sigma _e^Z}{2}} \right) \right) \left( \sigma _e^X \sigma _{e'}^X (-1)^{s_I} \right) \left( \left( \textrm{i}^{\frac{1 - \sigma _e^Z}{2}} \right) ^* \sigma _{e'}^X \right) \left( (-1)^{s_I} \sigma _{e'}^X \sigma _e^X \right) = \textrm{i}{{\,\mathrm{\mathbb {1}}\,}}, \end{aligned}$$

which implies $\epsilon (S, S) = \textrm{i}{{\,\mathrm{\mathbb {1}}\,}}$.

Fig. 16

Edges e and $e'$ playing a role in the computation of $\epsilon (S, S)$

We now use the braid equations (Lemma 2.13)

$$\begin{aligned} \epsilon (\rho , \sigma \otimes \tau )&= ({{\,\mathrm{\mathbb {1}}\,}}_{\sigma } \otimes \epsilon (\rho , \tau ))(\epsilon (\rho , \sigma ) \otimes {{\,\mathrm{\mathbb {1}}\,}}_{\tau }) \\ \epsilon (\rho \otimes \sigma , \tau )&= (\epsilon (\rho , \tau ) \otimes {{\,\mathrm{\mathbb {1}}\,}}_{\sigma }) ({{\,\mathrm{\mathbb {1}}\,}}_{\rho }) \otimes \epsilon (\sigma , \tau ) ) \end{aligned}$$

(where $I_{\rho }$ denotes the identity intertwiner from $\rho$ to itself) to compute

$$\begin{aligned} \epsilon ({\bar{S}}, S)&= \epsilon (S \times B, S) = \epsilon (S, S) {\overline{w}}_S( \epsilon (B, S) ) = \textrm{i}{{\,\mathrm{\mathbb {1}}\,}}\\ \epsilon (S, {\bar{S}})&= \epsilon (S, S \times B) = {\overline{w}}_S(\epsilon (S, B)) \epsilon (S, S) = -\textrm{i}{{\,\mathrm{\mathbb {1}}\,}}\\ \epsilon ({\bar{S}}, {\bar{S}})&= \epsilon (S \times B, {\bar{S}}) = \epsilon (S, {\bar{S}}) {\overline{w}}_S(\epsilon (B, S)) = - \textrm{i}{{\,\mathrm{\mathbb {1}}\,}}. \end{aligned}$$

Thus, we have computed all braiding intertwiners,see Table 2 for a summary.

Table 2 Braiding intertwiners $\epsilon (a, b)$ for the double semion state

The R-symbols are defined (26) by

$$\begin{aligned} \Omega (b, a) \epsilon (a, b) = R(a, b) \times \Omega (a, b). \end{aligned}$$

(81)

Since $\Omega (a, b) = \Omega (b, a)$ for all a, b, we find that the R-symbols are shown in Table 3.

Table 3 R-symbols R(a, b) for the double semion state

One can verify that the F and R-symbols indeed satisfy the pentagon and hexagon equations.

3.5 Anyons are Described by $\textrm{Rep}_f \mathcal {D}^{\phi }(\mathbb {Z}_2)$

In Sect. 3.2.5, we verified Assumptions 1–4 for the superselection sectors $\pi _a$ with $a \in I = \{1, S, {\bar{S}}, B\}$ of the double semion state. Let us denote by $\Delta _{\Lambda _0}^I$ the corresponding braided tensor category constructed in Sect. 2.2.

In this section, we show that $\Delta _{\Lambda _0}^I$ is braided monoidal equivalent to the representation category $\textrm{Rep}_f \mathcal {D}^{\phi }(\mathbb {Z}_2)$, where $\phi$ is a non-trivial 3-cocycle on $\mathbb {Z}_2$. We will do this by showing that both categories have equivalent fusion rings, F-symbols, and R-symbols, and appealing to Proposition 7.5.2 of [12].

This identification is relevant, because it partially verifies a conjecture about string-net models [16]. A given string-net model is defined by an input fusion category $\mathcal {F}$, and the topological order of the model is conjectured [17] to correspond to the Drinfeld centre $\mathcal {Z}(\mathcal {F})$ of $\mathcal {F}$. In the case where $\mathcal {F}= \textrm{Vec}_G^{\phi }$ for a finite group G and a 3-cocycle $\phi$ of G, we have $\mathcal {Z}(\mathcal {F}) = \textrm{Rep}_f \mathcal {D}^{\phi }(G)$ (cf. [18]), where $\mathcal {D}^{\phi }(G)$ is the twisted quantum double algebra first described in [4].

