Floquet codes and phases in twist-defect networks

The HFC consists of a periodically-repeating sequence of three measurement rounds acting on qubits arranged on the sites of a honeycomb. To define the model, label the three distinct orientations of bonds on the honeycombs by label $\alpha\in\{x,y,z\}$ as shown in Fig. 1. Introduce a three-coloring of plaquettes red (R), green (G), and blue (B). In addition to the bond-orientation ($x,y,z$) labels, also label bonds that connect $R,G,B$ plaquettes with $R,G,B$ respectively. For future convenience, we also number the sites around each hexagonal plaquette, $p$ by $1\dots 6$ starting from the middle-left corner and going around in the counter-clockwise direction.

Following the notation of Kitaev’s honeycomb model [8], define the bond-operators $P_{ij}=\sigma^{\alpha}_{i}\sigma^{\alpha}_{j}$ where $\sigma^{\alpha=x,y,z}$ are spin-1/2 Pauli operators for direction $\alpha$ that coincides with the $x,y,z$ label of bond $ij$. Further, for each site $i$, introduce the Majorana fermion operators $c_{i},b^{\alpha}_{i}$ related to the spin operators by $\sigma^{\alpha}=ib^{\alpha}_{i}c_{i}$. The physical spin model is recovered by projecting $ic_{i}b^{x}_{i}b^{y}_{i}b^{z}_{i}=1$ on each site. It is convenient to repackage these operators as Majorana “defects” $c_{i}$ on each site, and $\mathbb{Z}_{2}$ gauge connections $u_{ij}=ib^{\alpha}_{\langle ij\rangle_{1}}b^{\alpha}_{\langle ij\rangle_{2}}$ on each bond. Here, we choose an orientation for each bond, $\langle ij\rangle$, and $\langle ij\rangle_{1,2}$ label the start, end sites of the bond respectively (in this way $u_{ij}=u_{ji}$ and one may henceforth ignore the bond orientations). In this representation $P_{ij}=ic_{i}u_{ij}c_{j}$ are simply interpreted as the (gauge-invariant version of the) fermion parity of the pair of Majorana defects $c_{i},c_{j}$.

The HFC [1] then consists of a repeating sequence of three rounds of measurements: in round $1,2,3$, $P_{ij}$ are measured, or “checked”, for each $R,G,B$ bonds respectively. We refer to these measured bond operators as parity checks or simply “checks”²²2In the subsystem code literature $P_{ij}$ are often referred to as gauge operators or gauge checks; a notation that is ripe for confusion with standard gauge theory formulation of topological orders.. While the measurement schedule repeats periodically, the checks in round $r$ do not commute with those in round $r+1$, and hence the measurement outcomes are random and generically non-repeating. However, after three rounds, the products of checks $P_{ij}$ around any hexagon $P$ (and hence also any contractible loop), given by $\Phi_{P}=\prod_{\langle ij\rangle\in\hexagon_{p}}P_{ij}$, are determined. $\Phi_{P}$ then commute with all future measurement checks, to form persistent stabilizers of the HFC. In the gauged-fermion language, the persistent stabilizers are simply the $\mathbb{Z}_{2}$ gauge flux $\Phi_{p}=\prod_{\hexagon_{p}}u_{ij}=\pm 1$. Crucial to the performance of the code, the measurement sequence should be chosen to avoid measuring the flux through non-contractible loops, which represent logical operators.

