Symmetry fractionalization and twist defects

Our model is a ${{\mathbb{Z}}}_{n}$ gauge theory living on a certain oriented, quasi-2D lattice, presented in figure 1. Following [1], we start with a truly 2D oriented lattice, which can be taken to be a square lattice in the xy plane for all of the examples we consider, and stack $| G|$ identical copies of it. This stacking allows us to identify corresponding vertices and links in each copy. In particular, consider the set of $| G|$ vertices that all have the same $x,y$ coordinate. For any ordered pair (v, w) of such vertices, we add an oriented link connecting v to w. For clarity, we refer to these $| G| (| G| -1)$ links as vertical, as opposed to the links within layers, which will be called horizontal. The orientation of horizontal links is the same across all layers.

The set of $| G|$ corresponding vertices together with the $| G| (| G| -1)$ links connecting them will also be referred to as a supervertex ([1] calls this a Cayley graph). Likewise, the set of $| G|$ links which project to the same 2D link will be referred to as a superlink.

The Hilbert space is taken to be spanned by ${{\mathbb{Z}}}_{n}$ labellings of the links of our lattice. From now on we will identify ${{\mathbb{Z}}}_{n}$ with the set of nth roots of unity, i.e. complex numbers of the form ${{\rm{e}}}^{2\pi {\rm{i}}j/n},j=0,\ldots ,n-1$. Formally, we define an n dimensional link Hilbert space ${{ \mathcal H }}_{l}$ whose basis states are in one to one correspondence with such roots of unity:

$$\begin{eqnarray}&&{{ \mathcal H }}_{l}={\rm{span}}\{| {\eta }_{l}\rangle \;| {\eta }_{l}\in \{{{\rm{e}}}^{2\pi {\rm{i}}j/n},j\in {\mathbb{Z}}\}\}\end{eqnarray} \tag{ 5 }$$

and take the total Hilbert space ${ \mathcal H }$ to be the tensor product of these link Hilbert spaces (including both horizontal and vertical links):

$$\begin{eqnarray}&&{ \mathcal H }=\underset{l}{\otimes }{{ \mathcal H }}_{l}.\end{eqnarray} \tag{ 6 }$$

On each link Hilbert space we define the usual 'phase' and 'charge' measuring operators a_l and e_l:

$$\begin{eqnarray}&&{e}_{l}| {\eta }_{l}\rangle =| {{\rm{e}}}^{2\pi {\rm{i}}/n}{\eta }_{l}\rangle ,\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{a}_{l}| {\eta }_{l}\rangle ={\eta }_{l}| {\eta }_{l}\rangle .\end{eqnarray} \tag{ 8 }$$

By tensoring with the identity on all other links, we can think of a_l and e_l as being defined on the total Hilbert space ${ \mathcal H }$. Note that two such operators acting on different links l and $l^{\prime}$ commute.

The Hamiltonians we will work with contain terms which act on certain groupings of links, associated to vertices and plaquettes, and before we can write them down we need to establish some effective notation. First of all, as we mentioned above, our quasi-2D lattice is oriented, which means that there is a preferred choice of direction for each link. This orientation is efficiently encoded in a function s_v(l), where l is any link and v is one of the two endpoint vertices of this link:

$$\begin{eqnarray}{s}_{v}(l)=\left\{\begin{array}{ll}1 & \ \mathrm{if}\ l\ \mathrm{points}\ \mathrm{toward}\ v\\ -1 & \ \mathrm{if}\ l\ \mathrm{points}\ \mathrm{away}\ \mathrm{from}\ v.\end{array}\right.\end{eqnarray} \tag{ 9 }$$

We will assume that our orientation is consistent across the $| G|$ layers, i.e. ${s}_{v}(l)={s}_{v^{\prime} }(l^{\prime} )$ whenever $v,v^{\prime}$ are in the same supervertex, and horizontal links $l,l^{\prime}$ are in the same superlink.

Additionally, we now also assign an orientation to all plaquettes p (i.e. plaquettes involving any combination of horizontal and vertical links). This orientation is just a choice of direction, either clockwise or counterclockwise, along the links that border p. For each such link l bordering a plaquette p, this choice of direction could be the same or opposite to the one defined by equation (9). This distinction is encoded in a function s_p(l), where l is a link bordering the plaquette p:

$$\begin{eqnarray}{s}_{p}(l)=\left\{\begin{array}{ll}1 & \ \mathrm{if}\ l\ \mathrm{oriented}\ \mathrm{with}\ p\\ -1 & \ \mathrm{if}\ l\ \mathrm{oriented}\ \mathrm{against}\ p.\end{array}\right.\end{eqnarray} \tag{ 10 }$$

We will assign this plaquette orientation consistently across the layers, in that if p and $p^{\prime}$ are plaquettes made up entirely of horizontal links that project to the same plaquette in the xy plane, we assign them the same orientation. This just means that if l and $l^{\prime}$ are corresponding links of p and $p^{\prime}$ respectively (so that $l,l^{\prime}$ are in the same superlink), then ${s}_{p}(l)={s}_{p^{\prime} }(l^{\prime} )$. Besides this constraint, the plaquette orientations are chosen arbitrarily.

Now, in [1], G acts by permuting layers, and since such a permutation induces a one to one mapping of the underlying oriented quasi-2D lattice to itself, an example of a G-invariant Hamiltonian is:

$$\begin{eqnarray}&&{\bf{H}}=-\displaystyle \sum _{v}{A}_{v}-\displaystyle \sum _{p}{B}_{p}+{\rm{h.c.}},\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{A}_{v}=\displaystyle \prod _{l\sim v}{e}_{l}^{{s}_{v}(l)},\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{B}_{p}=\displaystyle \prod _{l\in p}{a}_{l}^{{s}_{p}(l)}.\end{eqnarray} \tag{ 13 }$$

Where the notation $l\sim v$ refers to all links l that begin or end at v, and $l\in p$ refers to all the links that border a given plaquette p. While equation (11) is adequate in the case where the symmetry does not permute the gauge theory quasiparticles (anyons), we will need a slightly different construction for a symmetry action which does permute the anyons.

In our model, G will act both by permuting the links and changing the ${{\mathbb{Z}}}_{n}$ labels on the links. The permutation of the links induced by $g\in G$ is the same as that in [1]: given a vertex v in layer h, we let gv denote the vertex in layer gh which is in the same supervertex as v. Then, for a link $l=\langle {vv}^{\prime} \rangle$, we define ${gl}=\langle {gv}\;{gv}^{\prime} \rangle$. The change in the ${{\mathbb{Z}}}_{n}$ label that goes together with this link permutation—which is the new feature of our model, and is referred to as a twisting—is encoded in an integer valued function $\rho (g)$, with each $\rho (g)$ relatively prime to n (i.e. having no common factors with n). ρ is required to satisfy $\rho ({gh})=\rho (g)\rho (h)$ (note that this is multiplication of integers modulo n) and allows us to define a permutation action of G on ${{\mathbb{Z}}}_{n}$, namely $\eta \to {\eta }^{\rho (g)}$. More explicitly, if $\eta ={{\rm{e}}}^{2\pi {\rm{i}}k/n}$, then this action just takes $k\to \rho (g)k\;{\rm{mod}}\;n$. An example that we will focus on in the rest of the paper is $G={{\mathbb{Z}}}_{2}$ and n = 4, and $\rho (g)=-1$ for the non-trivial generator g of ${{\mathbb{Z}}}_{2}$.

Using ρ, we define the global action of G on the Hilbert space as follows. With a slight abuse of notation, we denote the unitary action of g by U_g, regardless of what Hilbert space is being acting upon. For the link degrees of freedom we let:

$$\begin{eqnarray}&&{U}_{g}| {\eta }_{l}\rangle =| {\eta }_{{gl}}^{\rho (g)}\rangle .\end{eqnarray} \tag{ 14 }$$

This induces the action on operators:

$$\begin{eqnarray}&&{U}_{g}{a}_{l}{U}_{g}^{-1}={a}_{{gl}}^{\rho ({g}^{-1})},\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{U}_{g}{e}_{l}{U}_{g}^{-1}={e}_{{gl}}^{\rho (g)}.\end{eqnarray} \tag{ 16 }$$

We can immediately infer that the action on vertex and plaquette terms defined in equation (11) is

$$\begin{eqnarray}&&{U}_{g}{B}_{p}{U}_{g}^{-1}={B}_{{gp}}^{\rho ({g}^{-1})},\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{U}_{g}{A}_{v}{U}_{g}^{-1}={A}_{{gv}}^{\rho (g)},\end{eqnarray} \tag{ 18 }$$

where gp is the plaquette made up of the links gl, for all $l\in p$. Note that for $\rho (g)\ne 1$, the Hamiltonian defined in equation (11) is not invariant under this global action of g. Instead, we consider the following more general Hamiltonian:

$$\begin{eqnarray}&&{{\bf{H}}}_{\mathrm{SET}}=-\displaystyle \sum _{m=1}^{n}\left(\displaystyle \sum _{v}{A}_{v}^{m}+\displaystyle \sum _{p}{({\omega }_{p}^{-1}{B}_{p})}^{m}\right).\end{eqnarray} \tag{ 19 }$$

Here ${\omega }_{p}$ are phases—in fact, nth roots of unity—associated with each plaquette p, which satisfy

$$\begin{eqnarray}&&{\omega }_{{gp}}={\omega }_{p}^{\rho (g)}.\end{eqnarray} \tag{ 20 }$$

We can verify that ${{\bf{H}}}_{\mathrm{SET}}$ is G-invariant:

$$\begin{eqnarray}&&{U}_{g}{{\bf{H}}}_{\mathrm{SET}}{U}_{g}^{-1}=-\displaystyle \sum _{m=1}^{n}\left(\displaystyle \sum _{v}{A}_{{gv}}^{m\rho (g)}+\displaystyle \sum _{p}{({\omega }_{p}^{-1}{B}_{{gp}}^{\rho ({g}^{-1})})}^{m}\right)\\ &&\quad =-\displaystyle \sum _{m=1}^{n}\left(\displaystyle \sum _{v}{A}_{{gv}}^{m\rho (g)}+\displaystyle \sum _{p}{({\omega }_{p}^{-\rho (g)}{B}_{{gp}})}^{m}\right)\\ &&\quad ={{\bf{H}}}_{\mathrm{SET}},\ \mathrm{if}\ {\omega }_{{gp}}={\omega }_{p}^{\rho (g)}.\end{eqnarray} \tag{ 21 }$$

Thus, for ${\omega }_{p}$ which satisfy ${\omega }_{{gp}}={\omega }_{p}^{\rho (g)}$, equation (19) describes a Hamiltonian that is invariant under the twisted G action.

Let $| {\rm{\Psi }}\rangle$ be a state of ${ \mathcal H }$ corresponding to a specific ${{\mathbb{Z}}}_{n}$ labeling of links. Recalling that the spectrum of B_p consists of the roots of unity ${{\rm{e}}}^{2\pi {\rm{i}}k/n}$, we see that

$$\begin{eqnarray}\displaystyle \sum _{m=1}^{n}{({\omega }_{p}^{-1}{B}_{p})}^{m}| {\rm{\Psi }}\rangle =\{\;n| {\rm{\Psi }}\rangle & \ \mathrm{if}\ {B}_{p}| {\rm{\Psi }}\rangle ={\omega }_{p}| {\rm{\Psi }}\rangle \\ 0 & \ \mathrm{otherwise}.\end{eqnarray} \tag{ 22 }$$

Thus the operator defined on the left side of equation (22) is equal to n times a projector. We now describe a notation that will let us concisely express such operators; we emphasize that this formulation is nothing more than a notational convenience. First, recall that the regular representation of a group H is an $| H|$ dimensional vector space with basis $\{| h^{\prime} \rangle | h^{\prime} \in H\}$, where $h\in H$ acts by

$$\begin{eqnarray}&&h\;:| h^{\prime} \rangle \to | {hh}^{\prime} \rangle .\end{eqnarray} \tag{ 23 }$$

Thus $h\in H$ is represented by an $| H|$ by $| H|$ matrix M(h), where each column and row is labeled by a group element $g,k\in H$, and the matrix elements are

$$\begin{eqnarray}&&M{(h)}_{g,k}={\delta }_{g,{hk}}.\end{eqnarray} \tag{ 24 }$$

A feature of these matrices is that $\mathrm{Tr}\ M(h)=| H| {\delta }_{h,1}$. Now, recall that the operator a_l acts by the phase ${\eta }_{l}$ on $| {\eta }_{l}\rangle$. In our new notation, acting with the operator a_l yields the matrix $M(\eta )$, where now $H={{\mathbb{Z}}}_{n}$. In the case where n = 4 we have

$$\begin{eqnarray}M({{\rm{e}}}^{\frac{{\rm{i}}\pi }{2}})=\left(\begin{array}{cccc}0 & 0 & 0 & 1\\ 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\end{array}\right).\end{eqnarray} \tag{ 25 }$$

To construct the plaquette terms, we take a trace of the matrix produced by a closed loop of a_l's. For example, consider a triangular plaquette p with links $1,2,3$, and ${\omega }_{p}=1$, and suppose $| {\rm{\Psi }}\rangle$ is an eigenvalue ${\eta }_{j}$ eigenvector of each a_j. Then:

$$\begin{eqnarray}&&(\mathrm{Tr}\ {a}_{1}{a}_{2}{a}_{3})| {\rm{\Psi }}\rangle =(\mathrm{Tr}\ M({\eta }_{1})M({\eta }_{2})M({\eta }_{3}))| {\rm{\Psi }}\rangle ,\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}=\;\left\{\begin{array}{ll}n| {\rm{\Psi }}\rangle & \ \mathrm{if}\ {\eta }_{1}{\eta }_{2}{\eta }_{3}={\bf{1}}\\ 0 & \ \mathrm{otherwise}.\end{array}\right.\end{eqnarray} \tag{ 27 }$$

Thus, the operator in (22) can be rewritten in this new notation as:

$$\begin{eqnarray}&&\displaystyle \sum _{m=1}^{n}{({\omega }_{p}^{-1}{B}_{p})}^{m}\to \mathrm{Tr}\ {\omega }_{p}^{-1}{B}_{p}\end{eqnarray} \tag{ 28 }$$

yielding a notationally convenient way of writing n times the projector onto the eigenvalue 1 subspace of ${\omega }_{p}^{-1}{B}_{p}$. Here it is understood that the complex number ${\omega }_{p}$ is substituted with its regular representation matrix. Notice that the trace on the right-hand side of this equation is over the auxiliary regular representation, and not the many body Hilbert space; both sides are operators in the many body Hilbert space.