3.5.1 The Braided Fusion Ring of $\Delta _{\Lambda _0}^I$

We have extracted from the braided tensor category $\Delta _{\Lambda }^I$ its fusion ring, generated by the elements of $I = \{ 1, S, {\bar{S}}, B \}$ with abelian fusion rules given by the group structure on I described in Proposition 3.6. In other words, the fusion ring of $\Delta _{\Lambda _0}^I$ is isomorphic to $\mathbb {Z}(I)$. In Sect. 3.3, we obtained the F-symbols, and in Sect. 3.4 we obtained the R-symbols, Cf. Table 3 derived from the braided tensor category $\Delta _{\Lambda _0}^I$. It follows from Proposition 7.5.2 of [12] that these data completely determine the category $\Delta _{\Lambda _0}^I$ up to braided monoidal equivalence.

3.5.2 Description of $\textrm{Rep}_f \mathcal {D}^{\phi }(\mathbb {Z}_2)$

We describe the quasi Hopf algebra $\mathcal {D}^{\phi }(\mathbb {Z}_2)$ first introduced in [4]. We follow the presentation in [22].

Let $\phi : (\mathbb {Z}_2)^3 \rightarrow U(1)$ be the normalised representative of the non-trivial class in $H^3(\mathbb {Z}_2, U(1))$:

$$\begin{aligned} \phi (-, -, -) = -1, \quad \text {all other components equal to 1}. \end{aligned}$$

(82)

Let

$$\begin{aligned} c_x(f, g):= (\iota _x \phi )(f, g) = \frac{\phi (x, f, g)\phi (f, g, x)}{\phi (f, x, g)}. \end{aligned}$$

(83)

for all $x, f, g \in \mathbb {Z}_2$. For each $x \in \mathbb {Z}_2$ the map $c_x: (\mathbb {Z}_2)^2 \rightarrow U(1)$ is a 2-cocycle, it satisfies

$$\begin{aligned} c_x(f, g) c_x(fg, h) = c_x(f, gh) c_x(g, h). \end{aligned}$$

(84)

The quasi-quantum double $\mathcal {D}^{\phi }(\mathbb {Z}_2)$ is an algebra spanned by $\{ P_x f \}_{x, f \in \mathbb {Z}_2}$ with multiplication

$$\begin{aligned} (P_x f) (P_y g) = \delta _{x, y} (P_x fg) \, c_x(f, g). \end{aligned}$$

(85)

The unit for this multiplication is $\sum _{x \in \mathbb {Z}_2} (P_x 1)$.

The quasi-quantum double is, moreover, equipped with a coproduct $\Delta : \mathcal {D}^{\phi }(\mathbb {Z}_2) \rightarrow \mathcal {D}^{\phi }(\mathbb {Z}_2) \otimes \mathcal {D}^{\phi }(\mathbb {Z}_2)$ given by

$$\begin{aligned} \Delta (P_x f) = \sum _{y z = x} c_f(y, z) (P_y f) \otimes (P_z f). \end{aligned}$$

(86)

Associativity and quasicoassociativity follow readily from Eq. (84), in particular

$$\begin{aligned} (\textrm{id}\otimes \Delta ) \Delta (P_x f) = \varphi \cdot (\Delta \otimes \textrm{id}) \Delta (P_x f) \cdot \varphi ^{-1} \end{aligned}$$

(87)

with $\varphi = \sum _{f, g, h \in \mathbb {Z}_2} \, \phi ^{-1}(f, g, h) \, (P_f 1) \otimes (P_{g} 1) \otimes (P_{h} 1)$. That $\Delta$ is an algebra morphism follows from the identity

$$\begin{aligned} \frac{c_x(f, g) c_y(f, g)}{c_{xy}(f, g)} \times \frac{c_f(x, y) c_g(x, y)}{c_{fg}(x, y)} = 1. \end{aligned}$$

(88)

There is a counit $\epsilon : \mathcal {D}^{\phi }(\mathbb {Z}_2) \rightarrow \mathbb {C}$ and an antipode $S: \mathcal {D}^{\phi }(\mathbb {Z}_2) \rightarrow \mathcal {D}^{\phi }(\mathbb {Z}_2)$ given by

$$\begin{aligned} \epsilon ( P_x f ) = \delta _{x, 1}, \quad S( P_x f) = (P_{x^{-1}} f^{-1}) \, c_{x^{-1}}(f, f^{-1})^{-1} c_f(x, x^{-1})^{-1}.\nonumber \\ \end{aligned}$$

(89)

These give $\mathcal {D}^{\phi }(\mathbb {Z}_2)$ the structure of a quasi Hopf algebra. This quasi-Hopf algebra is, moreover, quasitriangular with universal R-matrix

$$\begin{aligned} R = \sum _{x, y} (P_x 1 ) \otimes (P_y x). \end{aligned}$$

(90)

3.5.3 Category of Representations and Its Fusion Ring

Since $\mathcal {D}^{\phi }(\mathbb {Z}_2)$ is a quasitriangular Hopf algebra, its category of finite-dimensional representations $\textrm{Rep}_f \mathcal {D}^{\phi }(\mathbb {Z}_2)$ is a braided tensor category. We extract the fusion ring, F-symbols, and R-symbols of this braided tensor category. See [18] for a more in depth analysis of this category of representations.