After each measurement round, the most-recently-measured parity checks and persistent gauge-flux stabilizers together form the stabilizers of a $\mathbb{Z}_{2}$ topological order that is topologically equivalent to a toric code. This is referred to as an instantaneous code space (ICS). The three different ICSs after the $R,G,B$ measurement rounds are respectively labeled ${\rm ICS}_{R,G,B}$. Each ICS is represented by a distinct, but topologically equivalent, pairing of the Majorana defects, immersed in a fixed gauge-flux configuration dictated by the persistent stabilizer values. The states of each ICS can be labeled by configurations of point-like anyon excitations with topological charges (a.k.a. topological super-selection sectors), $f,m,\text{ and }e=m\times f$. We associate a flipped bond stabilizer $P_{ij}=-1$ with a fermion ($f$) excitation. Since the bond-stabilizers are on different ($R,G,B$) bonds during each round, the resulting $e,m$ excitation labeling also depends on the round. For ${\rm ICS}_{R}$, we label the $\mathbb{Z}_{2}$ gauge fluxes, $\Phi_{p}=-1$ on the $p\in R$ plaquettes as $m$ excitations, and those on the $B,G$ plaquettes as $e$ excitations. To understand this color-dependent labeling, note that the local operator $S^{y}_{4}=ic_{4}b^{y}_{4}$ in Fig. 1 changes $u_{45}\rightarrow-u_{45}$, i.e. creates fluxes on plaquettes labeled $R$ and $G$ and also creates a fermion on the red bond touching site $4$. This shows that it would not be consistent to label all gauge fluxes as $m$ regardless of color. In our labeling convention this simply corresponds to the fusion rule $1=e\times m\times f$. Similarly for ${\rm ICS}_{G}$ (${\rm ICS}_{B}$) denote the $G$ ($B$) fluxes as $m$ excittions and the $R,B$ ($R,G$) plaquette fluxes as $e$ excitations.

The logical operators of each ICS can also be labeled by anyon types $e,m,f=e\times m$. At a coarse grained level, each type of logical operator, $a$, can be viewed as creating an $a$ particle/anti-particle pair, dragging them around a non-contractible cycle, and then annihilating them. These loop operators can be decomposed into a product of many short line segments whose ends create $a/\bar{a}$ anyon/anti-anyon pairs. Representative logical operators for the HFC on a torus are shown in Fig. 3. The $f$-loops through a cycle indicate the $\mathbb{Z}_{2}$ gauge flux $\prod_{ij}u_{ij}$ through any co-threaded cycle, and are persistent logical operators shared by each ICS. The $e,m$ loops of round $r$ are not persistent operators, but may be augmented by multiplying by parity checks $P_{ij}$ of round $r$ to become logical operators of the next round, $r+1$. In this way, the measurement dynamics map each logical operator from ${\rm ICS}_{r}$ to those of ${\rm ICS}_{r+1{\rm mod}~2}$. A key property of the HFC is that, after three cycles, the $e$ and $m$ loops are interchanged.

Many of the dynamical features of the HFC, especially the $e\leftrightarrow m$ exchanging property, are also found in the unitary dynamics of Honeycomb models of Floquet enriched topological orders (FETs) [18, 17, 15]. It is natural to ask whether these features are related. To establish a connection between the measurement-only FC dynamics and the unitary FET dynamics, we design a sequence of unitary circuit evolutions, $U_{1,2,3}=e^{-iH_{1,2,3}}$, that maps between the ICSs of the $R,G,B$ measurement rounds of the HFC. In other words, $U_{1,2,3}$ will respectively map each state of $\rm ICS_{R,G,B}$ to one in $\rm ICS_{G,B,R}$. A closely-realted circuit structure, dubbed Kramers-Wannier circuits, was introduced in [5], and used to construct an adiabatic path through the space of gapped Hamiltonians. Here, we focus instead on the non-equilibrium Floquet dynamics of the entire spectrum of excited states generated by repeated application of this circuit, which allows us to connect to well-established topological indices for unitary loops and quantum cellular automata [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21].

To construct the sequence of $U_{a}$’s, note that the parity-checks for the $R$ and $G$ measurement rounds of the HFC differ by a $60^{\circ}$ rotation of each $B$ plaquette, and the transitions $G\rightarrow B$ and $B\rightarrow R$ can be similarly accomplished by rotating the $R$ and $G$ plaquettes respectively. To implement this action with a unitary, we introduce local unitary operators that cyclically permute the Majorana operators counterclockwise about a given hexagonal plaquette $P$:

$\displaystyle C_{p}$

$\displaystyle=B_{12}B_{23}B_{34}B_{45}B_{56},$

(1)

where $B_{j,i}=e^{\frac{\pi}{4}c_{j}u_{ij}c_{i}}$ braids Majorana mode $c_{i}$ around $c_{j}$:

$\displaystyle B_{j,i}^{\dagger}\begin{pmatrix}c_{i}\\ c_{j}\end{pmatrix}B_{j,i}$

$\displaystyle=u_{ij}\begin{pmatrix}-c_{j}\\ +c_{i}\end{pmatrix}$

(2)

Referring to the numbering convention of sites on each plaquette shown in Fig. 2, this cyclic permutation operator, $C_{p}$, has the following action on the bond-parity checks:

$\displaystyle C_{p}:\begin{cases}P_{i,i+1}\rightarrow P_{i+1,i+2}&i\neq 5\\ P_{5,6}\rightarrow-\Phi_{p}P_{12}&\\ \end{cases}$

(3)

where we take site numbers $i$ modulo $6$. In other words, each of the bond parity checks is cyclically permuted around the edge of the hexagon. In addition, one bond $P_{6,5}$ parity check picks up a phase equal to the gauge flux, $\Phi_{p}$ through the $p$; this will be crucial to recovering the $e\rightarrow m$ exchanging dynamics of the HFC.

We then define:

$\displaystyle U_{1,2,3}=\prod_{p\in B,R,G}C_{p}.$

(4)

which can each be generated by local Hamiltonians $H_{1,2,3}=-i\log U_{1,2,3}$. Then, we define the continuous-time unitary Floquet evolution:

	$\displaystyle U(t)$	$\displaystyle=\mathcal{T}e^{-i\int_{0}^{t}H(s)ds}$
	$\displaystyle H(t)$	$\displaystyle=3\begin{cases}H_{1}&0\leq t<1/3\\ H_{2}&1/3\leq t<2/3\\ H_{3}&2/3\leq t<1\end{cases}$		(5)

where $\mathcal{T}$ denotes time-ordering, and we have set the Floquet period to $1$ for convenience.

Before embarking on this we highlight two subtleties in defining an equivalence between codes and loops that arise in the HFC example. In the following, we will restrict our attention to local codes and loops, which can be generated by strictly local measurements or Hamiltonians respectively.

First, note that the trajectories of individual states generally differ between the measurement-only or parameterized unitary dynamics, and only the evolution of the logical subspace of the codes match. For example, the local measurement outcomes in each step of the HFC are random, so that, after one Floquet cycle of measurements a state initially in ${\rm ICS}_{R}$ returns to a state in ${\rm ICS}_{R}$ but with a generically-different pattern of fermion excitations on the red bonds. In contrast, the unitary dynamics produce deterministic state evolutions. For this reason, we define criterion ii) above only in terms of the logical subspace.

Second, the correspondence between the HFC and ULs is one-to-many: there are multiple ULs that reproduce i) and ii) of the HFC, which differ by stacking with a non-topologically-ordered Floquet topological phase with rational CF index. The UL for this rational CF phase satisfies $U(t=1)=\mathbbm{1}$, implying that the eigenstates of $U(t=1)$ can be chosen to be short-range entangled. We will refer to ULs with this property as invertible, generalizing the terminology for “integer” topological phases of gapped ground-states. Here, by “stacking”, we mean that one can extend a unitary loop, $U(t+n)=U(t)$ by adding additional degrees of freedom and/or adding additional “trivial” dynamics within the period. Specifically, we allow modifying the generating Hamiltonian, $H_{0}(t+1)=H_{0}(t)$ by adding an extra step:

$\displaystyle H_{0}(t)\rightarrow H(t)=\begin{cases}2H_{0}(2t)&0\leq t<1/2\\ 2H_{1}(2t)&t/2\leq t<1\end{cases}$

(8)

where $H_{1}(t)$ generates a trivial unitary loop $U_{H_{1}}\equiv\mathcal{T}e^{-i\int_{0}^{1}H_{1}(t)dt}=\mathbbm{1}$. To get some intuition for this expression, note that the evolution for one period is: $U_{H}=U_{H_{1}}U_{H_{0}}=U_{H_{0}}$. With this in mind, we define an equivalence class of ULs by moding out stacking with invertible Floquet phases, and look for a stable equivalence between FCs and these equivalence classes of ULs.