With this new notation, our Hamiltonian takes the form

$$\begin{eqnarray}&&{{\bf{H}}}_{\mathrm{SET}}=-\displaystyle \sum _{v}\left(\displaystyle \sum _{m\in {{\mathbb{Z}}}_{n}}{A}_{v}^{m}\right)-\displaystyle \sum _{p}\mathrm{Tr}\ {\omega }_{p}^{-1}{B}_{p}.\end{eqnarray} \tag{ 29 }$$

One benefit of our new notation is that it makes it easy to generalize ${{\bf{H}}}_{\mathrm{SET}}$ to the case of an arbitrary abelian gauge group H, rather than just ${{\mathbb{Z}}}_{n}$. Indeed, to do this one just needs to let ρ be a map from G to ${\rm{Aut}}(H)$, the group of automorphisms of H. Nevertheless, we will stick to a ${{\mathbb{Z}}}_{n}$ gauge group in the remainder of this paper.

The next step is to discuss the phases ${\omega }_{p}$, different choices of which will give rise to different twisted symmetry enriched phases. To facilitate this discussion, it is useful to first group the degrees of freedom in our model in a slightly more convenient way. First, recall that a supervertex V is a collection of $| G|$ vertices which all project onto the same point in the xy plane, i.e. are vertically aligned. We now tensor the Hilbert spaces living on these links into a single supervertex Hilbert space ${{ \mathcal H }}_{V}$ spanned by $\{| {\eta }_{V}(g,h)\rangle \}$, where ${\eta }_{V}(g,h)$ is the ${{\mathbb{Z}}}_{n}$ label of the link ${l}_{g,{gh}}$ connecting layer g to layer gh (here h is necessarily different from the identity in G). We then denote by ${a}_{V}(g,h),{e}_{V}(g,h)$ the action of the operators ${a}_{{l}_{g,{gh}}},{e}_{{l}_{g,{gh}}}$, tensored with the identity on the remaining links, in the Hilbert space ${{ \mathcal H }}_{V}$.

Similarly, we define a superlink L to be the collection of $| G|$ horizontal links whose projections to the xy plane are all the same, and likewise define a superlink Hilbert space ${{ \mathcal H }}_{L}$ to be the tensor product of the associated $| G|$ link Hilbert spaces. We denote by ${a}_{L}(g),{e}_{L}(g)$ the action of the operators ${a}_{{l}_{g}},{e}_{{l}_{g}}$ on the link ${l}_{g}\in L$ in layer g, tensored with the identity on the remaining $| G| -1$ links.

Our total Hilbert space ${ \mathcal H }$ is thus a tensor product of the supervertex and superlink Hilbert spaces:

$$\begin{eqnarray}&&{ \mathcal H }=(\underset{V}{\otimes }{{ \mathcal H }}_{V})\otimes (\underset{L}{\otimes }{{ \mathcal H }}_{L}).\end{eqnarray} \tag{ 30 }$$

We now discuss the choice of U(1) phases ${\omega }_{p}$, which we also refer to as ${{\mathbb{Z}}}_{n}$ fluxes, since they are restricted to take values in the nth roots of unity. First of all, throughout this paper we will deal exclusively with the situation where the only non-trivial ${\omega }_{p}$ (i.e. ${\omega }_{p}\ne 1$) occur for plaquettes p that sit entirely within a single supervertex, or, in other words, contain no horizontal links. Let us focus on a specific supervertex V. Each vertical plaquette p within it is labeled by a triple $(f,g,h)$, where $g,h\ne 1$, indicating that it involves the links connecting layers f, fg and fgh. The corresponding term in the Hamiltonian (equation (29)) is:

$$\begin{eqnarray}\mathrm{Tr}\ {\omega }_{p}^{-1}{B}_{p} & \equiv & \mathrm{Tr}\ {\omega }_{p}^{-1}{B}_{V}(f,g,h)\\ & = & \mathrm{Tr}\ {\omega }_{p}^{-1}{a}_{V}{(f,{gh})}^{-1}{a}_{V}({fg},h){a}_{V}(f,g).\end{eqnarray} \tag{ 31 }$$

While most plaquettes will be three-edged, we are allowed to set $h={g}^{-1}$, producing a degenerate, two-edged plaquette. This situation can still be captured by equation (31) by defining ${a}_{V}(f,e)\equiv {\bf{I}}$ (the identity operator).

Using equation (20) we see that all of the ${\omega }_{p}$ are uniquely determined by the ${\omega }_{p}$ for $p=(e,g,h)$, i.e. for plaquettes p which start at the identity element of G. Letting $\omega (g,h)$ denote ${\omega }_{p}$ for $p=(e,g,h)$, we then have that for a general plaquette $p^{\prime} =(f,g,h)$, ${\omega }_{p^{\prime} }=\omega {(g,h)}^{\rho (f)}$ (figure 2).

Now, we would like to work with models which are unfrustrated, i.e. whose ground states are lowest energy eigenstates of all of the plaquette terms Consider the tetrahedron formed by the layers $\{e,f,{fg},{fgh}\}$, which contains four plaquettes: $(e,f,g)$, $(f,g,h)$, $(e,{fg},h)$, and $(e,f,{gh})$. A necessary and sufficient condition for the model to be unfrustrated is that the ${{\mathbb{Z}}}_{n}$ flux emanating out of any such tetrahedron be zero (figure 3), i.e.

$$\begin{eqnarray}&&\omega {(g,h)}^{\rho (f)}\omega (f,{gh})=\omega (f,g)\omega ({fg},h).\end{eqnarray} \tag{ 32 }$$

Indeed, if the ground state is unfrustrated, there must be some labeling $\{{\eta }_{V}(g,h)\}$ of the links in the supervertex V (namely one that corresponds to a configuration that enters the unfrustrated ground state with non-zero amplitude) such that

$$\begin{eqnarray}&&{\eta }_{V}(f,g){\eta }_{V}({fg},h){\eta }_{V}^{-1}(f,{gh})=\omega {(g,h)}^{\rho (f)},\forall f,g,h\in G.\end{eqnarray} \tag{ 33 }$$

Expressing ω in terms of the ${\eta }_{V}$ using this equation, we see that it satisfies equation (32). Conversely, given a choice of ω's that satisfy equation (32), we can simply set ${\eta }_{V}(g,h)=\omega (g,h)$. This link labeling satisfies all of the plaquette terms within the supervertex V. Later, we will see that it can be extended to an unfrustrated ground state of all of the vertex and plaquette terms in our model—indeed, we will explicitly solve the model for any choice of fluxes satisfying equation (32).

Certain different choices of $\omega (f,g)$ actually define Hamiltonians which can be made equivalent by redefining link variables:

$$\begin{eqnarray}&&{\eta }_{V}(g,h)\to \mu {(h)}^{\rho (g)}{\eta }_{V}(g,h),\end{eqnarray} \tag{ 34 }$$

where $\mu \;:G\to {{\mathbb{Z}}}_{n}$ is an arbitrary function. This redefinition then takes

$$\begin{eqnarray}&&\omega (g,h)\to \omega ^{\prime} (g,h)=\omega (g,h)\mu {(h)}^{\rho (g)}\mu (g)\mu {({gh})}^{-1}.\end{eqnarray} \tag{ 35 }$$

Note that the new $\omega ^{\prime} (g,h)$ also satisfy equation (32). In group cohomology terms this means that equivalence classes of non-frustrated Hamiltonians of the above form are parametrized by twisted group cohomology classes⁵ in${H}_{\rho }^{2}(G,{{\mathbb{Z}}}_{n})$.

For simple enough G, it is easy to compute these cohomology groups explicitly. For example, take $G={{\mathbb{Z}}}_{2}=\{1,-1\}$, n = 4. Then there is only a single plaquette, bounded by the links $1\to -1,-1\to 1$, which is pierced by a ${{\mathbb{Z}}}_{4}$ flux $\omega (-1,-1)$, subject to a twisting $\rho (-1)=-1$. Equation (32) produces a non-trivial constraint only for $f,g,h=-1$:

$$\begin{eqnarray}&&\omega {(-1,-1)}^{\rho (-1)}=\omega (-1,-1)\end{eqnarray} \tag{ 36 }$$

which implies $\omega (-1,-1)=\pm 1$. As we will see, these two choices of ω, which we call ${\omega }_{\pm }$, will produce two inequivalent SET Hamiltonians, which in turn yield two distinct non-abelian gauge theories once we gauge the ${{\mathbb{Z}}}_{2}$ symmetry.

Let us write out the final form of the SET Hamiltonian in a form convenient for gauging G, which we do in the next section. Recall (equation (30)) that our total Hilbert space ${ \mathcal H }$ is a tensor product of supervertex and superlink Hilbert spaces ${{ \mathcal H }}_{V}$ and ${{ \mathcal H }}_{L}$. Because all of the links in a superlink are oriented the same way we can define the orientation factor ${s}_{V}(L)={s}_{v}(l)=\pm 1$ where $v,l$ are any vertex and adjoining link in V and L respectively. Now, plaquettes containing only horizontal links can similarly grouped into superplaquettes P (figure 4). Again, because the plaquette orientations have been chosen so that all plaquettes p in a superplaquette P are oriented the same way, we can define ${s}_{P}(L)={s}_{p}(l)$ for any p in P and l in L bordering p. We then have the form of the Hamiltonian:

$$\begin{eqnarray}{{\bf{H}}}_{\mathrm{SET}} & = & -\displaystyle \sum _{g\in G}\left(\displaystyle \sum _{V}\left(\displaystyle \sum _{m\in {{\mathbb{Z}}}_{n}}{A}_{V}^{m}(g)\right)+\displaystyle \sum _{P}\mathrm{Tr}\ {B}_{P}(g)\right)-\displaystyle \sum _{{\rm{vertical}}\;p}\mathrm{Tr}\ {\omega }_{p}^{-1}{C}_{p},\end{eqnarray} \tag{ 37 }$$

where V and P range over supervertices and superplaquettes respectively, and the various terms in the above sum are defined as follows. We've rewritten our vertex and plaquette operators as

$$\begin{eqnarray}{A}_{V}(g) & = & \displaystyle \prod _{h\in G}{e}_{V}{(g,h)}^{-1}{e}_{V}({gh},{h}^{-1})\displaystyle \prod _{L\sim V}{e}_{L}{(g)}^{{s}_{V}(L)},\end{eqnarray} \tag{ 38 }$$

which denotes the term acting on the vertex on layer g of the supervertex V, and

$$\begin{eqnarray}&&{B}_{P}(g)=\displaystyle \prod _{L\in P}{a}_{L}^{{s}_{P}(L)}(g)\end{eqnarray} \tag{ 39 }$$

denotes the term acting on the plaquette on layer g of the superplaquette P. Finally, for any plaquette p which is not composed solely of horizontal links, which we refer to as 'vertical' in equation (37) above

$$\begin{eqnarray}&&{C}_{p}=\displaystyle \prod _{l\in p}{a}_{l}^{{s}_{p}(l)}.\end{eqnarray} \tag{ 40 }$$

Note that there are two distinct kinds of vertical plaquettes: ones contained entirely in a single supervertex, and ones involving a superlink and the adjoining two supervertices. Only for the ones contained entirely in a single supervertex can we have ${\omega }_{p}\ne 1$.