There are four irreducible representations of $\mathcal {D}^{\phi }(\mathbb {Z}_2)$, labelled by pairs $(x, \chi ) \in \mathbb {Z}_2 \times \mathbb {Z}_2^*$. ($\mathbb {Z}_2^*$ consists of the characters of $\mathbb {Z}_2$, namely 1 and $\textrm{sgn}$.) They are given by

$$\begin{aligned} \Pi _{(x, \chi )}(P_y f) = \delta _{x, y} \, \varepsilon _x(f) \chi (f) \end{aligned}$$

(91)

with

$$\begin{aligned} \varepsilon _x(f):= \exp \left( \frac{\pi \textrm{i}}{2} [x]. [f] \right) \end{aligned}$$

(92)

where [x], [f] are the additive representation of x and f. i.e. $\varepsilon _{-}(-) = \textrm{i}$ and all other components are equal to one. $\varepsilon _x$ is a cocycle and

$$\begin{aligned} c_x(f, g) = (d \varepsilon _x)(f, g) = \frac{\varepsilon _{x}(fg)}{\varepsilon _x(f) \varepsilon _x(g)}. \end{aligned}$$

(93)

Since we have a coproduct, we have the following tensor product of representations:

$$\begin{aligned} (\Pi _1 \otimes \Pi _2)(P_x f):= ((\Pi _1 \otimes \Pi _2) \circ \Delta )(P_x f) = \sum _{yz = x} c_f(y, z) \, \Pi _1(P_y f) \otimes \Pi _2(P_z f).\nonumber \\ \end{aligned}$$

(94)

One easily verifies that

$$\begin{aligned} \Pi _{(x, \chi )} \otimes \Pi _{(y, \sigma )} = \Pi _{(xy, \chi \sigma )}. \end{aligned}$$

(95)

The representation $\Pi _{(1, 1)}$ is an identity for this tensor product (with trivial left and right unitors). In particular, the fusion ring of the representation category is $\mathbb {Z}(G)$ with G the abelian group with elements $\{ (x, \chi ) \}_{(x, \chi ) \in \mathbb {Z}_2 \times \mathbb {Z}_2^*}$ and group multiplication given by $(x, \chi ) \cdot (y, \sigma ) = (xy, \chi \sigma )$.

With this tensor product, the representations of $\mathcal {D}^{\phi }(\mathbb {Z}_2)$ form a tensor category with simple objects $\Pi _{(x, \chi )}$ and associators between simple objects

$$\begin{aligned} \alpha _{(x, \chi ), (y, \sigma ), (z, \tau )}: ( \Pi _{(x, \chi )} \otimes \Pi _{(y, \sigma )} ) \otimes \Pi _{(z, \tau )} \rightarrow \Pi _{(x, \chi )} \otimes (\Pi _{(y, \sigma )} \otimes \Pi _{(z, \tau )})\nonumber \\ \end{aligned}$$

(96)

given by multiplication with $\phi (x, y, z)$. This shows that the F-symbols of the representation category $\textrm{Rep}_f \mathcal {D}^{\phi }(\mathbb {Z}_2)$ are given by the 3-cocycle $\alpha$ on G given by $\alpha ((x, \chi ), (y, \sigma ), (z, \tau )) = \phi (x, y, z)$.

The braiding $\epsilon _{(x, \chi ), (y, \sigma )}: \Pi _{(x, \chi )} \otimes \Pi _{(y, \sigma )} \rightarrow \Pi _{(y, \sigma )} \otimes \Pi _{(x, \chi )}$ of simple objects of $\mathcal {D}^{\phi }(\mathbb {Z}_2)$ is given by multiplication with

$$\begin{aligned} (\Pi _{(x, \chi )} \otimes \Pi _{(y, \sigma )}) (R) = \varepsilon _y(x) \sigma (x), \end{aligned}$$

(97)

where R is the universal R-matrix given in Eq. (90). These braidings are summarised in Table 4, and it follows from (95) that the R-symbols of $\mathcal {D}^{\phi }(\mathbb {Z}_2)$ are given by the same table.

