The theory of topological quantum computation is underpinned by two important classes of models. One is based on non-abelian Chern–Simons theory, which yields the so-called $ SU ( 2 ) k$ anyon models that often appear in the context of electrically charged quantum fluids. The physics of the other is captured by symmetry broken Yang–Mills theory in the absence of a Chern–Simons term and results in the so-called quantum double models. Extensive resources have been invested into the search for $ SU ( 2 ) k$ anyon quasi-particles, in particular, the so-called Ising anyons (*k* = 2) of which Majorana zero modes are believed to be an incarnation. In contrast to the $ SU ( 2 ) k$ models, quantum doubles have attracted little attention in experiments despite their pivotal role in the theory of error correction. Beyond topological error correcting codes, the appearance of quantum doubles has been limited to contexts primarily within mathematical physics, and as such, they are of seemingly little relevance for the study of experimentally tangible systems. However, recent works suggest that quantum double anyons may be found in spinor Bose–Einstein condensates. In light of this, the core purpose of this article is to provide a self-contained exposition of the quantum double structure, framed in the context of spinor condensates, by constructing explicitly the quantum doubles for various ground state symmetry groups and discuss their experimental realisability. We also derive analytically an equation for the quantum double Clebsch–Gordan coefficients from which the relevant braid matrices can be worked out. Finally, the existence of a particle-vortex duality is exposed and illuminated upon in this context.

## I. INTRODUCTION

As one ventures beyond the realm of classical phases of matter, the classification paradigm due to Landau^{1} ceases to apply. Such quantum systems often exhibit phenomena of a long-range collective nature connected to a non-trivial underlying topological structure. Consequently, topology may serve as a better fingerprint to categorize such phases. Owing to its inherent relationship to the connectivity of space, the spin-statistics theorem generally breaks down on non-trivial topologies, thus clearing the way for more exotic particle species to emerge that may not be classified as bosons or fermions. Such peculiar particles exhibit fractional statistics and are, therefore, referred to as *anyons*^{2–5} (*any* as in *any* statistics). Permutation of anyons will, in contrast to fermions and bosons, generally implement a more complex unitary transformation than a simple change of sign of the wavefunction.

While anyons are very interesting from a fundamental viewpoint, most research on the topic concerns quantum information processing.^{6,7} A strong interest in anyons was sparked when Kitaev suggested that they may hold the key to the realization of fault tolerant quantum computation.^{8} The prospects for quantum computation are tantalizing, but in order to not succumb to decoherence, the effects of environmental noise must be considered. This is naturally addressed in a quantum computer based on anyons. Owing to the intrinsic properties of topology, qubit states based on anyons possess a natural shield against various unwanted interactions with the environment. Error-immune computers of this kind are known as *topological quantum computers* (TQCs).^{8–12}

Anyons may be realized in planar fermionic fluids, such as an electron gas in the fractional quantum Hall effect,^{13} which effectively are governed by the Chern–Simons theory.^{14–16} The Chern–Simons term is responsible for a type of charge–flux attachment, akin to that in the Aharanov–Bohm experiment,^{17,18} which further results in quasi-particles with fractional statistics. These models are known as $ SU ( 2 ) k$ models,^{19,20} where the parameter $k$ is an integer referring to a particular deformed representation of $ SU ( 2 )$. A strong interest for Majorana quasi-particle zero modes has been developed in recent years since they are believed to realize the $ SU ( 2 ) 2$ Ising anyons.^{20–22} Majorana zero modes have been predicted to emerge in the vortex quasi-particle spectrum of chiral p-wave superfluids such as in ^{3}He superfluids or cold atom Fermi gases.^{23–27} They are also expected to emerge in solid state systems such as in fractional quantum Hall fluids with filling fraction $ \nu = 5 / 2$,^{28–30} in superconducting–semiconducting nanowires,^{31–38} and in the vortex cores of certain topological superconductors.^{39}

Moreover, anyons are also believed to emerge in certain bosonic systems, such as Bose–Einstein condensates (BECs),^{40} as a result of spontaneous breaking of the initial continuous gauge symmetry to a discrete residual subgroup. Models based on spontaneously broken gauge symmetries are known as *quantum doubles.*^{8,41–44} A quantum double can be viewed as an emergent low-temperature symmetry algebra, where the group structure is “doubled” by combining it with its Fourier dual. Mathematically they constitute, just like the $ SU ( 2 ) k$ models, examples of so-called *quantum groups.*^{43,45} In particular, quantum group symmetry emerges when the degrees of freedom pertaining to (generalized) electric charges interact with those of (generalized) magnetic fluxes. Consequently, the particle content of an anyon model is labeled by the irreducible representations of the pertinent quantum group. This is in contrast to conventional quantum field theory, where the particles are labeled by those of an undeformed group. For the sake of completeness, a detailed exposition of the particular quantum group relevant to this work is provided in Appendix C and is also thoroughly discussed in Ref. 41.