The goal of this somewhat technical appendix is to derive equations (50) and (57), which describe our SET Hamiltonian in equation (37) coupled to a G gauge field. First we introduce G gauge field degrees of freedom, which are just $| G|$ dimensional Hilbert spaces which we insert between any supervertex and an outgoing superlink (see figure A1). The $| G|$ degrees of freedom are incorporated by enlarging each superlink Hilbert space ${{ \mathcal H }}_{L}\to {{ \mathcal H }}_{L}\otimes {{\mathbb{C}}}^{| G| }$, with states in this larger Hilbert space carrying an extra G gauge field label g_L (figure 5):

$$\begin{eqnarray}&&| {\{{\eta }_{L}(g)\}}_{g}\rangle \to | {\{{\eta }_{L}(g)\}}_{g},{g}_{L}\rangle .\end{eqnarray} \tag{ 41 }$$

The minimal coupling prescription we use is as follows. Given any local term in the original ungauged Hamiltonian, for a G gauge field configuration which is gauge equivalent to the trivial configuration in the vicinity of this local term (i.e. has no G fluxes), the form of the corresponding minimally coupled term is completely fixed by G gauge invariance. For a G gauge field configuration which does contain non-zero G fluxes in the vicinity of this local term, we simply set the corresponding minimally coupled term to 0. This actually only occurs for the superplaquette term, when there is a non-zero G flux through it. Since the Hamiltonian consists of commuting terms which all have negative eigenvalue on the ground state, setting this minimally coupled term to 0 is actually an energetic penalty for the non-zero G flux. Lastly, we include 'vertex' terms which make the G gauge field fluctuate and thus energetically impose G gauge invariance. These vertex terms are the only ones which alter the G gauge field configuration.

The above paragraph specifies the minimally coupled Hamiltonian uniquely, but to actually write it out in a compact form it is useful to introduce some additional notation. It is easiest to start with the vertex terms, given by equation (38). A particular such term involves a specific vertex v, which is part of a supervertex V. To minimally couple it, we have to modify each e_L(g) term corresponding to an outgoing superlink L from V (that is, one with ${s}_{V}(L)=-1$) to take account of the gauge field g_L, and replace it with:

$$\begin{eqnarray}&&{e}_{L}{(g)}^{R}| {\{{\eta }_{L}(g^{\prime} )\}}_{g^{\prime} },{g}_{L}\rangle \;\equiv | {\{{\eta }_{L}(g^{\prime} ){{\rm{e}}}^{-2\pi {\rm{i}}\frac{\rho ({g}_{L})}{n}{\delta }_{g,g^{\prime} {g}_{L}}}\}}_{g^{\prime} },{g}_{L}\rangle .\end{eqnarray} \tag{ 42 }$$

Here R is just a superscript. To keep the notation compact, we define a superscript-valued function $s{^{\prime} }_{V}(L)=R$ when ${s}_{V}(L)=-1$, and $s{^{\prime} }_{V}(L)$ being trivial otherwise, so that we can write the minimally coupled version of A_V(g) simply as:

$$\begin{eqnarray}{\tilde{A}}_{V}(g) & = & \displaystyle \prod _{h\in G}{e}_{V}{(g,h)}^{-1}{e}_{V}({gh},{h}^{-1})\times \displaystyle \prod _{L\sim V}{e}_{L}{(g)}^{s{^{\prime} }_{V}(L)}.\end{eqnarray} \tag{ 43 }$$

Now let us minimally couple the horizontal plaquette terms B_P(g). When there is a non-trivial G flux through P, we simply set ${\tilde{B}}_{P}(g)=0$. When this flux is trivial, we first define an auxiliary 'book-keeping' operator

$$\begin{eqnarray}&&{\alpha }_{L}(| {\{{\eta }_{L}(g)\}}_{g},{g}_{L}\rangle \otimes \ldots )\quad =M({g}_{L}){U}_{{gL}}^{-1}(| \{{\eta }_{L}(g)\}{}_{g},{g}_{L}\rangle \otimes \ldots ),\end{eqnarray} \tag{ 44 }$$

where $M({g}_{L})$ is the $| G|$ by $| G|$ matrix representing g_L in the regular representation of G (see the discussion around equation (23)), U${g}_{L}$ is the global action of g_L (as defined in equation (14)), and the ellipses denote the rest of the Hilbert space ${ \mathcal H }$. Now define

$$\begin{eqnarray}&&{\tilde{B}}_{P}(g)=\displaystyle \prod _{L\in P}{({a}_{L}(g){\alpha }_{L})}^{{s}_{P}(L)}.\end{eqnarray} \tag{ 45 }$$

Note that even though the operators ${\alpha }_{L}$ are non-local (because they involve the global U${g}_{L}$ operator, when the G flux through P is trivial their combined action in ${\tilde{B}}_{P}(g)$ cancels away from the plaquette P, and we end up with a local operator. Also recall that it is the trace of ${\tilde{B}}_{P}(g)$ over the auxiliary regular representation space which appears in the Hamiltonian.

Next, let us minimally couple the term C_p which involves both a superlink and the adjacent two supervertices. Any such p is a rectangle, and C_p is a product of four link terms:

$$\begin{eqnarray}&&{C}_{p}={({a}_{L}({gh}){a}_{V}(g,h))}^{-1}{a}_{V^{\prime} }(g,h){a}_{L}(g).\end{eqnarray} \tag{ 46 }$$

Minimally coupling this inserts the gauge field, in the form of the ${\alpha }_{L}$ operators, along the horizontal links:

$$\begin{eqnarray}&&{\tilde{C}}_{p}={({a}_{L}({gh}){\alpha }_{L}{a}_{V}(g,h))}^{-1}{a}_{V^{\prime} }(g,h){a}_{L}(g){\alpha }_{L}.\end{eqnarray} \tag{ 47 }$$

Recall that there is never any G flux through such a plaquette p. Finally, the minimal coupling of C_p for a plaquette p entirely within a supervertex V does not involve the gauge field at all, ${\tilde{C}}_{p}={C}_{p}$.

In summary, the minimal coupling is done by starting with the Hamiltonian in equation (37) and making the modifications

$$\begin{eqnarray}&&{a}_{L}(g)\to {a}_{L}(g){\alpha }_{L},\end{eqnarray} \tag{ 48 }$$

$$\begin{eqnarray}{s}_{V}(L)\to s{^{\prime} }_{V}(L)=\left\{\begin{array}{cc}{\bf{1}} & \ \mathrm{if}\ L\to V\\ R & \ \mathrm{if}\ L\leftarrow V,\end{array}\right.\end{eqnarray} \tag{ 49 }$$

resulting in a minimally coupled Hamiltonian ${{\bf{H}}}_{{\rm{m.c.}}}$:

$$\begin{eqnarray}{{\bf{H}}}_{{\rm{m.c.}}} & = & -\displaystyle \sum _{g\in G}\left(\displaystyle \sum _{V}\left(\displaystyle \sum _{m\in {{\mathbb{Z}}}_{n}}{\tilde{A}}_{V}^{m}(g)\right)+\displaystyle \sum _{P}\mathrm{Tr}\ {\tilde{B}}_{P}(g)\right)-\displaystyle \sum _{{\rm{vertical}}\;p}\mathrm{Tr}\ {\tilde{C}}_{p},\end{eqnarray} \tag{ 50 }$$

$$\begin{eqnarray}{\tilde{A}}_{V}(g) & = & \displaystyle \prod _{h\in G}{e}_{V}{(g,h)}^{-1}{e}_{V}({gh},{h}^{-1})\displaystyle \prod _{L\sim V}{e}_{L}{(g)}^{{s}_{V}^{\prime }(L)},\end{eqnarray} \tag{ 51 }$$

$$\begin{eqnarray}&&{\tilde{B}}_{P}(g)=\displaystyle \prod _{L\in P}{({a}_{L}(g){\alpha }_{L})}^{{s}_{P}(L)}\end{eqnarray} \tag{ 52 }$$

$$\begin{eqnarray}&&{\tilde{C}}_{p}=\left\{\begin{array}{c}{\omega }_{p}^{-1}{a}_{V}{(f,{gh})}^{-1}{a}_{V}({fg},h){a}_{V}(f,g)\\ \unicode{U+21B3} {\rm{if}}\ p\ {\rm{is}}\;{\rm{in}}\ V\\ {({a}_{L}({gh}){\alpha }_{L}{a}_{V}(g,h))}^{-1}{a}_{V^{\prime} }(g,h){a}_{L}(g){\alpha }_{L}\\ \unicode{U+21B3} \mathrm{if}p\ \mathrm{is}\ \mathrm{in}\ {\rm{L}}=\langle {\rm{V}}\ {\rm{V}}^{\prime} \rangle .\end{array}\right.\end{eqnarray} \tag{ 53 }$$

Here the traces are over both G and ${{\mathbb{Z}}}_{n}$. The Hamiltonian ${{\bf{H}}}_{{\rm{m.c.}}}$ describes our SET in a fixed background G gauge field configuration, corresponding to some set of extrinsic defects. We will also want to have a Hamiltonian where the G gauge field is dynamical. To obtain it, first define operators which change the value of g_L on a single superlink L. We will need two such operators, ${\epsilon }_{L}(g)$ and ${\epsilon }_{L}^{R}(g)$:

$$\begin{eqnarray}&&{\epsilon }_{L}(g)| {\{{\eta }_{L}(g^{\prime} )\}}_{g^{\prime} },{g}_{L}\rangle =| {\{{\eta }_{L}{({g}^{-1}g^{\prime} )}^{\rho (g)}\}}_{g^{\prime} },{{gg}}_{L}\rangle ,\end{eqnarray} \tag{ 54 }$$

$$\begin{eqnarray}&&{\epsilon }_{L}{(g)}^{R}| {\{{\eta }_{L}(g^{\prime} )\}}_{g^{\prime} },{g}_{L}\rangle =| {\{{\eta }_{L}(g^{\prime} )\}}_{g^{\prime} },{g}_{L}{g}^{-1}\rangle .\end{eqnarray} \tag{ 55 }$$

Using these we define:

$$\begin{eqnarray}&&{{ \mathcal A }}_{V}(g)={U}_{g}^{-1}(V)(\displaystyle \prod _{L\sim V}{\epsilon }_{L}{(g)}^{s{^{\prime} }_{V}(L)}),\end{eqnarray} \tag{ 56 }$$

where U_g(V) is the action of g on the supervertex V. The Hamiltonian with dynamical G gauge field then becomes

$$\begin{eqnarray}&&{{\bf{H}}}_{{\rm{gauged}}}={{\bf{H}}}_{{\rm{m.c.}}}-\displaystyle \sum _{V}\displaystyle \sum _{g\in G}{{ \mathcal A }}_{V}(g).\end{eqnarray} \tag{ 57 }$$

This is the model whose topological order we have to analyze.

We claim that the gauged model defined by equation (57) is equivalent to—i.e. has the same topological order as—an E gauge theory, where the finite group E is a particular extension of G by ${{\mathbb{Z}}}_{n}$. We derive this equivalence carefully in appendix A, while in this section we just write down the result. First, let us discuss the extension E. As a set, it is just the product ${{\mathbb{Z}}}_{n}\times G$, so we can label its elements by $(\eta ,g)$, with η an nth root of unity and $g\in G$. On the other hand, the group multiplication law is determined by both by ρ and $\omega (g,h)$:

$$\begin{eqnarray}&&({\eta }_{1},{g}_{1})\cdot ({\eta }_{2},{g}_{2})=({\eta }_{1}{({\eta }_{2})}^{\rho ({g}_{1})}\omega ({g}_{1},{g}_{2}),{g}_{1}{g}_{2}).\end{eqnarray} \tag{ 58 }$$

Now we write down an E gauge theory on a single copy of the 2D lattice we considered above (as opposed to $| G|$ stacked copies of it). Since the vertices and links of this 2D lattice are in one to one correspondence with the supervertices and superlinks of the $| G|$ stacked lattice, we will just label them by V and L respectively, and keep calling them supervertices and superlinks to avoid confusion. Then the degrees of freedom are just pairs ${x}_{L}=({\eta }_{L},{g}_{L})$ (with ${\eta }_{L}$ an nth root of unity and ${g}_{L}\in G$, and x_L treated as an element of the group E) defined on the superlinks L. The Hamiltonian consists of two kinds of terms, vertex terms and plaquette terms.