Table 4 Braiding isomorphisms of $\textrm{Rep}_f \mathcal {D}^{\phi }(\mathbb {Z}_2)$ for simple objects

3.5.4 Braided Monoidal Equivalence of $\Delta _{\Lambda _0}^I$ and $\textrm{Rep}_f \mathcal {D}^{\phi }(\mathbb {Z}_2)$

By the identification

$$\begin{aligned} (1, 1) \leftrightarrow 1, \quad (-1, 1) \leftrightarrow S, \quad (-1, \textrm{sgn}) \leftrightarrow {\bar{S}}, \quad (1, \textrm{sgn}) \leftrightarrow B. \end{aligned}$$

(98)

we see that the groups G and I and therefore the fusion rings $\mathbb {Z}(G)$ and $\mathbb {Z}(I)$ of $\textrm{Rep}_f \mathcal {D}^{\phi }(\mathbb {Z}_2)$ and $\Delta _{\Lambda _0}^I$ are isomorphic.

Under this identification, the F-symbols of $\Delta _{\Lambda _0}^I$ computed in Sect. 3.3 match precisely with the F-symbols $\alpha$ of the representation category. Furthermore, comparing Tables 3 and 4 we see that also the R-symbols match precisely. Thus, the braided tensor category $\Delta _{\Lambda _0}^I$ and the representation category $\textrm{Rep}_f \mathcal {D}^{\phi }(\mathbb {Z}_2)$ have the same fusion rings $\mathbb {Z}(G) \simeq \mathbb {Z}(I)$, the same F-symbols, and the same R-symbols. It follows from Proposition 7.5.2 of [12] that $\Delta _{\Lambda _0}^I$ and $\textrm{Rep}_f \mathcal {D}^{\phi }(\mathbb {Z}_2)$ are isomorphic as braided monoidal categories.

Appendices

Purity of the Double Semion State

In appendix, we prove Theorem 3.1, stating that the double semion state $\omega$ constructed in Sect. 3.1 is pure.

1.1 Restrictions of $\omega$ to finite regions

Denote by $\Pi ^E_n$ the set of edges belonging to some hexagon of $\Pi _n$ and set $\mathcal {M}_n = \mathcal {A}_{\Pi ^E_n}$. We investigate the restrictions $\omega |_n:= \omega |_{\mathcal {M}_n}$.

Lemma A.1

For any $m > n$, we have that $\omega |_n = \omega _m|_n$.

Proof

It is sufficient to note that $\omega _m$ has the same expectation value for any operator in $\mathcal {M}_n$ as $\omega$ does. Indeed, for any $A \in \mathcal {M}_n$ we have $\omega (A) = \lim \omega _m(A)$, and the latter sequence becomes constant as soon as $\Pi _m$ contains all hexagons containing edges in the support of A. i.e. $\omega _m(A) = \omega (A)$ for all $m \ge n + 1$.

$\square$

Thus, we can restrict our attention to states $\omega _{m}|_n$. Recall that $\omega _m$ is given by the expectation value in the vector state

$$\begin{aligned} \Omega _m = \sqrt{\frac{1}{2^{\left| \Pi _m\right| }}} \sum _{\Pi \subset \Pi _m} (-1)^{\sharp \Pi } A_{\Pi } \Omega _0 = \sqrt{ \frac{1}{2^{\left| \Pi _m\right| }} } \sum _{\Pi \subset \Pi _m} (-)^{\sharp \partial \Pi } | \partial \Pi \rangle \end{aligned}$$

(99)

where $\sharp \partial \Pi$ is the number of closed loops in the loop soup $\partial \Pi$, and we chose to write $A_{\Pi }$ instead of $\pi _0(A_{\Pi })$ because the representation $\pi _0$ is faithful, and $| \partial \Pi \rangle$ is the product state with all degrees of freedom spin up, except those on the edges along the path $\partial \Pi$, which are spin down.

Note that every closed path $\alpha$ supported on $\Pi ^E_m$ is of the form $\partial \Pi$ for a unique $\Pi \subset \Pi _m$, so we have written $\Omega _m$ as a uniform superposition over all closed-loop soups supported on $\Pi ^E_m$. Moreover, for closed paths $\alpha$ and $\beta$ we have $| \alpha \rangle = | \beta \rangle$ if and only if $\alpha = \beta$, and these states are orthogonal otherwise.

We will show that $\omega _m|_n$ is a mixed state which is an equal-weight convex combination of pure states $\eta _n(b)$ where b is a boundary condition, namely an assignment of up-or down to each out edge of the region $\Pi _{n}$ such that an even number of edges are up, see Fig. 17

The state $\eta _n(b)$ is then given by a uniform superposition of all loop soups that satisfy the boundary condition b, weighed by $\pm 1$ depending on whether a fixed ‘closure’ of the boundary condition has an even or an odd number of closed loops.

Lemma A.2

There are $2^{\left| \Pi _n\right| }$ 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 $\Pi ^E_n$.

To obtain loop soups for an arbitrary boundary condition b, act on any closed-loop soup with $A_p$ on the hexagons 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, every 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.