In general, Floquet codes and loops are not equivalent in the sense defined above.

The Kitaev honeycomb model can be described as a gauged fermion system with Majorana “defects” $c_{i}$ on each site $i$ of the honeycomb, and $\mathbb{Z}_{2}$ gauge connections $u_{ij}$ on the edges $\langle ij\rangle$ of the honeycomb.

The Floquet honeycomb code has a particularly simple structure in this representation: the persistent stabilizers are the gauge flux through the plaquette, $\prod_{\langle ij\rangle\in\hexagon}u_{ij}$, and the instantaneous check are projectors onto parity of Majorana pairs: $P_{ij}=ic_{i}u_{ij}c_{j}$. After one cycle of measurements, the fluxes become frozen into a fixed non-dynamical pattern, and subsequently only the fermion degrees of freedom have non-trivial dynamics. In this case, we can regard $u_{ij}$ as a background, non-dynamical gauge field and consider just the system of Majorana fermions.

This structure can be extended to a network of Majoranas on vertices of a general graph (with even number of sites) with gauge links on edges. We define a paired-Majorana network code on such a graph as one stabilized by the flux through each face of the graph, and the fermion parities $P_{ij}$ of a fixed pairing of the Majoranas. Without loss of generality, we can consider only nearest-neighbor pairings (possibly by adding extra edges to the graph). Similarly, define a paired-Majorana network Floquet code as an FC for which each ICS is a paired-Majorana network code.

For this structure, there is a succinct condition for whether a given schedule of local Majorana-pair measurements results in the measurement of a logical operator. Each pairing defines a dimer cover of the graph, with each dimer representing an edge connecting the paired Majoranas. We can represent any dimer cover as a $\mathbb{Z}_{2}$ module on the edges of the graph, i.e. a formal sum of edges with $\mathbb{Z}_{2}$ coefficients. Consider the dimer covers $D_{i}$ and $D_{i+1}$ corresponding to subsequent ICSs in a FC. The sum $D_{i}+D_{i+1}$ defines a loop configuration on the graph. After subsequent measurement rounds, the gauge flux through each loop is also measured, since $\prod_{\langle ij\rangle\in\text{loop}}P_{ij}=i^{L}\prod_{\langle ij\rangle\in\text{loop}}u_{ij}\sim\Phi_{\text{loop}}$. Since logical operators of the code are given by gauge fluxes through non-contractible loops, the requirement that a measurement sequence does not measure a logical is that $D_{i}+D_{i+1}$ does not contain any non-contractable loops for any measurement round $i$. We refer to this as the “no-long-loops” condition.

It is straightforward to establish an equivalence between Floquet codes, loops, and Floquet MBL phases for gauged Majorana defect networks satisfying this no-long-loops condition.

As an application of this formalism, we next show how the equivalence between FCs and ULs places dynamical-anomaly constraints on the type of allowed boundaries of the qubit HFC. Since any paired Majorana FC, such as the HFC or any modification that preserves this structure, is equivalent to a UL, they are both governed by the bulk-boundary correspondence and dynamical chiral-Floquet edge anomalies of the UL. Specifically, for ULs bulk $e\leftrightarrow m$ exchanging dynamics, implies a radical CF index $\nu(U)\in\sqrt{2}\mathbb{Q}$, which implies that there is no way to form a gapped/localized boundary [17] that preserves the time-translation symmetry of the bulk dynamics. The equivalence between FCs and ULs of paired-defect-network form, implies that there is similarly an obstruction to forming a gapped/logical edge of the HFC with a boundary measurement schedule of the same period as the bulk.