Intuitively, we define the vertex term at V corresponding to $x\in E$ by multiplying all of the link variables on links L terminating at V by x. However, there is a complication due to the fact that some of the links are outgoing and some are incoming: in one case we want to left multiply by x and in the other we want to right multiply by ${x}^{-1}$. We handle both cases by defining the link multiplication operator:

$$\begin{eqnarray}&&{e}_{V,L}(x)| {x}_{L}\rangle =| {{xx}}_{L}\rangle \;{\rm{if}}\;{\rm{}}{s}_{V}(L)=1,\end{eqnarray} \tag{ 59 }$$

$$\begin{eqnarray}&&=\;| {x}_{L}{x}^{-1}\rangle \;{\rm{if}}\;{\rm{}}{s}_{V}(L)=-1.\end{eqnarray} \tag{ 60 }$$

With it we then define

$$\begin{eqnarray}&&{A}_{V}(x)=\displaystyle \prod _{L\sim V}({e}_{V,L}(x)).\end{eqnarray} \tag{ 61 }$$

The plaquette term B_P associated to any (super)plaquette P is defined by assigning an energetic penalty for E flux through P. More formally, in the basis of link labellings, we define:

$$\begin{eqnarray}&&{B}_{P}=1\;\mathrm{if}\ \displaystyle \prod _{L\in P}{{x}_{L}}^{{s}_{P}(L)}=e,\end{eqnarray} \tag{ 62 }$$

$$\begin{eqnarray}&&=\;0\;{\rm{otherwise.}}\end{eqnarray} \tag{ 63 }$$

Using these operators, the Hamiltonian becomes:

$$\begin{eqnarray}&&{{\bf{H}}}_{E}=-\displaystyle \sum _{V}\displaystyle \sum _{x\in E}{A}_{V}(x)-\displaystyle \sum _{P}{B}_{P}.\end{eqnarray} \tag{ 64 }$$

The equivalence between the Hamiltonian in equation (57) and that in equation (64) is non-trivial, and derived in appendix A. Here we will just make a couple of comments about how these two Hamiltonians are related. First of all, the G-flux through any plaquette, given by

$$\begin{eqnarray}&&\displaystyle \prod _{L\in P}{{g}_{L}}^{{s}_{P}(L)}\end{eqnarray} \tag{ 65 }$$

in the notation of equation (57), is given by the projection from E to G of

$$\begin{eqnarray}&&\displaystyle \prod _{L\in P}{{x}_{L}}^{{s}_{P}(L)}.\end{eqnarray} \tag{ 66 }$$

If we represent each x_L as ${x}_{L}=({\eta }_{L},{g}_{L})$, then this just reduces to the expression in equation (65).

Now let us examine configurations with trivial G gauge field, i.e. g_L = 1 for all L. Then the Hamiltonian in equation (57) just reduces to the original ${{\mathbb{Z}}}_{n}$ gauge theory on the $| G|$ fold stacked lattice. On the other hand, the E gauge theory given in equation (64) reduces to a ${{\mathbb{Z}}}_{n}$ gauge theory on the ordinary 2D lattice. The identification between these two is non-trivial, but in particular a ${{\mathbb{Z}}}_{n}$ flux in the latter corresponds to that same flux penetrating all $| G|$ layers in the stacked lattice.

Now we will examine some consequences of this equivalence, for a particular example.

In this subsection we analyze the example of a ${{\mathbb{Z}}}_{4}$ gauge theory with $G={{\mathbb{Z}}}_{2}=\{1,g\}$ acting by $\rho (g)\;:j\to -j$, where $j\in \{0,1,2,3\}$. As we discussed earlier, after gauging G the topological order given by equation (64) is just that of the gauge theory of the extension E, where E can be either ${{\mathbb{D}}}_{8}$ or ${{\mathbb{Q}}}_{8}$.

Analyzing extrinsic defects in the ungauged theory is the same as including a non-dynamical background G gauge field, which just amounts to the following modification of the Hamiltonian in equation (64):

$$\begin{eqnarray}&&{{\bf{H}}}_{E}^{\mathrm{non}-\mathrm{dynam}.}=-\displaystyle \sum _{V}\displaystyle \sum _{\eta \in {{\mathbb{Z}}}_{n}}{A}_{V}(\eta )-\displaystyle \sum _{P}{B}_{P}.\end{eqnarray} \tag{ 67 }$$

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Note that we are now working with a purely 2D square lattice. Let us understand these two terms for a fixed background gauge field configuration, namely that of two widely separated defects, illustrated in figure 6. In this case, the two terms in the Hamiltonian of equation (67) reduce to the ordinary ${{\mathbb{Z}}}_{4}$ gauge theory vertex and plaquette term everywhere except for the plaquettes intersected by the dashed blue branch cut line and the vertices directly above them. For these intersected plaquettes, we have a modified plaquette term where a_L is inverted for the upper horizontal link in the product over plaquette links (recall that a_L is the 'phase' operator in the ${{\mathbb{Z}}}_{4}$ gauge theory). Likewise, just above the dashed blue line we have modified vertex terms, where e_L (the charge operator in the ${{\mathbb{Z}}}_{4}$ gauge theory) is inverted for the lower vertical link (the one bisected by the dashed blue line) in the product over vertex links.

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Now, the topological superselection sector of a ${{\mathbb{Z}}}_{2}$ defect can be altered by fusing ${{\mathbb{Z}}}_{4}$ charges and fluxes into it. However, as shown in figure 6, we can create a charge of $2\in {{\mathbb{Z}}}_{4}=\{0,1,2,3\}$ or a flux of $2\in {{\mathbb{Z}}}_{4}$ using a local operator near a ${{\mathbb{Z}}}_{2}$ defect. So the only non-trivial superselection sectors near the ${{\mathbb{Z}}}_{2}$ defect correspond to the parity of charges or fluxes bound to the ${{\mathbb{Z}}}_{2}$ defect, and there are only 4 such superselection sectors (as opposed to the 16 anyon types). This also implies that when we fuse a pair of defects, the fusion product can be changed by an even charge or even flux with a local operator. This implies that a pair of defects can fuse into any even charge or even flux, so the fusion rules are non-abelian. This is a general feature of defects in permuting theories. Indeed, the local operator which creates an even charge in the presence of the defect can be thought of as creating a pair (fundamental charge, fundamental anti-charge), braiding the anti-charge around the defect to turn it into a charge, and re-fusing with the leftover charge to make a charge of 2, and similarly for the operator that creates an even flux. In the next section we will generalize this idea to arbitrary permutation actions.

Notice that none of the above discussion depended on whether we chose $E={{\mathbb{D}}}_{8}$ or $E={{\mathbb{Q}}}_{8}$. We will see in the next section that this is a general feature: the superselection sectors of a defect, as well as its fusion rules with anyons, depend only on the permutation ρ, and not on $\omega (g,h)$ or any other data. In order to discriminate between $E={{\mathbb{D}}}_{8}$ and $E={{\mathbb{Q}}}_{8}$ we have to perform a more subtle measurement, involving braiding anyons and defects. We will discuss this in detail in the next section; for now just note that this kind of braiding measurement really comes down to computing commutation relations of the string operator that creates defects with other string operators, and is not something that can be done in a purely fixed background G gauge field configuration.

Let us first give a condensed review of symmetry fractionalization in the case that G does not permute the anyons, i.e. fixes all topological superselection sectors. When G does not permute the anyons, we assume the existence of an approximate 'local' action of G on particular anyons—see appendix B for details on how to construct it. This local action is just an operator ${U}_{\mathrm{loc}}^{a}$ acting on the spins within a distance r of an anyon a, whose commutation relations with all local operators acting near the anyon are approximately the same as those of U_g. Here the quality of the approximation is controlled by ${e}^{-r/\xi }$, where ξ is the correlation length; henceforth we take r sufficiently large and neglect the exponentially small errors. These properties imply that the ${U}_{\mathrm{loc}}^{a}$ satisfy the group relations up to a possible phase ambiguity ${\omega }_{a}(g,h)$ (here a is the anyon acted on):

$$\begin{eqnarray}&&{U}_{g}^{a}{U}_{h}^{a}={\omega }_{a}(g,h){U}_{{gh}}^{a}.\end{eqnarray} \tag{ 68 }$$

Associativity then gives

$$\begin{eqnarray}&&{\omega }_{a}(g,h){\omega }_{a}(f,{gh})={\omega }_{a}({fg},h){\omega }_{a}(f,g),\end{eqnarray} \tag{ 69 }$$

which, together with the uniqueness of the ${U}_{g}^{a}$ up to overall phase, imply that we can extract from them a unique cohomology class $[{\omega }_{a}]\in {H}^{2}(G,U(1))$. Furthermore, for every allowed fusion channel $a\times b\to c$, we must have ${\omega }_{a}(g,h){\omega }_{b}(g,h)={\omega }_{c}(g,h)$, since such a fusion is a local operation and carries integral symmetry quantum numbers (see appendix B for a detailed argument). Now a technical result of category theory (lemma 3.31 of [22]) shows that any U(1)-valued function of the anyons which satisfies this property can be represented as

$$\begin{eqnarray}&&{\omega }_{a}(g,h)={S}_{\omega (g,h),a},\end{eqnarray} \tag{ 70 }$$

where $\omega (g,h)$ is an abelian anyon, and ${S}_{b,a}$ denotes the full braiding phase of the abelian anyon b around a general anyon a. From equation (69), we then have that $[\omega ]\in {H}^{2}(G,{{ \mathcal A }}_{\mathrm{abelian}})$, where ${{ \mathcal A }}_{\mathrm{abelian}}$ is the subset of abelian anyons viewed as an additive group. Thus we see that the symmetry fractionalization data is encoded in a cohomology class in $[\omega ]\in {H}^{2}(G,{{ \mathcal A }}_{\mathrm{abelian}})$.

4.1.1. Defect fusion rules in the non-permuting case

The case where G permutes the topological superselection sectors is significantly more complicated. For one thing, there is now no canonical 'trivial' SET, as opposed to the untwisted case, where we can construct a trivial SET simply by defining G to act trivially on the microscopic degrees of freedom for any Hamiltonian realizing the desired intrinsic topological order. Thus, we should not expect to be able to assign group cohomology classes to permuting SETs, since there is now no preferred choice of a trivial SET corresponding to the trivial cohomology class. Instead, given a particular choice of intrinsic topological order and twisted action of G, our approach will be to start with an arbitrary SET realizing this twisted action, and look for all possible gauge inequivalent ways of deforming the defect fusion rules. This approach is rooted in the general idea of classifying SETs via braided G-crossed categories [12, 20, 23], where constructing defect fusion rules—formally, a tensor product of bimodule categories—is one step in a systematic construction of the braided G-crossed category. For now, however, let us examine a single g-defect in the twisted case where $\rho (g)$ non-trivially permutes the anyons.

To appreciate the complexity of this twisted case, note that the g-defects are now generically non-abelian. For example, consider a defect of a ${{\mathbb{Z}}}_{2}$ symmetry which exchanges e and m in a model with toric code topological order (for an example of such a model see [14, 21]). By nucleating a pair of e's, braiding one around the defect to turn it into an m, and fusing with the remaining e, we perform a local process that nucleates the $e\times m=f$ particle in the vicinity of the defect. This is reminiscent of an Ising defect, and indeed there is a two-fold degeneracy for widely separated defects associated with absorbing and emitting the f fermion (also, the defects actually become Ising anyons upon gauging the ${{\mathbb{Z}}}_{2}$ symmetry [12, 15, 20]).

To get a handle on this complexity, suppose first that we are given a Hamiltonian corresponding to a single fixed g-defect, and suppose further that we know nothing about the location of the branch cut or the action of the symmetry on our microscopic degrees of freedom. What kind of data can we extract about the symmetry enriched phase, using only braiding and fusion operations of anyons in the background of this fixed defect configuration? Certainly we can perform experiments that braid anyons around the defect and measure the topological charge before and after braiding: this is just measuring the permutation action of g:

$$\begin{eqnarray}&&a\to g\cdot a.\end{eqnarray} \tag{ 71 }$$

Furthermore, we can also measure the action of g on anyon fusion spaces:

$$\begin{eqnarray}&&g\;:{V}_{c}^{a,b}\to {V}_{g\cdot c}^{g\cdot a,g\cdot b},\end{eqnarray} \tag{ 72 }$$

$$\begin{eqnarray}&&X\in {V}_{c}^{a,b}\to g\cdot X\in {V}_{g\cdot c}^{g\cdot a,g\cdot b}.\end{eqnarray} \tag{ 73 }$$

A physical process that can measure this action on anyon fusion spaces is as follows. Given an operator X which splits c into the pair $a,b$, i.e. $X\in {V}_{c}^{a,b}$, define $g\cdot X$ be the operator that acts as follows: first braid $g\cdot a$ and $g\cdot b$ around the defect to turn them into $a,b$ respectively, apply X, and then braid the fusion product c in the opposite direction to get $g\cdot c$. Note that because braiding an anyon a around the defect is only defined up to a phase ${\alpha }_{a}$, $g\cdot X$ is only well defined up to a phase ${\alpha }_{a}{\alpha }_{b}{\alpha }_{c}^{-1}$.

A permutation action on anyons together with a compatible action by unitary linear transformations on anyon fusion spaces is called a braided tensor autoequivalence of ${ \mathcal A }$ (in [20], this also goes under the name of topological symmetry, discussed in section 3.1). There is also a notion of two braided autoequivalences being the same: this occurs when the two autoequivalences have the same permutation action, and their actions on the fusion space ${V}_{c}^{a,b}$ differ by ${\alpha }_{a}{\alpha }_{b}{\alpha }_{c}^{-1}$, where the $\{{\alpha }_{a}\}$ is some fixed set of U(1) phases. Hence, by performing braiding experiments around a fixed g defect configuration we uniquely recover precisely a braided tensor autoequivalence corresponding to g, up to this notion of braided tensor autoequivalences being the same⁷ .