$\square$

Write $\mathcal {P}_n^{(b)}$ for the loop soups in $\Pi ^E_n$ that satisfy the boundary condition b. For a given boundary condition b, any $\alpha \in \mathcal {P}_n^{(b)}$ can be ‘closed up’ in precisely two ways by connecting neighbouring marked edges using edges in $\Pi ^E_{n+1} {\setminus } \Pi ^E_n$, see Fig. 17. Pick one such ‘pairing’ of marked boundary edges, and let $\sharp \alpha$ be the number of loops of $\alpha$ closed up with the chosen pairing. Then we have normalised vectors

$$\begin{aligned} |\eta _n^{(b)} \rangle = \sqrt{\frac{1}{2^{\left| \Pi _n\right| }}} \sum _{\alpha \in \mathcal {P}_n^{(b)}} (-1)^{\sharp \alpha } | \alpha \rangle . \end{aligned}$$

(100)

We have $\langle \eta _n^{(b)}, \eta _n^{(b')} \rangle = \delta _{b, b'}$, i.e. these vectors form an orthonormal set. Denote by $\eta _n^{(b)}$ the pure state on $\mathcal {M}_n$ corresponding to the vector $| \eta _n^{(b)} \rangle$.

Fig. 17

A loop soup $\alpha \in \mathcal {P}_4^{(b)}$ 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 (color figure online)

Proposition A.3

For $m > n \ge 1$,

$$\begin{aligned} \omega _m|_n = \frac{1}{2^{6n - 1}} \sum _{b} \eta _n^{(b)}. \end{aligned}$$

(101)

Since the $| \eta _n^{(b)} \rangle$ form an orthonormal set, this is a Schmidt decomposition of $\omega _m|_n$.

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 $\omega _{n+1}$ on $\Pi ^E_{n+1}$ is a uniform superposition of closed-loop soups in $\Pi ^E_{n+1}$. Any such loop soup $\alpha$ defines a boundary condition $b(\alpha )$ by the outer edges of $\Pi _n$ that are occupied by strings of $\alpha$. We can therefore organise the $\alpha$ according to which boundary condition they induce:

$$\begin{aligned} |\Omega _{n+1} \rangle = \sqrt{\frac{1}{2^{\left| \Pi _{n+1}\right| }}} \sum _{b} \sum _{\alpha : b(\alpha ) = b} \, (-1)^{\sharp \alpha } | \alpha \rangle . \end{aligned}$$

(102)

The states $| \alpha \rangle$ are orthonormal product states. If O is supported on $\Pi ^E_n$, then the matrix elements $\langle \beta , O \alpha \rangle$ only depend on the configuration of $\alpha$ and $\beta$ on $\Pi ^E_n$. This information still allows us to deduce the boundary conditions $b(\alpha )$ and $b(\beta )$. Moreover, the matrix element vanishes if $b(\alpha ) \ne b(\beta )$, hence

$$\begin{aligned} \omega _{n+1}(O)&= \langle \Omega _{n+1}, O \Omega _{n+1} \rangle = \frac{1}{2^{\left| \Pi _{n+1}\right| }} \sum _{\alpha , \beta } (-1)^{\sharp \alpha + \sharp \beta } \langle \beta , O \alpha \rangle \\&= \frac{1}{2^{\left| \Pi _{n+1}\right| }} \sum _{b} \sum _{\begin{array}{c} \alpha : b(\alpha ) = b \\ \beta : b(\beta ) = b \end{array}} (-1)^{\sharp \alpha + \sharp \beta } \langle \beta , O \alpha \rangle \\&= \frac{2}{2^{\left| \Pi _{n+1}\right| }} \sum _{b} \sum _{\alpha ', \beta ' \in \mathcal {P}_n^{(b)}} (-1)^{\sharp \alpha ' + \sharp \beta '} \langle \beta ', O \alpha ' \rangle \\&= \frac{1}{2^{6n - 1}} \sum _b \, \frac{1}{2^{\left| \Pi _n\right| }} \sum _{\alpha ', \beta ' \in \mathcal {P}_n^{(b)}} (-1)^{\sharp \alpha ' + \sharp \beta '} \langle \beta ', O \alpha ' \rangle \\&= \frac{1}{2^{6n - 1}} \sum _b \eta _n^{(b)}(O). \end{aligned}$$

The factor of 2 appearing in the third line is the number of choices of completing a loop soup $\alpha '$ in $\Pi ^E_{n}$ with boundary condition b to a closed-loop soup $\alpha$ in $\Pi ^E_{n+1}$. The phase $(-)^{\sharp \alpha ' + \sharp \beta '}$ does not depend on which (common) completion is chosen. Indeed, changing the completion changes both $\sharp \alpha '$ and $\sharp \beta '$ 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 $\left| \Pi _{n+1}\right| - \left| \Pi _n\right| = 6n - 1$. $\square$

Lemma A.4

Given $\alpha \in \mathcal {P}_n^{(b)}$, denote by $\sharp _1 \alpha$ and $\sharp _2 \alpha$ the number of loops in the two possible completions. Then $\sharp _1 \alpha - \sharp _2 \alpha$ is odd if the number of marked points for b is a multiple of 4, and even otherwise.