Previous constructions for a planar HFC with gapped/logical boundary [3] modified the bulk measurement sequence by doubling the periodicity and measuring $R\rightarrow G\rightarrow B\rightarrow G\rightarrow B\rightarrow R^{\prime}$ bonds where $R$ and $R^{\prime}$ differed at the edge. This sequence effectively performs the non-contractible loop $R\rightarrow G\rightarrow B$, and then undoes this loops by reversing its direction. The full 6-step sequence has no overall $e\leftrightarrow m$ exchanging action on logical operators. The unitarization of this process would be to consider alternating between chiral unitary evolution $U_{1}$, $\nu(U_{1})=\sqrt{2}$ and antichiral unitary evolution, $U_{2}$ with $\nu_{2}=1/\sqrt{2}$, to get an overall trivial index, $\nu(U_{2}U_{1})=1$ for the full evolution.

A related perspective comes from the equivalence of FCs and ALs for this class of systems. Namely, if we have an AL that exchanges $e\leftrightarrow m$ excitations, then there is no gapped boundary Hamiltonian with the same periodicity as the bulk. Specifically, there are two types of gapped boundaries of the $\mathbb{Z}_{2}$ gauge theory corresponding to condensing either $e$ or $m$ at the edge. Since one period of the HFC evolution exchanges $e$ and $m$ particles, it also exchanges these types of gapped boundaries. Hence, there is no periodic boundary-Hamiltonian that is both gapped and invariant under this $e\leftrightarrow m$ exchange.

However, this argument suggests that it should be possible to gap the boundary by simply doubling the periodicity of the boundary without altering the bulk. Schematically, we could choose $H_{\text{boundary}}(t)=\begin{cases}H_{e}&0\leq t<1\\ H_{m}&1\leq t<2\end{cases}$ where $H_{e,m}$ respectively represent boundary Hamiltonians for the $e,m$ condensed boundaries. Returning to the FC setting, we can confirm that such a period-doubled gapped/logical boundary is indeed possible for the HFC. An example is drawn in Fig. 6. Here, we consider a zig-zag edge of the Honeycomb, modified to a trivalent graph as shown in Fig. 6(a). The check operators of each round are highlighted in Fig. 6(d); the bulk measurements are unchanged ( $R\to G\to B$) and boundary measurement schedule has period six ( $\text{solid }R\to\text{solid }G\to\text{solid }B\to\text{dashed }R\to\text{dashed }G\to\text{dashed }B$). By inspection, one can see that each adjacent measurement round contains only short loops such that no logical operators are measured. By contrast, one can verify by inspection, that repeating the solid-line boundary conditions would result in the measurement of a long $f$-loop around the boundary upon going from the $B\rightarrow R$ step.

We note that, for this construction, one needs to directly measure gauge fluxes through the downward facing boundary triangle plaquettes shown in Fig. 6(c), as these are not accumulated as persistent stabilizers of the other measurements. Additionally, the boundary Majorana pair measurements are not nearest neighbor on the original lattice, and require several-spin measurements. These boundary measurements spoil the two-body measurement structure of the original HFC code. While potentially of practical importance for error correction thresholds, for the purposes of exploring general topological features of FCs, we view such details as non-universal engineering challenges.

One can also explicitly track the evolution of logical operators that terminate on the boundary as shown in Fig. 6. The logical operators are either $e$ or $m$ strings that terminate on the open boundary. To form a logical qubit one needs to consider a plane with multiple holes punched out to form multiple non-contractible loops, however, for simplicity we simply show the end points of operators on one boundary. Starting with the solid-red ICS, we can label the logical operators as $e$ or $m$ strings depending on whether or not they terminate within or outside the red edge bond. To track the evolution of this operator to the next round, one needs to relabel it by tacking on $R$ stabilizers to form an operator that commutes with the $G$ measurements. Following the evolution through one bulk Floquet period, we see that the edge operator evolves as shown in Fig. 6.

This example illustrates that, while the direct construction of gapped boundaries of the HFC can be complicated to construct, the insights from the ALs and ULs can identify possible universal mechanisms for their construction, rather than trying to build them by (potentially tedious) trial and error.

To illustrate the construction of FCs and phases from twist-defect networks, we next construct a generalization of the Honeycomb code of Haah and Hastings to arbitrary non-chiral, Abelian topological order with non-trivial anyon permutation symmetry $\sigma$. ⁵⁵5We note that, at an abstract level, this construction can also describe adiabatic loops and Floquet codes with chiral or non-Abelian topological order.