Now, the goal of the classification program of SETs is to construct a larger algebraic structure—the braided G-crossed category—which describes the fusion and braiding rules of both anyons and defects. How much of this larger structure can one recover from knowing the braided tensor autoequivalence corresponding to each $g\in G$? The answer is given by theorem 5.2 of [19]: the braided tensor autoequivalence uniquely determines an 'invertible bimodule category' corresponding to the g-defect. Roughly, the 'invertible bimodule category' corresponding to the g-defect is the subset of the braiding and fusion data of the theory that involves a single g-defect and an arbitrary number of anyons, modulo gauge equivalences. The intuition is that, while it may be possible to find a gauge equivalence between such a subset of braiding and fusion data for two different theories, it might be impossible to extend this gauge equivalence to a gauge equivalence of the full theories, involving braiding and fusion data of an arbitrary number of defects and anyons. So the invertible bimodule categories associated to the various defects constitute coarse data that only partially classifies different SETs. In particular, in the context of the lattice models from the previous section, this data is sensitive only to the permutation ρ, and not to the co-cycle ω.

Let us now define invertible bimodule categories in a little more detail.

4.2.1. Invertible bimodule categories

First of all, let us put all of our defects and anyons on a line. Then we can hold the defect at a fixed position, and imagine fusing anyons into and out of it. We can do this either from the left or the right; the corresponding structures, consisting of the superselection sectors of the g-defect, together with the F-move isomorphisms for these fusion processes, are called left and right module categories respectively. More precisely, a (say) left module category involves this fusion and F-move data for anyons fusing in from the left, modulo gauge equivalence corresponding to basis redefinitions within the anyon-defect fusion spaces, and satisfying pentagon equation constraints involving three anyons and one defect. The data defining a left module category is shown in figure 7.

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Now, physically we can extract the data of both a left and right module category structure for a single fixed defect. However, in a physical system these data are not independent: there are additional data and compatibility constraints between fusion from the left and right, beyond those encoded in the left and right module structures alone. This is the associativity of a process where one anyon fuses in from the left and another fuses in from the right, constrained by the appropriate pentagon equations. A left and right module category together with such additional data and constraints is called a bimodule category. Thus a single defect gives rise to a bimodule category over the anyons.

Additionally though, all of the bimodule categories that we encounter have further properties due to the existence of a braiding structure. For now this braiding structure involves leaving the g-defect fixed, and just braiding anyons around it. Let us imagine that the branch cut emanating from the defect goes towards the back of the page (away from the line where all the quasiparticles sit). Then braiding an anyon a in front of the defect, so as to avoid its branch cut, gives an isomorphism of anyon defect fusion spaces:

$$\begin{eqnarray}&&{R}_{{y}_{g}}^{a,{x}_{g}}:{V}_{{y}_{g}}^{a,{x}_{g}}\to {V}_{{y}_{g}}^{{x}_{g},a}.\end{eqnarray} \tag{ 74 }$$

Here x_g and y_g are defect superselection sectors and a is an anyon. We can identify the two fusion spaces ${V}_{{y}_{g}}^{a,{x}_{g}}$ and ${V}_{{y}_{g}}^{{x}_{g},a}$ using the isomorphism in equation (74), as illustrated in figure 8. Then the compatibility of braiding and fusion, formally expressed in terms of the hexagon equations, uniquely determines all the F-move isomorphisms involving two anyons and one defect in terms of those where the anyons only fuse in from the left, illustrated in figure 7. Mathematically one says that in this case the bimodule category structure is induced from a left module category via the braiding in ${ \mathcal A }$. Thus a single defect gives rise to a bimodule category over the anyons, where the bimodule structure is induced form a left module structure by using the anyon braiding.

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A second property of these bimodule categories comes from the fact that anyons can braid behind the defect as well. Various hexagon equation constraints⁸ turn out to determine the half-braiding of an anyon behind a defect up to a U(1) phase which depends on the anyon being braided, but not on the superselection sector of the defect [19]. These are just the phases ${\alpha }_{a}$ mentioned earlier. Mathematically, these extra constraints imply that the bimodule category is 'invertible'—see [19] for a precise definition of invertibility, which we will not state here. Physically, however, invertibility turns out to mean that for any defect, there exists an anti-defect that can annihilate with it, under the defect fusion product defined in the next section.

Thus, in summary, just from braiding experiments involving a single static g-defect one can determine the action of g by braided tensor autoequivalences, which is equivalent to all fusion and braiding rules involving a single g-defect, with the exception of braiding behind the defect, which is determined only up to a phase that depends on the anyon being braided (but not the superselection sectors of the defect). The uniqueness here is up to a gauge freedom corresponding to a change of orthonormal bases in the anyon-defect and defect-anyon fusion spaces. Mathematically, this collection of data is an invertible bimodule category with bimodule structure induced from braiding in the anyons ${ \mathcal A }$.

Example:${{\mathbb{Z}}}_{4}\;$openup-->gauge theory with${{\mathbb{Z}}}_{2}\;$openup-->symmetry

Let us give a non-trivial example of the above structure. Let $G={{\mathbb{Z}}}_{2}$, and consider ${{\mathbb{Z}}}_{4}$ gauge theory, whose anyon content is ${{\mathbb{Z}}}_{4}\times {{\mathbb{Z}}}_{4}$. This is the theory we discussed in the previous section. We will think of the first ${{\mathbb{Z}}}_{4}$ as charge and the second ${{\mathbb{Z}}}_{4}$ as flux, and denote anyons by (i, j), $0\leqslant i,j\lt 4$. We will take the non-trivial generator $g\in {{\mathbb{Z}}}_{2}$ to act on the anyons by $(i,j)\to (4-i,4-j)$. What is the invertible bimodule category structure associated to a g-defect?

First of all, by the argument above, we can nucleate anyons of the form $(2i,2j)$ in the presence of the g-defect. Thus the g-defect has only four superselection sectors, namely ${(\alpha ,\beta )}_{g}$, $0\leqslant \alpha ,\beta \leqslant 1$. The fusion rules with the anyons are simply given by addition mod 2

$$\begin{eqnarray}&&(i,j)\times {(\alpha ,\beta )}_{g}={(\alpha +i{\rm{mod}}2,\beta +j{\rm{mod}}2)}_{g}.\end{eqnarray} \tag{ 75 }$$

These are the same fusion rules that we found directly in the previous section.

The F symbols involving one defect and two anyons are non-trivial. There is a choice of basis in which they can all be set equal to ±1, and they can be determined explicitly using a general equivalence between such invertible bimodule categories and Lagrangian subgroups of $H\otimes {H}^{*}$ for any abelian H gauge theory—see the discussion in section 10.3 of [19]. More explicitly, in a particular gauge they can be taken as follows. Given even anyons ${a}_{1}=(2{i}_{1},2{j}_{1}),{a}_{2}=(2{i}_{2},2{j}_{2})$, let

$$\begin{eqnarray}&&{F}_{y}^{{a}_{1},{a}_{2},x}={(-1)}^{{i}_{1}{j}_{2}-{i}_{2}{j}_{1}}.\end{eqnarray} \tag{ 76 }$$

Anyons not of this form can always be uniquely written as ${a}_{1}=(2{i}_{1}+{\mu }_{1},2{j}_{1}+{\nu }_{1}),{a}_{2}=(2{i}_{2}+{\mu }_{1},2{j}_{2}+{\nu }_{1})$, where ${\mu }_{\mathrm{1,2}},{\nu }_{\mathrm{1,2}}=0,1$. For these anyons we define the F-symbol the same way, ${F}_{y}^{{a}_{1},{a}_{2},x}={(-1)}^{{i}_{1}{j}_{2}-{i}_{2}{j}_{1}}$. F-matrices for fusion of anyons from the right are then uniquely determined using the rule in figure 8 and ordinary anyon fusion rules. The R-move corresponding to braiding an anyon a behind a defect is also determined uniquely up to a phase that depends on a but not on the superselection sector of the defect, as is the action on the anyon fusion spaces, which turns out to be trivial.

Let us emphasize that the F-matrices we have written down here, involving one defect and two anyons, will be valid for both of the SETs with this permutation action on the anyons. The data which distinguishes between the two SETs, namely the cohomology classes ${\omega }_{\pm }(g,h)$ discussed in the previous section, will show up only in quantities involving two defects. Below we will discuss this quantity ω in our more general context.

Above we saw that the gauge invariant data contained in a single g-defect is entirely determined by the action of g by braided autoequivalences on the anyons. In particular, we cannot distinguish among SETs with the same braided autoequivalence action of G by considering only single defects. In this section, we will see that such SETs can be distinguished by considering pairs of defects and their splitting and fusion rules.

First let us fix a pair $g,h\in G$, and consider a gh defect splitting into a g defect and an h defect. One can again look at associativity relations for processes which start with the gh defect and end in a g defect, an h defect, and an additional anyon a. There are three types of processes, depending on whether a appears to the left, to the right, or in between the defects in our chosen line ordering; they are illustrated in figure 9.

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The coherence conditions on the F-matrices illustrated in figure 9 are highly constraining. Indeed, these F-matrices must satisfy compatibility conditions with each other, with the invertible module category structure, and with the braiding and fusion structure of the original anyons. These all take the form of pentagon equations and variants of the hexagon equations involving 0, 1, or two defects. Despite the complicated nature of these equations [19], again proves a theorem that constrains the possible solutions. Specifically, proposition 7.3(ii) of [19] shows that, for a fixed pair ($g,h$), and having fixed all of the one defect data (namely, the map from G into the braided autoequivalences of ${ \mathcal A }$), the set of gauge equivalence classes of solutions for the fusion rules of g and h defects into a gh defect together with the associated F-matrices is in one to one correspondence with the set of abelian anyons ${ \mathcal A }$. Indeed, given one solution [19], shows how to 'deform' it by any abelian anyon ${a}_{g,h}$. Intuitively, one can think of this deformation as splitting off ${a}_{g,h}$ from the gh defect before it splits into g and h—see figure 10. We stress that this intuitive picture is just graphical shorthand for a precise formula, written down in appendix C, that expresses the new, deformed fusion rules and F-matrices in terms of the original ones.

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Now, suppose we have a complete solution for all of the data in our theory, including F-matrices with any number of anyons or defects. We can try to find a new solution by leaving alone all data involving at most one defect (in particular the fusion and braiding data of the anyons and the invertible bimodule category structure), deforming, for each pair $(g,h)$, the fusion and F-matrix data involving two defects g and h by some anyon ${a}_{g,h}$, as in figure 10 and attempting to solve for the rest of the structure (including associativity conditions with three defects, and the defect braiding matrices). That is, we perform the deformation discussed above for every pair of defects g and h. Now, there is no guarantee that if we do this then we will be able to solve all of the remaining coherence conditions and obtain a consistent theory. A basic requirement on ${a}_{g,h}$ is

$$\begin{eqnarray}&&{\rho }_{f}({a}_{g,h}){a}_{f,{gh}}={a}_{{fg},h}{a}_{f,g},\end{eqnarray} \tag{ 77 }$$

where ${\rho }_{f}$ represents the permutation action of f on the anyon labels. In the non-permuting case this condition is simply the requirement that we obtain the same topological superselection sector regardless of the order in which we fuse three defects $f,g,h$. In the twisted case, equation (77) is not as obvious, but rather follows from imposing all pentagon equations involving three defects and one anyon: a solution for the three defect F-matrix can be found only if equation (77) is satisfied. In the language of [19], these pentagon equations translate to the statement that the tensor product of bimodule categories referred to previously is associative, and the condition follows from their theorem 8.5. We note that even when this is so, this associativity isomorphism might not satisfy the four-defect pentagon equation and fail to give a consistent theory [24]—however, we will not be concerned with this issue here.

Because deformations of the form ${a}_{g,h}={\rho }_{g}({b}_{h}){b}_{{gh}}^{-1}{b}_{g}$, where the b_g are abelian, can be undone by a re-labeling of defect superselection sectors, equation (77) shows that the potentially non-trivial deformations correspond to non-zero twisted cohomology classes in $[\omega ]\in {H}_{{\rm{twisted}}}^{2}(G,{ \mathcal A })$, represented by the functions $\omega (g,h)={a}_{g,h}$. In section 4 of [20], there is defined a collection of U(1)-valued two-cocycles ${\tau }_{a}(g,h)$, where a ranges over all of the anyons in the theory (not just abelian ones). This ${\tau }_{a}(g,h)$ is related to our $\omega (g,h)$ by:

$$\begin{eqnarray}&&{\tau }_{a}(g,h)={S}_{a,\omega (g,h)},\end{eqnarray} \tag{ 78 }$$

where ${S}_{a,\omega (g,h)}$ is the full U(1) braiding phase of the two anyons.

Of course, given such a non-zero cohomology class, it is not a priori clear how many consistent theories it leads to, if any. Nevertheless, assuming there does exist at least one consistent deformed solution for a given $[\omega ]$, one can ask what gauge invariant quantity distinguishes this deformed solution from the un-deformed one. In principle, one can simply compute all of the relevant (two defect, one anyon) F-matrices for both solutions and check if there is a gauge equivalence between them. However, in practice this is difficult, both because of the large number of F-matrices involved, and also because extracting universal data associated to F-matrices is significantly more subtle than extracting braiding data for a given physical system.