Proof

Assume first that $\alpha$ has no closed loops. Let the number of marked points be 2n. The following construction is illustrated in Fig. 18. Abstract the region $\Pi _n$ 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 $\alpha$ connects these n intervals into groups. The number of groups g is the number of closed loops in this completion, say $\sharp _1 \alpha = g$. 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 $\sharp _2 \alpha$, and we find

$$\begin{aligned} \sharp _1 \alpha - \sharp _2 \alpha = g - (1 + n - g) = 2g -n - 1, \end{aligned}$$

(103)

which is odd if n is even and vice versa.

Any closed loops of $\alpha$ remain connected components of both completions, so internal loops do not contribute to $\sharp _1 \alpha = \sharp _2 \alpha$.

Fig. 18

Red dotted paths completing $\alpha$ 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 $\alpha$ to a closed-loop soup (color figure online)

$\square$

We further show

Lemma A.5

For any boundary condition b and any O supported on $\Pi ^E_{n-1}$, we have $\eta _{n}^{(b)}(O) = \omega (O)$.

Proof

From Lemma A.1, it is sufficient to show that $\eta _n^{(b)}(O) = \omega _n(O)$. Note that $\omega _n = \eta _n^{\emptyset }$, where $\emptyset$ stand for the trivial boundary condition.

For any other boundary condition b, let $A_b$ be the product of $A_p$ operators over hexagons between pairs of marked edges of b. Clearly, $A_b$ is supported outside $\Pi ^E_{n-1}$, so $A_b^* O A_b = O$, and since $A_b$ bijectively maps loop soups satisfying b to closed-loop soups, we find

$$\begin{aligned} \eta _n^{(b)}(O) = \eta _n^{(b)}(A_b^* O A_b) = \eta _n^{\emptyset }(O) = \omega _n(O) = \omega (O). \end{aligned}$$

(104)

$\square$

1.2 Purity of the Limit State

We will now show that $\omega$ is a pure state by making use of the following lemma, which is a special case of Lemma 2.1. of [13].

Lemma A.6

(Lemma 2.1 of [13]). A state $\omega$ on a UHF algebra realised as the inductive limit of a sequence of finite matrix algebras $\{\mathcal {M}_m\}$ is pure if the following holds:

For each n, there exists $m > n$ such that if $\rho$ is a linear functional on $\mathcal {M}_m$ that satisfies

$$\begin{aligned} \omega |_{\mathcal {M}_m} \ge \rho \ge 0, \end{aligned}$$

(105)

then

$$\begin{aligned} \rho |_{\mathcal {M}_n} = \lambda \omega |_{\mathcal {M}_n} \end{aligned}$$

(106)

for some $\lambda \in \mathbb {R}$.

In applying this theorem to our setting, we take $\mathcal {M}_n$ to be the algebra supported on $\Pi ^E_n$.

Fix n and take $m \ge n+1$. Let $\rho$ be a linear functional on $\mathcal {M}_m$ such that Eq. (105) is satisfied. From Proposition A.3 and Lemma A.1, we have

$$\begin{aligned} \omega |_m = \omega _{m+1}|_m = \frac{1}{2^{6m - 1}} \sum _b \eta _m^{(b)}, \end{aligned}$$

(107)

which is a Schmidt decomposition for $\omega |_m$. The assumption $\omega |_m \ge \rho \ge 0$ implies that $\rho$ is a mixture of pure states in the span of the $\eta _m^{(b)}$’s (Lemma A.7). It then follows from Lemma A.5 and $m > n$ that

$$\begin{aligned} \rho |_n = \lambda \omega |_n \end{aligned}$$

(108)

for some $0 \le \lambda \le 1$.

We conclude by Lemma A.6 that $\omega$ is pure.

We have used the following lemma:

Lemma A.7

Let $\omega$ and $\rho$ be linear functionals on a finite matrix algebra such that $\omega \ge \rho \ge 0$. Suppose

$$\begin{aligned} \omega = \sum _{\alpha } p_{\alpha } | \psi _{\alpha } \rangle \langle \psi _{\alpha } | \end{aligned}$$

(109)

is a Schmidt decomposition of $\omega$. Then any Schmidt decomposition

$$\begin{aligned} \rho = \sum _{\beta } q_{\beta } | \phi _{\beta } \rangle \langle \phi _{\beta } | \end{aligned}$$

(110)

satisfies

$$\begin{aligned} \textrm{span}\{ | \phi _{\beta } \rangle \}_{\beta } \subset \textrm{span}\{ | \psi _{\alpha } \rangle \}_{\alpha }. \end{aligned}$$

(111)

i.e. the Schmidt states of $\rho$ span a subspace of the space spanned by the Schmidt states of $\omega$.