We start by forming a honeycomb by gluing together hexagonal plaquettes of a given topological order with twist defects $\sigma$ on each of the $A$ sublattice sites and anti-defects $\bar{\sigma}$ on each of the $B$ sublattice sites ⁶⁶6Here, we use $\sigma$ to denote both the defect and its associated anyon automorphism.. We then three-color the plaquettes with labels $R,G,B$, as was done with the qubit HFC. We choose the branch-cuts for the twist defects to reside along the red links, meaning that an anyons’ topological charge gets transformed from $a$ to $\sigma(a)$ upon crossing a red link (with orientation shown in Fig. 7(a).

The measurement schedule follows that of the qubit HFC, measuring $R$ then $G$ then $B$ bonds. The Majorana parity measurements of Sec II.1 are replaced with “braiding check” measurements: we measure the phase that results from creating an anyon/anti-anyon pair, wrapping the anyon around a small loop enclosing the bond and then re-annihilating it with the anti-anyon. This anyon interferometry measurement partially determines the fusion channel of the $\sigma,\bar{\sigma}$ pair on each bond. In order to completely determine the fusion channel it is sufficient but not necessary (see Appendix B) to measure the aforementioned braiding process for each generating anyon $\{a_{1},a_{2},...a_{N}\}$ of the TO. In each round $r=R,G\text{ or }B$ a minimal set of braiding checks are performed around each bond of type $r$, collapsing the degeneracy introduced by the presence of the defects, and leaving behind a state of the ICS with only the topological degeneracy due to the underlying Abelian TO.

After each measurement round, logical operators of the ICS are anyon string operators that wrap around non-contractible loops which do not intersect any defect lines. They will need to be modified constantly by measured braiding checks in order to commute with the next round of measurements. The evolution of logicals is depicted in Fig. 7, it is clear that after a full cycle of measurement an $a$ anyon string is converted into a $\sigma(a)$ string.

The above continuum description of the defect networks can be converted into an exactly-solvable lattice model whenever the underlying topological order admits such a lattice model description. As a brute-force construction, one could consider a network of patches of string-net models [7]). However, the resulting lattice model will generally be cumbersome and involve measurement of many-spin terms.

In this section we present an alternative defect network construction which naturally furnishes a lattice model description. The construction is based on the twist-defect parton (a.k.a. “slave-genon”) approach introduced by [35]. This approach yields generalized HFC models with non-chiral Abelian topological order with order-two twist defects (i.e. for which twisting twice restores the anyons). For this family of generalized HFC models we derive a measurement schedule corresponding to two-site nearest-neighbor Pauli measurements and describe the automorphism generated by measurements. Unlike the continuum description the parton construction only implements anyon automorphisms with order two ($\sigma^{2}=1$).

Schematically, the twist-defect parton construction follows that of Kitaev’s Majorana representation of spins-1/2, but replaces the Majorana defects of the $\mathbb{Z}_{2}$ topological order, with arbitrary twist defects, $\sigma$, of a general abelian topological order, $\mathsf{TO}_{0}$. Parton constructions describe a local Hilbert space as a projection from a larger auxiliary Hilbert space. Following [35], the auxiliary Hilbert space can be viewed as a small island of topological order, $\mathsf{TO}_{0}$, with two twist defect/anti-defect ($\sigma/\bar{\sigma})$ pairs. The physical Hilbert space is then obtained by projecting each island (site) into the sector with trivial total topological charge. This projection plays the role of the gauge-constraint $c_{i}b^{x}_{i}b^{y}_{i}b^{z}_{i}=1$ in the spin-1/2 Kitaev model, which forces the four Majorana defects to have overall trivial fermion parity, yielding a bosonic spin model. Here, a notable distinction from the original spin-1/2 Kitaev honeycomb model arises: though in Kitaev’s Honeycomb model, paired phases of the Majorana defects also had the same type of $\mathbb{Z}_{2}$ topological order, the Abelian (paired-defect) phase of generalized Kitaev models may have a completely different type of topological order, $\mathsf{TO}$, distinct from $\mathsf{TO}_{0}$. For example,  [35] used islands of fractional quantum Hall (FQH) bilayers, with interlayer-genon defects as the twist-defect partons, to construct a generalized Kitaev honeycomb model with $\mathbb{Z}_{N}$ topological order.