The quantum double structure has been studied extensively from a mathematical perspective, yet little efforts have been made to reconcile it with real physical systems. It is not well known that low-temperature phases of spinor BECs^{46–52} with discrete residual symmetry are underpinned by a quantum double structure. Hence, this work primarily seeks to introduce the concept of quantum doubles to the cold atoms community and to provide concrete examples demonstrating how the various components of the quantum double structure might materialize in spinor BECs. There are mainly two types of quantum double excitations referred to as *fluxons* and *chargeons*. While the fluxons, as per homotopy theory, correspond to quantized vortices, the physical interpretation of chargeons is not that clear. By illustrating that fluxons and chargeons are in fact Fourier duals of one another, we find that the chargeons may be associated with delocalized waves. In particular, by conducting an analysis of the normal modes, the chargeons appear to be represented as spin rotations and spin waves, also known as *magnons.*^{53–56} This agrees with results obtained from numerical simulations carried out in Ref. 57, where spin waves were observed as remnants after fusion events involving non-abelian vortices. The fact that spinor BECs can be routinely produced in the laboratories^{58–62} brings further justification supporting the study of non-abelian anyons in such systems. Before moving on, we wish to highlight that the field theory descriptions of anyons in fermionic and bosonic systems are not always of a Chern–Simons and Yang–Mills type, respectively. Indeed, there exist systems that are based on spin-1/2 particles, which are underpinned by quantum doubles, for example, Kitaev's toric code^{8} and honeycomb model.^{63} Similarly, there are systems whose constituent particles are bosons and are described by Chern–Simons theory. A Chern–Simons gauge gravity is an example of the latter.^{64,65}

## II. CHERN–SIMONS ANYONS

Before discussing the quantum double structure in spinor BECs, we describe anyons in fermionic systems of a Chern–Simons type for the purpose of providing a more cohesive perspective on anyons. In addition, the classification scheme for topological defects is described, which is later applied to spinor BECs in Secs. III E and III F to derive their topological charge.

### A. From Aharonov–Bohm effect to topological quantum field theory

Let us consider the Aharonov–Bohm setup.^{17} In this experiment, an electron encircles an (effectively) infinitely long tube of magnetic flux, see Fig. 1, which is piercing through three-dimensional (3D) space. As the electron traverses around the flux-tube, the wave-function accumulates a complex phase due to the non-zero vector (gauge) potential originating from the flux.

*c*is the speed of light, and $A$ is the magnetic vector potential, then the electric and magnetic fields, respectively, can be computed as

*ρ*is the electric charge density and $ J = \rho v$ is the current density. Variation of this action yields the sourceful Maxwell's equations of electrodynamics. The phase acquired by the electron as it encircles the flux may be computed using Feynman's path integral picture. From the electrons perspective, all fields in the action are zero except the vector potential so the only contribution to the phase is given by

*α*. This phase is an example of a

*Berry's phase.*

^{18}Note that the contour $ \u2202 A$ is arbitrary so the only thing of relevance is the flux through the enclosed surface and not the particular shape of the contour. The value of the path integral only depends on whether the flux was enclosed by the loop or not. Note that the existence of the gauge field $A$ can be attributed to the invariance under the gauge transformation

^{66,67}and as we shall see, we may generalize this concept further to more complex spaces, which will allow for the implementation of topological quantum computation protocols. The magnetic field can also be regarded as the curvature of parameter space since the field strength tensor is the analog of the Riemannian curvature tensor in the context of gauge theory. A non-zero magnetic field is, thus, inducing a non-trivial holonomy as the particle is parallel transported around the flux tube. Now, let us descend to the two dimensional (2D) plane. One of the key differences between two and three spatial dimensions is that the curl of a vector is a scalar, thus implying that the magnetic field $ B i = \epsilon ijk \u2202 j A k$ is a scalar in the plane. This transition opens new doors since the most general Maxwell Lagrangian now accept new terms. In particular, Poincaré and gauge symmetry is respected by the additional term

*k*is a coupling constant. This Chern–Simons term has played a pivotal role in the development of our understanding of two-dimensional quantum physics, and in particular, the quantum Hall effects. Here, $ a \mu $ is an emergent statistical gauge field originating from the punctures caused by the flux lines (vortices). Just like the flux tube penetrates the space in the Aharonov–Bohm experiment, topological excitations may puncture the system in two dimensions, which gives rise to an analogous phenomenon (see Fig. 2). Mathematically, what this entails is that the space no longer is simply connected since any closed loops around the puncture are non-contractible. In three dimensions, however, the loop can get wrapped around the hole through the third dimension, after which it can be contracted to a single point. In terms of exchange operators, we must, therefore, enforce that two exchanges is equal to doing nothing, i.e., $ P \u0302 2 = I$, which implies that the eigenvalue of $ P \u0302$ can only be ±1, which correspond to fermions $ ( \u2212 1 )$ and bosons $ ( + 1 )$. However, when the loop is non-contractible, the eigenvalue of $ P \u0302$ may be