Fortunately, it turns out that once we specify the deformation $[\omega ]$, various coherence conditions—namely pentagon and analogues of the hexagon equations—force much of the remaining fusion and braiding structure. Indeed, according to the general classification of [20, 23], the SET is then actually determined almost uniquely: the only ambiguity is the stacking of an extra G-SPT (symmetry protected topological phase) on top of the system. In particular, the braiding of anyons around defects is completely determined. As we will see in the example below, the gauge invariants that can be constructed from this braiding data contain a much more convenient signature of the deformation.

Let us determine the defect fusion rules in the example discussed above. The only non-trivial case is a g-defect fusing with itself, and in this case, since we know that braiding around two such g-defect induces the trivial permutation on anyon labels and the trivial map on anyon fusion spaces, from theorem 5.2 of [19] it follows that the product of two g-defects is a trivial defect. Recall that we are representing the anyons by pairs $(i,j)$, where i denotes charge ${{\mathbb{Z}}}_{4}$ charge and j denotes ${{\mathbb{Z}}}_{4}$ flux, that the superselection sectors of the non-trivial g-defect are labeled by pairs ${(\alpha ,\beta )}_{g}$, with $\alpha ,\beta$ defined modulo 2, and that the fusion rules are:

$$\begin{eqnarray}&&{(\alpha ,\beta )}_{g}\times {(\alpha ^{\prime} ,\beta ^{\prime} )}_{g}={\oplus }_{i,j}(i,j),\end{eqnarray} \tag{ 79 }$$

where the sum on the right is over $0\leqslant i,j\lt 4$ with $\alpha +\alpha ^{\prime} =i\;{\rm{mod}}\;2$ and $\beta +\beta ^{\prime} =j\;{\rm{mod}}\;2$. Notice that these fusion rules are non-abelian.

Now, according to the above discussion, the potential deformations of this defect product are parametrized by ${H}_{{\rm{twisted}}}^{2}({{\mathbb{Z}}}_{2},{{\mathbb{Z}}}_{4}\times {{\mathbb{Z}}}_{4})={{\mathbb{Z}}}_{2}\times {{\mathbb{Z}}}_{2}$. Let us now focus on the deformations where the anyons ${a}_{g,h}$ are fluxes of the ${{\mathbb{Z}}}_{4}$ gauge theory, which restricts us to a ${{\mathbb{Z}}}_{2}$ subgroup of ${{\mathbb{Z}}}_{2}\times {{\mathbb{Z}}}_{2};$ the non-trivial co-cycle is then ${a}_{g,g}=(0,2)$. We do this because this ${{\mathbb{Z}}}_{2}$ is the one that can be accessed in the particular class of exactly-solvable models given in the previous section. Indeed, these models, like those of [1] on which they are based, were constructed to allow for fractional symmetry charge assignments to gauge theory charges, since it is the gauge charges that are sensitive to the Berry flux in the vertical plaquette terms The other ${{\mathbb{Z}}}_{2}$ would correspond to a phase where instead the gauge fluxes carry fractional symmetry charge, and would require a different lattice construction⁹ .

Note that deforming by this co-cycle does not change the fusion rules in equation (79). Thus it is now much more difficult to distinguish the deformed and undeformed products of bimodule categories. Indeed, as opposed to the untwisted case, where the distinction can be ascertained by just examining the fusion rules for defect superselection sectors, we now have to examine subtle gauge invariant F-matrix data for processes involving two defects and one anyon. Fortunately, as discussed above, this distinction appears also in the braiding data of anyons with the defect in the full solutions ('braided G-crossed categories') corresponding to the deformed and undeformed data. As shown in appendix C, imposing hexagon coherence conditions shows that the deformed and undeformed cases differ in the sign of the gauge invariant braiding quantity

$$\begin{eqnarray}&&{R}_{{x}_{g},(3,0)}{R}_{(\mathrm{3,0}),{x}_{g}}{R}_{{x}_{g},(1,0)}{R}_{(\mathrm{1,0}),{x}_{g}},\end{eqnarray} \tag{ 80 }$$

where x_g is an arbitrary superselection sector of the g-defect, and ${R}_{a,b}$ is the R-matrix corresponding to the exchange of a and b. Recall that $(i,j)$ represents an anyon which is a composite of a charge i and flux $j,i,j=0,\ldots ,3$. Thus equation (80) simply gives the braiding phase for the process of braiding a fundamental ${{\mathbb{Z}}}_{4}$ charge (1,0) twice around the g-defect. Notice that braiding it only once turns it into the opposite charge (3,0) and hence does not lead to a gauge invariant phase.

Now recall that in the previous section we had an exactly-solvable lattice construction for precisely this topological order and action of symmetry, and found two distinct SETs. We claim that these are lattice realizations of the two cases—deformed and un-deformed—that we are discussing right now. In order to argue this, we will just show that the two distinct lattice SETs we constructed in the previous section have a different sign of the gauge invariant quantity defined in equation (80). Recall that upon gauging G, the two SETs we constructed in the previous section resulted in ${{\mathbb{D}}}_{4}$ and ${{\mathbb{Q}}}_{8}$ topological orders, respectively.

First, let us establish some more notation. The second group cohomology ${H}_{\rho }^{2}({{\mathbb{Z}}}_{2},{{\mathbb{Z}}}_{4})={{\mathbb{Z}}}_{2}$, where ${{\mathbb{Z}}}_{2}=\{1,g\}$ acts as $\rho \;:x\to {x}^{-1}$ on ${{\mathbb{Z}}}_{4}=\{1,i,-1,-i\}$. We found two representative co-cycles, ${\omega }_{+}$ and ${\omega }_{-}.{\omega }_{+}$ was equal to 1 for all values of its arguments, and ${\omega }_{-}(g,g)=-1$. According to appendix D, ${\omega }_{+}$ corresponds to the dihedral group ${{\mathbb{D}}}_{4}$, and ${\omega }_{-}$ to the quaternion group ${{\mathbb{Q}}}_{8}$. The first of these just consists of the symmetries of the square, with a ${{\mathbb{Z}}}_{4}$ rotation subgroup which we refer to as ${{\mathbb{Z}}}_{4}=\{1,i,-1,-i\}$, while the second consists of the elements $\{\pm 1,\pm i,\pm j,\pm k\}$ which follow the multiplication rules of the quaternion group. Here we again take the ${{\mathbb{Z}}}_{4}$ subgroup to be $\pm 1,\pm i$. The key difference between these two extensions, captured by the respective co-cycles, is that in ${{\mathbb{D}}}_{4}$, any lift of g squares to +1, whereas in ${{\mathbb{Q}}}_{8}$, any such lift squares to −1.

We now perform the double braid computation in equation (80) in the fully gauged theories. Here, the charges (1, 0) and (3, 0) (obtained from it by acting with g) pair up into the unique two-dimensional irreducible representation of either ${{\mathbb{D}}}_{4}$ or ${{\mathbb{Q}}}_{8}$ depending on whether we are in the ${\omega }_{+}$ or ${\omega }_{-}$ case, respectively. Now, the gauge theories (or 'quantum doubles') of these two groups can be constructed using a general prescription outlined e.g. in section 3.2 of [25]. The key formula for computing braiding is the general formula for the universal R matrix given there. Here we are interested in moving a pure charge—our two-dimensional irreducible representation of ${{\mathbb{D}}}_{4}$ or ${{\mathbb{Q}}}_{8}$—around a g-flux twice. Using the formula for the R matrix given in section 3.2 of [25], the action of such a double braid operation is simply given by g² acting within the two-dimensional irreducible representation of E that contains the fundamental ${{\mathbb{Z}}}_{4}$ charge. This is just the identity in ${{\mathbb{D}}}_{4}$, and −1 in ${{\mathbb{Q}}}_{8}$, which acts as −1 in the two-dimensional irreducible representation, so that the value of the double braid in equation (80) differs by a sign for the two theories, as desired.

A.1.1. Supervertex constraints

We first impose as constraints the plaquette terms ${\tilde{C}}_{p}$ for plaquettes p contained entirely within a single supervertex V: that is, we restrict to the states $| {\rm{\Psi }}\rangle$ for which ${\tilde{C}}_{p}| {\rm{\Psi }}\rangle =| {\rm{\Psi }}\rangle$. For each plaquette triple $(f,g,h)$, this constraint reads

$$\begin{eqnarray}&&{\eta }_{V}(f,g){\eta }_{V}({fg},h){\eta }_{V}{(f,{gh})}^{-1}=\omega {(g,h)}^{\rho (f)}.\end{eqnarray} \tag{ A1 }$$

Using equation (32), we see that one solution to this equation is $\bar{\eta }(g,h)=\omega (g,h)$. Now, any other solution to equation (A1) must be related to $\bar{\eta }(g,h)$ by a gauge transformation¹⁰ :

$$\begin{eqnarray}&&{\eta }_{V}(g,h)={\theta }_{V}^{-1}({gh})\bar{\eta }(g,h){\theta }_{V}(g)\end{eqnarray} \tag{ A2 }$$

We now define an unconstrained Hilbert space by trading in the ${{\mathbb{Z}}}_{n}$ link variables ${\eta }_{V}(g,h)$, i.e. those with links entirely within supervertex V, for ${{\mathbb{Z}}}_{n}$ valued 'clock' variables ${\theta }_{V}(g)$. We introduce the usual operator algebra associated to each ${{\mathbb{Z}}}_{n}$ clock: ${{\rm{\Theta }}}_{V}(g)$ is the operator that has ${\theta }_{V}(g)$ as an eigenvalue, and ${{\rm{\Phi }}}_{V}(g)$ acts as $| {\theta }_{V}(g^{\prime} )\rangle \to | {e}^{\frac{2\pi i}{n}{\delta }_{g,g^{\prime} }}{\theta }_{V}(g^{\prime} )\rangle$. One advantage of representing ${\eta }_{V}(g,h)$ in terms of the ${\theta }_{V}(g)$ as in equation (A2) is that it simplifies the vertex term ${\tilde{A}}_{V}(g)$. Indeed, the contributions to ${\tilde{A}}_{V}(g)$ coming from links within the supervertex V amount to just ${{\rm{\Phi }}}_{V}^{-1}(g)$, so the vertex term becomes:

$$\begin{eqnarray}&&{\tilde{A}}_{V}(g)\to {{\rm{\Phi }}}_{V}^{-1}(g)\prod _{L\sim V}{e}_{L}{(g)}^{s{^{\prime} }_{V}(L)}.\end{eqnarray} \tag{ A3 }$$

Note that $\{{\theta }_{V}(g)\}$ form a redundant parametrization of the original link degrees of freedom, in that shifting all ${\theta }_{V}(g)$ simultaneously by the same g-independent amount does not change any ${\eta }_{V}(g,h)$ defined in equation (A2). Thus the Hilbert space of the gauged Hamiltonian in equation (57) with the supervertex constraints ${\tilde{C}}_{p}| {\rm{\Psi }}\rangle =| {\rm{\Psi }}\rangle ,p\in V$ imposed, is recovered by projecting onto the subspace where

$$\begin{eqnarray}&&\prod _{g\in G}{{\rm{\Phi }}}_{V}^{m}(g)| {\rm{\Psi }}\rangle =| {\rm{\Psi }}\rangle ,\forall \;m\in {{\mathbb{Z}}}_{n}.\end{eqnarray} \tag{ A4 }$$

This is the reparametrization redundancy condition we mentioned earlier.

What about the Hamiltonian? We can certainly write down a Hamiltonian which reduces to that in equation (57) when acting on the subspace satisfying the condition in equation (A4), and which also includes projectors onto this subspace, such that these projectors commute with the rest of the Hamiltonian. However, before we do this we will impose the second set of constraints in the next subsection. For now, however, we just note that such a Hamiltonian will necessarily have the same topological order as the original one in equation (50). To see this, we argue as follows. First of all, one may worry that there are non-trivial topological excitations in the Hamiltonian of equation (50) which are removed from the spectrum once the constraint ${\tilde{C}}_{p}=1$ for $p\in V$ is imposed. However, it is fairly easy to see that this is not the case. Indeed, any violation of ${\tilde{C}}_{p}$ associated to V can be removed with a local operator supported only on V. This operator may create other excitations which are violations of ${\tilde{C}}_{p^{\prime} }$ for $p^{\prime}$ containing some horizontal links, but crucially, it will not create violations of any ${\tilde{C}}_{p^{\prime \prime} }$ for $p^{\prime \prime}$ contained entirely in some supervertex. Now, the violations of ${\tilde{C}}_{p^{\prime} }$ for $p^{\prime}$ containing horizontal links can again be removed by acting on the appropriate horizontal link variables. These will not create excitations at any other ${\tilde{C}}_{p}$. Thus, any local excitation can be pushed off to violating only the ${B}_{p}(g)$ and remaining terms in the Hamiltonian using a local process, so that its topological superselection sector remains in the spectrum even after the constraint is imposed. A similar argument shows that imposing the reparametrization redundancy condition in equation (A4) does not introduce new topological superselection sectors.