In the parton description, the local, gauge-invariant “spin” operators on each site: $T^{a}_{i,\alpha}$ are defined by braiding anyon $a$ around the pair $\alpha=x,y,z$ of defects on site/island $i$, as shown in Fig. 9. The local operators obey the algebra:

	$\displaystyle T^{a}_{z}T^{b}_{x}=e^{i\theta_{a,b\sigma(\overline{b})}}T^{b}_{x}T^{a}_{z}$		(11)
	$\displaystyle T^{a}_{x}T^{b}_{y}=e^{i\theta_{a,b\sigma(\overline{b})}}T^{b}_{x}T^{a}_{y}$		(12)
	$\displaystyle T^{a}_{y}T^{b}_{z}=e^{i\theta_{a,b\sigma(\overline{b})}}T^{b}_{y}T^{a}_{z}$		(13)
	$\displaystyle T^{a}_{\alpha}T^{b}_{\alpha}=T^{ab}_{\alpha},~T^{a\dagger}_{\alpha}=T^{\overline{a}}_{\alpha}.$		(14)

The gauge constraint $T^{a}_{x}T^{a}_{y}T^{a}_{z}=1$, implies that in the physical subspace $T^{a}_{y}\equiv T^{a\dagger}_{x}T^{a\dagger}_{z}$. To form a local, on-site Hilbert space, we then construct a representation of this algebra for any abelian $\mathsf{TO}_{0}$ and $\sigma$.

The anyons that are invariant under $\sigma$ form a subgroup of $\mathsf{TO}_{0}$⁸⁸8we use $\mathsf{TO}_{0}$ for both the abelian topological order itself and the group of its fusion rules, which is a finite abelian group. that we denote by $\text{Inv}(\sigma):=\{a\in\mathsf{TO}_{0},\sigma(a)=a\}$. Denote the quotient group $\mathsf{TO}_{0}/\text{Inv}(\sigma)$ by $\mathsf{TO}_{0}^{\sigma}$ and its elements, the equivalent class of $a$, by $[a]$. Let $\mathcal{H}=\mathbb{C}[\mathsf{TO}_{0}^{\sigma}]$ be the group algebra over $\mathsf{TO}_{0}^{\sigma}$, that is, the formal linear combinations of group elements with complex coefficients. $\mathcal{H}$ has a standard basis $\{~\left|[a]\right\rangle;~a\in\mathsf{TO}_{0}\}$ and has dimension $D=|\mathsf{TO}_{0}|/|\text{Inv}(\sigma)|$. Define $T^{a}_{x},T^{a}_{z}$ by their action on the basis as:

$\displaystyle T^{a}_{z}|[g]\rangle=e^{i\theta_{a,\bar{g}\sigma(g)}}|[g]\rangle,~T^{a}_{x}|[g]\rangle=|[ag\rangle]\rangle.$

(15)

This representation is well-defined, since the phase factor of $T^{a}_{z}$ does not depend on the choice of $g$: $\theta_{a,\bar{g}\sigma(g)}$ is zero for any $g\in\text{Inv}(\sigma)$, and it forms the desired representation.

Note that, in this representation, $T^{a}_{\alpha}=1$ for any $a\in\text{Inv}(\sigma)$. Namely: $T^{a}_{z}=1$ because the phase factor in Eq. 15 satisfies $\theta_{a,\overline{g}\sigma(g)}=\theta_{g,\overline{a}\sigma(a)}=0$, and $T^{a}_{x}=1$ since $[ag]=[a][g]=1[g]=[g]$. In all examples known to us, the single site Hilbert space $\mathcal{H}$ is a tensor product $\mathbb{C}_{d_{1}}\otimes\mathbb{C}_{d_{2}}\otimes\cdots$ and the operators $T^{a}_{\alpha}$ are products of generalized Paulis. For examples we refer the reader to appendix D.