Proof

Suppose the conclusion is false, so one of the $\phi _{\beta }$, say $\phi _1$, lies outside of $\mathcal {V}= \textrm{span}\{\psi _{\alpha }\}$. Then there is a vector $\chi$ orthogonal to $\mathcal {V}$ and such that $c = \langle \phi _1, \chi \rangle \ne 0$.

Consider now the positive operator $P = | \chi \rangle \langle \chi |$. We have $\omega (P) = 0$ and

$$\begin{aligned} \rho (P) = \sum _{\beta } q_{\beta } |\langle \phi _{\beta }, \chi \rangle |^2 > 0 \end{aligned}$$

(112)

where all terms are non-negative and at least the term $\beta = 1$ is strictly positive.

It follows that $(\omega - \rho )(P) < 0$, violating the assumption. $\square$

Properties of String Operators

For each vertex v, let

$$\begin{aligned} A_v:= \frac{1}{2} \left( 1 + \prod _{e \sim v} \sigma _e^Z \right) \end{aligned}$$

(113)

where the product runs over the three edges connected to v. For each hexagon p, regarded as a closed loop with counterclockwise orientation, let

$$\begin{aligned} B_p:= \frac{1}{2} \left( 1 + W_S[\partial p] \right) \left( \prod _{v \in p} A_v \right) . \end{aligned}$$

(114)

The operators $B_p$ and $A_v$ are orthogonal projections, and they all commute with each other.

Let $\Pi _n$ be a finite set of hexagons as in Fig. 9. Let $\Pi ^V_n$ be the set of vertices belonging to some hexagon in $\Pi _n$. We set

$$\begin{aligned} H_{\Pi _n}:= \sum _{v \in \Pi ^V_n} (1 - A_v) + \sum _{p \in \Pi _n} B_p. \end{aligned}$$

(115)

Let us also introduce terms imposing boundary conditions:

$$\begin{aligned} H_{\partial \Pi _n} = \sum _{e \in \partial \Pi _n} \frac{1}{2}\left( 1 - \sigma _e^Z \right) . \end{aligned}$$

(116)

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 $H_{\Pi _n}$.

We now consider the commuting projection Hamiltonians

$$\begin{aligned} H_n:= H_{\Pi _n} + H_{\partial \Pi _n}. \end{aligned}$$

(117)

Let $\widetilde{\Pi }_n$ be the collection of edges that have an endpoint in $\Pi ^V_n$. Then $\mathcal {H}_n \in \mathcal {A}_{\widetilde{\Pi }_n}$. Moreover, the state $\omega _n$ restricts to $\mathcal {A}_{\widetilde{\Pi }_n}$ as a pure state. Let us continue to denote this restriction by $\omega _n$. We have

Lemma B.1

The state $\omega _n$ on $\mathcal {A}_{\widetilde{\Pi }_n}$ is the unique ground state of $H_n$.

Proof

The state $\omega _n$ is defined by the expectation in the vector state

$$\begin{aligned} \Omega _n = \sqrt{\frac{1}{2^{\left| \Pi _n\right| }}} \, \sum _{\Pi \subset \Pi _n} \, (-1)^{\sharp (\Pi )} \, A_{\Pi } \Omega _0 \end{aligned}$$

(118)

where $\Omega _0$ has all $\sigma _e^Z = 1$.

The state $\Omega _n$ is a superposition of closed string configurations in $\Pi _n$. Each such closed string configuration satisfies

$$\begin{aligned} (1 - A_v) A_{\Pi } \Omega _0 = 0, \quad \frac{1}{2}(1 - \sigma _e^Z) A_{\Pi } \Omega _0 = 0 \end{aligned}$$

(119)

for all $v \in \Pi ^V_n$, all $e \in \partial \Pi _n$ and all $\Pi \subset \Pi _n$.

To see that $\Omega _n$ is a ground state of $H_n$ it remains to show that it is in the kernel of all $B_p$ for $p \in \Pi _n$. One can check that

$$\begin{aligned} W_S[\partial p] A_{\Pi } \Omega _0 = \phi (p, \Pi ) A_{p \triangle \Pi } \Omega _0 \end{aligned}$$

(120)

where $\phi (p, \Pi ) = -1$ if $p \triangle \Pi$ has the same parity of connected components as $\Pi$, and $\phi (p, \Pi ) = 1$ otherwise. i.e. 

$$\begin{aligned} \phi (p, \Pi ) = (-1)^{\sharp \Pi + \sharp (p \triangle \Pi ) + 1} \end{aligned}$$

(121)

It follows that $W_S[\partial p] \Omega _n = - \Omega _n$ for any $p \in \Pi _n$, hence $B_p \Omega _n = 0$.