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A.1.2. Superlink constraints

Let us now also impose the constraints ${\tilde{C}}_{p}| {\rm{\Psi }}\rangle =| {\rm{\Psi }}\rangle$ for p involving a superlink $L=\langle {VV}^{\prime} \rangle$. The definition of ${\tilde{C}}_{p}$ in equation (53) shows that ${\tilde{C}}_{p}$ just measures the ${{\mathbb{Z}}}_{n}$ flux in a certain rectangular plaquette involving the superlink L. A slight rearrangement of equation (53) shows that setting this ${{\mathbb{Z}}}_{n}$ flux to 1 is equivalent to:

$$\begin{eqnarray}&&{\eta }_{L}({gh}){\eta }_{V}{({g}_{L}^{-1}g,h)}^{\rho ({g}_{L})}={\eta }_{V^{\prime} }(g,h){\eta }_{L}(g).\end{eqnarray} \tag{ A5 }$$

Now inserting the representation of ${\eta }_{V}(g,h)$ in terms of the ${\theta }_{V}(g)$ variables introduced in the previous section (see equation (A2)) we obtain:

$$\begin{eqnarray}&&\eta {^{\prime} }_{L}({gh},{g}_{L})\bar{\eta }{({g}_{L}^{-1}g,h)}^{\rho ({g}_{L})}=\bar{\eta }(g,h)\eta {^{\prime} }_{L}(g,{g}_{L}),\end{eqnarray} \tag{ A6 }$$

$$\begin{eqnarray}&&\mathrm{where}\ \eta {^{\prime} }_{L}(g,{g}_{L})\equiv {\theta }_{V^{\prime} }(g){\eta }_{L}(g){\theta }_{V}{({g}_{L}^{-1}g)}^{-\rho ({g}_{L})}.\end{eqnarray} \tag{ A7 }$$

This equation shows that once we specify $\eta {^{\prime} }_{L}(g,{g}_{L})$ for a single g, it is then determined uniquely for all g. A useful one to fix is $\eta {^{\prime} }_{L}({g}_{L},{g}_{L})$, simplifying equation (A6) to

$$\begin{eqnarray}&&\eta {^{\prime} }_{L}({g}_{L}h,{g}_{L})=\omega ({g}_{L},h)\eta {^{\prime} }_{L}({g}_{L},{g}_{L}).\end{eqnarray} \tag{ A8 }$$

From this point forward, we will refer to $\eta {^{\prime} }_{L}({g}_{L},{g}_{L})$ as $\eta {^{\prime} }_{L}$, i.e. drop the two arguments. Imposing the superlink plaquette constraint thus leads to a Hilbert space which can be parametrized by introducing a single ${{\mathbb{Z}}}_{n}$ clock variable $\eta {^{\prime} }_{L}$ for each superlink L, in place of the link variables ${\eta }_{L}(g)$. Note that there is no additional gauge redundancy introduced by this parametrization. We let $a{^{\prime} }_{L}$ and $e{^{\prime} }_{L}$ be the evaluation and shift operators corresponding to the $\eta {^{\prime} }_{L}$ variable.

We have thus constructed a Hilbert space parametrized by ${\{{\theta }_{V}(g)\}}_{V},{\{\eta {^{\prime} }_{L},{g}_{L}\}}_{L}$. Our goal here is to write out the Hamiltonian in this Hilbert space. We begin with the plaquette term ${\tilde{B}}_{P}(g)$ in equation (52), assuming, without loss of generality, that all links are oriented positively relative to the plaquette p, which we take to be j-sided. Order the j links $1,\ldots ,j$ starting from an arbitrary point, and for convenience let ${g}_{i}={g}_{{L}_{i}}$ denote the G gauge field variable on the ith link, $1\leqslant i\leqslant j$. In appendix E we show that the following term reproduces ${\tilde{B}}_{P}(g)$ in the new Hilbert space:

$$\begin{eqnarray}&&{{\rm{Tr}}}_{G,{{\mathbb{Z}}}_{n}}{\tilde{B}}_{p}(g)\to {{\rm{Tr}}}_{{{\mathbb{Z}}}_{n}}{{\rm{\Omega }}}_{p}B{^{\prime} }_{p}{\mathrm{Tr}}_{G}{{ \mathcal B }}_{p},\end{eqnarray} \tag{ A9 }$$

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{p}=\prod _{i=1}^{j}\omega ({\prod }_{r=1}^{i-1}{g}_{r},{g}_{i}),\end{eqnarray} \tag{ A10 }$$

$$\begin{eqnarray}&&B{^{\prime} }_{p}=\prod _{i=1}^{j}{(a{^{\prime} }_{l})}^{\rho ({\prod }_{r=1}^{i-1}{g}_{r})},\end{eqnarray} \tag{ A11 }$$

$$\begin{eqnarray}&&{{ \mathcal B }}_{p}=\prod _{i=1}^{j}{\alpha }_{i},\end{eqnarray} \tag{ A12 }$$

where ${{\rm{Tr}}}_{G}$ refers to tracing over the regular representation of G.

Now, for each superlink L, the pair $({\eta }_{L}^{\prime },{g}_{L})$ can naturally be thought of as an element of the extension E of G by ${{\mathbb{Z}}}_{n}$ determined by $\omega (g,h)$, since the extension is just a twisted product of ${{\mathbb{Z}}}_{n}$ and G (see appendix D for some background on group extensions). The twisting in this product is manifested only in the non-trivial nature of the group multiplication law in E:

$$\begin{eqnarray}&&(\eta {^{\prime} }_{1},{g}_{1})\cdot ({\eta }_{2}^{\prime },{g}_{2})=(\eta {^{\prime} }_{1}{\eta {^{\prime} }_{2}}^{\rho ({g}_{1})}\omega ({g}_{1},{g}_{2}),{g}_{1}{g}_{2}).\end{eqnarray} \tag{ A13 }$$

This way of thinking about the degrees of freedom on superlinks is especially convenient because the plaquette term in equation (A9) precisely just enforces that the pairs $({\eta }_{L}^{\prime },{g}_{L})$ around a plaquette multiply to the identity in the extension E. Indeed, repeated applications of the above group multiplication law in E give

$$\begin{eqnarray}&&\prod _{i}(\eta {^{\prime} }_{i},{g}_{i})=\left(\prod _{i}{({\eta }_{i}^{\prime })}^{{\prod }_{j=1}^{i-1}\rho ({g}_{j})}\prod _{i}\omega \left(\prod _{r=1}^{i-1}{g}_{r},{g}_{i}\right),\prod _{i}{g}_{i}\right)\end{eqnarray} \tag{ A14 }$$

which is precisely the quantity that the term in equation (A9) energetically prefers to be trivial. Thus we just have the plaquette term of a non-abelian E gauge theory.

As for the vertex term, a quick calculation shows the action of ${\tilde{A}}_{V}(g)$ within the constrained Hilbert space is the same as that of the operator ${\rm{\Phi }}{^{\prime} }_{V}(g)$ defined by:

$$\begin{eqnarray}&&{\rm{\Phi }}{^{\prime} }_{V}(g)(\underset{g^{\prime} \in G}{\otimes }| {\theta }_{V}(g^{\prime} )\rangle \otimes \underset{L\sim V}{\otimes }| \eta {^{\prime} }_{L},{g}_{L}\rangle )\\ &&=\underset{g^{\prime} \in G}{\otimes }| {e}^{\frac{2\pi i}{n}{\delta }_{g,g^{\prime} }}{\theta }_{V}(g^{\prime} )\rangle \otimes \underset{L\sim V}{\otimes }| \eta {^{\prime} }_{L},{g}_{L}\rangle .\end{eqnarray} \tag{ A15 }$$

This is a great simplification, because ${\rm{\Phi }}{^{\prime} }_{V}(g)$ does not act at all on the $\eta {^{\prime} }_{L}$ degrees of freedom. On the other hand, the constraint ((A4)), which now also has to be imposed energetically, is less trivial. We have:

$$\begin{eqnarray}&&\prod _{g\in G}{{\rm{\Phi }}}_{V}(g)(\underset{g^{\prime} \in G}{\otimes }| {\theta }_{V}(g^{\prime} )\rangle \otimes \underset{L\sim V}{\otimes }| \eta {^{\prime} }_{L},{g}_{L}\rangle )\\ &&\quad =\;\left\{\begin{array}{c}\otimes | {\theta }_{L}(g){{\rm{e}}}^{\frac{2\pi {\rm{i}}}{n}}\rangle \otimes \otimes | {{\rm{e}}}^{\frac{2\pi {\rm{i}}}{n}}\eta {^{\prime} }_{L},{g}_{L}\rangle ,L\to V\\ \otimes | {\theta }_{L}(g){{\rm{e}}}^{\frac{2\pi {\rm{i}}}{n}}\rangle \otimes \otimes | \eta {^{\prime} }_{L}{{\rm{e}}}^{-2\pi {\rm{i}}\frac{\rho ({g}_{L})}{n}},{g}_{L}\rangle ,L\leftarrow V.\end{array}\right.\end{eqnarray} \tag{ A16 }$$

This is equal to the action of the operator

$$\begin{eqnarray}&&A{^{\prime} }_{V}\prod _{g\in G}{\rm{\Phi }}{^{\prime} }_{V}(g)\end{eqnarray} \tag{ A17 }$$

with

$$\begin{eqnarray}&&A{^{\prime} }_{V}\equiv \prod _{L\sim V}{(e{^{\prime} }_{L})}^{s{^{\prime} }_{V}(L)},\end{eqnarray} \tag{ A18 }$$

where

$$\begin{eqnarray}&&e{^{\prime} }_{L}| \eta {^{\prime} }_{L},{g}_{L}\rangle =| {{\rm{e}}}^{\frac{2\pi {\rm{i}}}{n}}\eta {^{\prime} }_{L},{g}_{L}\rangle \end{eqnarray} \tag{ A19 }$$

$$\begin{eqnarray}&&{(e{^{\prime} }_{L})}^{R}| \eta {^{\prime} }_{L},{g}_{L}\rangle =| \eta {^{\prime} }_{L}{{\rm{e}}}^{-2\pi {\rm{i}}\frac{\rho ({g}_{L})}{n}},{g}_{L}\rangle .\end{eqnarray} \tag{ A20 }$$

Now, since ${\prod }_{g}{\rm{\Phi }}{^{\prime} }_{V}(g)$ already appears in the Hamiltonian, and since all the terms in the Hamiltonian will commute, we can replace it with 1 in the expression $A{^{\prime} }_{V}{\prod }_{g\in G}{\rm{\Phi }}{^{\prime} }_{V}(g)$, so ultimately the constraint (A4) is imposed by including the terms $-{\sum }_{m}A{^{\prime} }_{V}^{m}$ in the Hamiltonian. Now, according to equation (A18), $A{^{\prime} }_{V}^{m}$ is just the gauge transformation by $m\in {{\mathbb{Z}}}_{n}\subset E$ at supervertex V, when we interpret the degrees of freedom as those of an E gauge theory. Thus so far we have recovered the plaquette term of the E gauge theory, and a term that imposes gauge invariance with respect to the ${{\mathbb{Z}}}_{n}$ subgroup of E.

Finally, we must write down terms corresponding to ${{ \mathcal A }}_{V}(g)$, which make the G gauge field fluctuate. There is some ambiguity in the choice of these terms because the action on the portion of the Hilbert space violating the constraint (A4) is not uniquely determined. However, one valid choice, which commutes with this constraint and the rest of the terms in the Hamiltonian, is to simply define ${{ \mathcal A }}_{V}(g)$ to act as the gauge transformation by $(1,g)\in E;$ the arbitrariness is due to the fact that we could have equally chosen $({{\rm{e}}}^{2\pi {\rm{i}}m/n},g)\in E$.