To see that $\Omega _n$ is the unique ground state, observe that any ground state must be in the kernel of all the $1 - A_v$ for $v \in \Pi ^V_n$ and all the $\frac{1}{2}(1 - \sigma _e^Z)$ for $e \in \partial \Pi _n$. The space of states that are simultaneously in the kernels of all these projections is spanned by the closed string states

$$\begin{aligned} A_{\Pi } \Omega _0, \quad \Pi \subset \Pi _n. \end{aligned}$$

(122)

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 $W_S[\partial p]$ for $p \in \Pi _n$. Consider a general state

$$\begin{aligned} \Psi = \sum _{\Pi \subset \Pi _n} \psi (\Pi ) A_{\Pi } \Omega _0 \end{aligned}$$

(123)

where $\psi (\Pi ) \in \mathbb {C}$ are arbitrary. Then

$$\begin{aligned} W_S[\partial p] \Psi = \sum _{\Pi \subset \Pi _n} \psi (\Pi ) \phi (p, \Pi ) A_{p \triangle \Pi } \Omega _0, \end{aligned}$$

(124)

so $W_S[\partial p] \Psi = -\Psi$ only if

$$\begin{aligned} \psi (\Pi ) (-1)^{\sharp \Pi } - \psi (p \triangle \Pi ) (-1)^{\sharp (p \triangle \Pi )}. \end{aligned}$$

(125)

If any of the $\psi (\Pi )$ is non-zero (which must be the case, otherwise $\Psi = 0$), then this enforces

$$\begin{aligned} \psi (\Pi ') = (-1)^{\sharp \Pi + \sharp \Pi '} \psi (\Pi ') \end{aligned}$$

(126)

for all $\Pi ' \subset \Pi _n$. Indeed, and $\Pi$ can be related to any $\Pi '$ by a sequence of symmetric differences with elementary hexagons p. This shows that $\Psi \simeq \Omega _n$, so $\Omega _n$ is indeed the unique ground state of $H_n$ on $\mathcal {A}_{\widetilde{\Pi }_n}$. $\square$

Lemma B.2

If P is a closed path entirely contained in $\Pi _n$, 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 [16]. 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 $H_{\partial \Pi _n}$ is obvious. Let us show that $W_S[P]$ commutes with $B_p$ for $p \in \Pi _n$.

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

$$\begin{aligned} W_S[P] W_S[\partial p] W_S[P]^* = W_S[\partial p] U[Q] \end{aligned}$$

(127)

where the string operators U[Q] are defined in Eq. (55). Since $U[Q](\prod _{v \in p} A_v) = \prod _{v \in p} A_v$, and all $A_v$’s commute with $W_S[P]$ we find

$$\begin{aligned} W_S[P] \left( W_S[\partial p] \big ( \prod _{v \in p} A_v \big ) \right) W_S[P]^*= & {} W_S[\partial p] U[Q] \left( \prod _{v \in p} A_v \right) \nonumber \\= & {} W_S[\partial p] \left( \prod _{v \in p} A_v \right) . \end{aligned}$$

(128)

The claim for semion string operators $W_S[P]$ follows.

The required result is easy to verify for bound state strings $W_B[P]$, and since $W_{{\bar{S}}}[P] = W_{S}[P] W_{B}[P]$, the claim also holds for the anti-semion string operators. $\square$

Lemma B.3

If P is a finite closed string, then

$$\begin{aligned} \omega \circ w_a[P] = \omega \end{aligned}$$

(129)

for any $a \in \{1, S, {\bar{S}}, B\}$.

Proof

By Lemmas B.1 and B.2, we have

$$\begin{aligned} W_a[P] \Omega _n \sim \Omega _n \end{aligned}$$

(130)

for n sufficiently large. Hence, $\omega _n \circ w_a[P] = \omega _n$ for n sufficiently large. The double semion state $\omega$ is by definition the weak-* limit of the sequence $\omega _n$ so

$$\begin{aligned} \omega \circ w_a[P] = \lim _{n \uparrow \infty } \, \omega _n \circ w_a[P] = \lim _{n \uparrow \infty } \omega _n = \omega . \end{aligned}$$

(131)

$\square$

We are now ready to give the

Proposition 3.2

Let O be a strictly local observable. Then we can find a finite closed loop $P'$ such that $w_a[P](O) = w_a[P'](O)$. From Lemma B.3, we then find $(\omega \circ w_a[P])(O) = (\omega \circ w_a[P'])(O) = \omega (O)$. Since the strictly local operators are dense in $\mathcal {A}$, it follows that $\omega \circ w_a[P] = \omega$.