So, our Hamiltonian, when written in terms of operators that act on $\{{\theta }_{V}(g),\eta {^{\prime} }_{L},{g}_{L}\}$, becomes

$$\begin{eqnarray}{\bf{H}}{^{\prime} }_{E} & = & -\displaystyle \sum _{V}\displaystyle \sum _{m\in {{\mathbb{Z}}}_{n}}{\rm{\Phi }}{^{\prime} }_{V}^{m}(g)-\displaystyle \sum _{P}{{\rm{Tr}}}_{H}{{\rm{\Omega }}}_{P}B{^{\prime} }_{P}\ {\mathrm{Tr}}_{G}{{ \mathcal B }}_{P}\\ & & -\displaystyle \sum _{V}\displaystyle \sum _{m}A{^{\prime} }_{V}^{m}-\displaystyle \sum _{V}\displaystyle \sum _{g\in G}{{ \mathcal A }}_{V}(g).\end{eqnarray} \tag{ A21 }$$

This is almost the Hamiltonian of an E gauge theory. Indeed, the first term simply enforces that the ground state is symmetrized over the $\theta$'s, and the $\theta$ degrees of freedom do not appear anywhere else in the Hamiltonian. Thus we can remove them from the Hamiltonian without changing the low energy physics or the topological order. As far as the rest of the Hamiltonian, we can make it exactly equal to that of an E gauge theory by adding in terms of the form $A{^{\prime} }_{V}^{m}{{ \mathcal A }}_{V}(g);$ these do not introduce any new conditions:

$$\begin{eqnarray}{{\bf{H}}}_{E} & = & -\displaystyle \sum _{V}\displaystyle \sum _{\displaystyle \genfrac{}{}{0em}{}{g\in G}{m\in {{\mathbb{Z}}}_{n}}}A{^{\prime} }_{V}^{m}{{ \mathcal A }}_{V}(g)\\ & & -\displaystyle \sum _{P}{{\rm{Tr}}}_{{{\mathbb{Z}}}_{n}}{{\rm{\Omega }}}_{P}B{^{\prime} }_{P}\ {\mathrm{Tr}}_{G}{{ \mathcal B }}_{P}.\end{eqnarray} \tag{ A22 }$$

This is simply the Hamiltonian of the discrete lattice gauge theory based on the group E, the extension of G by ${{\mathbb{Z}}}_{n}$ defined by our group cohomology class $\omega (g,h)$. In particular, $A{^{\prime} }_{V}$ and ${{ \mathcal A }}_{V}(g)$ satisfy

$$\begin{eqnarray}&&{{ \mathcal A }}_{V}({g}_{1}){{ \mathcal A }}_{V}({g}_{2})=A{^{\prime} }_{V}^{q({g}_{1},{g}_{2})}{{ \mathcal A }}_{V}({g}_{1}{g}_{2}),\end{eqnarray} \tag{ A23 }$$

where ${{\rm{e}}}^{\frac{2\pi {\rm{i}}}{n}q({g}_{1},{g}_{2})}=\omega ({g}_{1},{g}_{2})$

$$\begin{eqnarray}&&{{ \mathcal A }}_{V}(g)A{^{\prime} }_{V}^{m}=A{^{\prime} }_{V}^{\rho (g)m}{{ \mathcal A }}_{V}(g).\end{eqnarray} \tag{ A24 }$$

The Hamiltonian ${{\bf{H}}}_{E}$ is equivalent to that in equation (64). While the arguments we have given were written down only for ${{\mathbb{Z}}}_{n}$ gauge theory, it is quite easy—though notationally cumbersome—to generalize to the gauge theory of any finite abelian group.

Throughout this subsection, we assume that G fixes the topological superselection sectors. Suppose we have a state with anyon a at position x, and its anti-particle somewhere far away at position y. In order to extract the fractional quantum numbers, we would like to define a local action of $g\in G$ in a neighborhood of x—not including y—so that we are sensitive to the Berry phases associated with acting on a, without also picking up the opposite phases from $\bar{a}$. More precisely, for each $g\in G$ we would like to have an operator ${U}_{g}^{x}$ that acts only on spins near x, that commutes with the Hamiltonian, and that has the same commutation relations with operators localized near x as the global action U_g. The naive proposal would be to act only on some subregion R_x of spins near x, but this is clearly unsatisfactory: such an action does not commute with the Hamiltonian at the boundary of this subregion, and so creates spurious excitations at this boundary. Also, it clearly has no projective character, since the underlying spins are honest linear representations of G. Instead, we define ${U}_{g}^{x}$ as follows.

Our operator ${U}_{g}^{x}$ will be designed to act non-trivially only when the state being acted on looks like the ground state at the boundary of R_x—i.e. its reduced density matrix is the same as the ground state's; if not, we define ${U}_{g}^{x}$ to act as 0. When the state does look like the ground state at the boundary of R_x, we act by braiding a twist defect of g around this boundary and acting with U^g inside R_x. To explain this, let us first explain what we mean by a twist defect.

A twist defect is a modification of the Hamiltonian along a branch cut terminating at the defect. The local terms in the Hamiltonian which straddle the branch cut then get twisted by g, as illustrated in figure B2. For example, if $G={{\mathbb{Z}}}_{2}$ is the Ising symmetry, a term ${\sigma }_{i}^{z}{\sigma }_{j}^{z}$ coupling two sites i and j on opposite sides of the branch cut flips sign: ${\sigma }_{i}^{z}{\sigma }_{j}^{z}\to -{\sigma }_{i}^{z}{\sigma }_{j}^{z}$. Note that this procedure is ambiguous near the endpoints of the cut, where the 'sides' of the cut are not well defined—such microscopic choices turn out to lead to discrete gauge ambiguities in the definitions below.

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Now to define ${U}_{g}^{x}$, we simply nucleate a pair consisting of a g-defect and a ${g}^{-1}$ defect, connected by a branch cut, and then grow the branch cut to encircle the region R_x (figure B1). This process defines a continuous path in the space of gapped Hamiltonians, and ${U}_{g}^{x}$ acting on a state $| \psi \rangle$ is then defined by adiabatically following $| \psi \rangle$ through the branch cut nucleation and growth process, and then acting by U_g on all of the spins inside the region R_x. We fix the overall phase of ${U}_{g}^{x}$ by demanding that ${U}_{g}^{x}| 0\rangle =| 0\rangle$, where $| 0\rangle$ is a state containing no anyons. Because the Hamiltonian is gapped, we expect that the adiabatic continuation can be approximated exponentially well by an operator acting in a thin ribbon (several correlation lengths thick) around the boundary of R_x. Hence ${U}_{g}^{x}$ acts like U_g in the interior of R_x, like the identity outside R_x, and interpolates smoothly between these options within a few correlation lengths of the boundary of R_x.

Because we made some arbitrary choices—e.g. the nature of the Hamiltonian at the defect core—we want to see to what extent ${U}_{g}^{x}$ is well defined. A naive argument would say that two different such constructions of ${U}_{g}^{x}$ have the same commutation relations with all local operators near x, and their overall phases are fixed to be the same, so they must be equal. This is not quite correct, however, because there exist excitations near x—namely anyons—which cannot be created with local operators, and are sensitive, via braiding, to possible topological charge bound to the encircling g-defect. That is, if we had chosen a different nucleation process with different defect cores, it is possible that it would differ from the first by an additional abelian anyon stuck on the defect, which could measure topological charge at x via braiding (a non-abelian anyon stuck to the defect would be more problematic, and may not lead to a well-defined ${U}_{g}^{x};$ we evade this possibility by insisting on a fully gapped defect). Now, there is no way to unambiguously measure the abelian anyon charge on a single defect, so we cannot specify the abelian anyon charge carried by the defect in our procedure—the best we can do is to specify the local G-action up to the gauge equivalence:

$$\begin{eqnarray}&&{U}_{g}^{x}| a\rangle \to {{\rm{e}}}^{2\pi {\rm{i}}\langle a,\alpha (g)\rangle }{U}_{g}^{x}| a\rangle .\end{eqnarray} \tag{ B1 }$$

Here $| a\rangle$ is any state with total anyon charge a inside the branch cut loop, $\alpha (g)$ is an arbitrary abelian anyon, and the angular brackets denote the full braiding phase of $\alpha (g)$ around a (i.e. ${{\rm{e}}}^{2\pi {\rm{i}}\langle \alpha (g),a\rangle }={S}_{\alpha (g),a}$ is their S-matrix element).

While the global action of G satisfies the group relation ${U}_{g}{U}_{h}={U}_{{gh}}$, the corresponding local operators ${U}_{g}^{x},{U}_{h}^{x},{U}_{{gh}}^{x}$ might only satisfy it up to a Berry's phase, which depends on $g,h$, and the anyon charge a near x. To demonstrate this, let us form the operator that would measure this Berry's phase:

$$\begin{eqnarray}&&{V}_{g,h}^{x}\equiv {U}_{{gh}}^{x}{({U}_{h}^{x})}^{-1}{({U}_{g}^{x})}^{-1}.\end{eqnarray} \tag{ B2 }$$

Now, consider any single site operator S_i acting on a spin i well inside the local neighborhood R_x of x—i.e. inside the encircling branch cut. Its commutation relations with ${U}_{g}^{x}$ are the same as its commutation relations with U_g—this is just the statement that ${U}_{g}^{x}$ acts just like U_g inside the branch cut region. S_i must commute with ${V}_{g,h}^{x}$, since the latter acts as ${U}_{{gh}}{({U}_{h})}^{-1}{({U}_{g})}^{-1}=1$ well inside R_x. Thus, since ${V}_{g,h}^{x}$ commutes with all observables inside the cut, we can approximate it by an operator that acts only on spins in a thin ribbon region around the cut itself, insofar as its action on excitations well inside the cut is concerned—see figure B3 (the approximation error is exponentially small in the thickness of the ribbon divided by the correlation length).

When an interior excitation is created by a local operator acting on the vacuum, i.e. is of the form $| \mathrm{loc}\rangle =Y| 0\rangle$, with Y a local operator acting only on spins inside the cut, we have

$$\begin{eqnarray*}&&{V}_{g,h}^{x}| \mathrm{loc}\rangle ={V}_{g,h}^{x}Y| 0\rangle ={{YV}}_{g,h}^{x}| 0\rangle =| \mathrm{loc}\rangle ;\end{eqnarray*}$$

${V}_{g,h}^{x}$ and Y commute because they are spatially separated. Thus there are no Berry's phases for excitations that can be made locally, as expected. However, consider now a state $| a\rangle$ which contains an anyon a at x (and no other quasiparticles within the branch cut region). $| a\rangle$ cannot be nucleated locally from the ground state, but rather must be created using a string operator that intersects the branch cut, and hence might fail to commute with ${V}_{g,h}^{x}$, resulting in a Berry's phase

$$\begin{eqnarray}&&{V}_{g,h}^{x}| a\rangle ={{\rm{e}}}^{{\rm{i}}\xi (g,h;a)}| a\rangle .\end{eqnarray} \tag{ B3 }$$

To compute this action of ${V}_{g,h}^{x}$, note first that it must be by an a-dependent phase, since ${V}_{g,h}^{x}$ must be proportional to the identity within each topological superselection sector, by the previous argument. Furthermore, if c is in the fusion product of a and b, the phase for c must be the sum of the phases for a and b, because there exists a local operator that fuses a and b into c. Mathematically, any such function from the anyons to U(1) must be equivalent to the braiding of an abelian anyon—which we call $\omega (g,h)$—around the anyon in question. Note that ${V}_{g,h}^{x}$ has support on a thin ribbon surrounding the anyon in question, so this is certainly an allowed possibility. We thus have

$$\begin{eqnarray}&&{V}_{g,h}^{x}| a\rangle ={{\rm{e}}}^{2\pi {\rm{i}}\langle a,\omega (g,h)\rangle }| a\rangle ={S}_{\omega (g,h),a}| a\rangle ,\end{eqnarray} \tag{ B4 }$$

where ${S}_{\omega (g,h),a}$ is the full braid S-matrix element.

Note that there are constraints among the $\omega (g,h)$. Specifically, consider the operator $W={U}_{{fgh}}^{x}{({U}_{h}^{x})}^{-1}{({U}_{g}^{x})}^{-1}{({U}_{f}^{x})}^{-1}$. We can write it in the following two ways:

$$\begin{eqnarray*}&&W={V}_{{fg},h}^{x}\;{V}_{f,g}^{x}\end{eqnarray*}$$

and

$$\begin{eqnarray*}&&W={V}_{f,{gh}}^{x}\;{U}_{f}^{x}{V}_{g,h}^{x}{({U}_{f}^{x})}^{-1}.\end{eqnarray*}$$

The first tells us that W acts by braiding $\omega (f,g)$ and then braiding $\omega ({fg},h)$, while the second says that it is equal to the composition of braiding operations of $\omega (g,h)$ and $\omega (f,{gh})$ (the conjugation by ${U}_{f}^{x}$ does not affect anything, because ${U}_{f}^{x}$ does not change the topological charge a that is being acted on). Thus, we have

$$\begin{eqnarray*}&&\omega ({fg},h)\omega (f,g)=\omega (f,{gh})\omega (g,h)\end{eqnarray*}$$

or

$$\begin{eqnarray*}&&\omega (g,h)\omega {({fg},h)}^{-1}\omega (f,{gh})\omega {(f,g)}^{-1}=1.\end{eqnarray*}$$

Those acquainted with group cohomology will recognize the above as the cocycle equation

$$\begin{eqnarray}&&d\omega (f,g,h)=1.\end{eqnarray} \tag{ B5 }$$

Furthermore, we can use the definition (B2) of ${V}_{g,h}^{x}$ to see that under the inherent gauge ambiguity (B1), $\omega (g,h)$ must transform as

$$\begin{eqnarray*}&&\omega (g,h)\to \omega (g,h)\alpha (g)\alpha (h)\alpha {({gh})}^{-1}\end{eqnarray*}$$

which can also be re-written in group cohomology terms as

$$\begin{eqnarray}&&\omega (g,h)\to \omega (g,h)\cdot d\alpha (g,h).\end{eqnarray} \tag{ B6 }$$

Functions $\omega (g,h)$ satisfying equation (B5), modulo the gauge equivalences in equation (B6), form a finite abelian group called the second group cohomology ${H}^{2}(G,{ \mathcal A })$. Here ${ \mathcal A }$ is the abelian group formed by anyons of quantum dimension 1. The equivalence class of $\omega$ in this set is denoted $[\omega ]\in {H}^{2}(G,{ \mathcal A })$, and is an invariant which encapsulates the projective properties of the anyons in our symmetry enriched topological phase.