*any*complex phase $ e i \theta $, which is why such quasi-particle excitations are known as anyons. Note that only the overall topology is relevant here and not the particular shape of the loop, which is reflected in Eq. (6) by the fact that it is independent of the metric. This is one of the defining properties of

*topological field theories.*

^{14,68,69}

#### 1. Charge–flux attachment

^{70}The Gauss's law

*ρ*does not only give rise to a diverging electric field, but to a scalar magnetic field as well. If we further integrate over the charge distribution and convert the $ \u2207 \xb7 E$ integral to a surface integral we see that it vanishes at large distance scales since the $E$ decays as $ r \u2212 1$. Consequently, we are left with

*Q*is the total charge and $\Phi $ the magnetic flux. Thus, every charged particle carries magnetic flux and one cannot have one without the other. This is exactly what happens in quantum Hall fluids.

^{71,72}As the electrons are subjected to increasingly strong external magnetic fields, they become trapped to tight cyclotron orbits around the magnetic flux lines and the Chern–Simons term begins to dominate the physics of the system. Note the similarity to the Aharanov–Bohm experiment where the electron couples to the flux tube via the gauge potential and acquires a phase that depends on the magnetic flux. In the quantum Hall systems, the electrons couple to the flux quanta via the Chern–Simons gauge field, and similarly, the phase acquired by the wave function depends on the filling fraction $ \nu = N e / N \varphi $, which is the ratio of the number of electrons

*N*to the number of flux quanta $ N \Phi $. Different values of

_{e}*ν*correspond to different anyon models determined by the Chern–Simons coupling constant

*k*. For instance, the $ \nu = 5 / 2$ system is expected to host the long sought-after Ising anyons, which we shall discuss further in Sec. II C. Moreover, systems with $ \nu = 12 / 5$ are predicted to be inhabited by the so-called Fibonacci anyons,

^{73–76}which set the golden standard for topological quantum computation. In Sec. III A 1, we shall see that discrete gauge theories give rise to a similar phenomenon described above without the additional Chern–Simons term added. The mathematical structures describing such models are known as quantum doubles, which is the main focus of our discussion.

### B. Symmetry classification of topological particles

The emergent charge–flux attachment concept discussed in Sec. II A is unique to (2 + 1)-dimensions. Imagine then that there are *n* flux tubes enclosed by a loop. The loop may be continuously deformed such that it can be decomposed into *n* distinct loops, each of which encircles one tube of magnetic flux [see Figs. 3(d)–3(f)]. This means that the total flux *αB* calculated in Eq. (4) now is $ n \Phi $. The integer *n* counts the phase winding, which is why it is known as the *winding number*, and as illustrated in Fig. 3, it must be a topological invariant. Similarly, we illustrate in Figs. 3(a)–3(c) how loops can be attached, which allows for the combined flux to be deduced.

*homotopy theory.*

^{77,78}In this framework, the classification is carried out by calculating the eigenvalues of loop operators corresponding to the encirclement

*γ*of excitations. These loop operators are known as

*Wilson loops*$ W \gamma $ and are defined as

^{79}

*π*

_{1}, see Appendix A for definition, also known as the fundamental group, over the $ U ( 1 ) \u2243 S 1$ manifold. This yields $ \pi 1 ( U ( 1 ) ) \u2243 \pi 1 ( S 1 ) \u2243 \mathbb{Z}$, since one can only encircle a circle

*S*

^{1}an integer number of times (see Fig. 4). Considering the total winding of two fluxons $ | n \u27e9$ and $ | m \u27e9$, where $ n , m \u2208 \mathbb{Z}$, their joint state is given by

The group of integers possesses abelian structure, which is why the flux–charge composites considered here are abelian anyons. However, we may also consider more general gauge theories in which the topology of the gauge group is more complicated and for a gauge group *G*, the fundamental group $ \pi 1 ( G )$ need not be isomorphic to the group of integers. Contrary to bosons and fermions, if two flux-charge composites in such a theory were to be exchanged twice, the wave function may not return to its original state as an arbitrary phase factor $ P \u0302 2 = e i \theta $ may be acquired.

*θ*can be traced back to the Chern–Simons action in Eq. (7), which is an abelian gauge theory. Therefore, more interesting theories can be constructed with a ground state degeneracy by promoting the Chern–Simons field $ a \mu $ of the $ U ( 1 )$ theory to a $ SU ( N )$ gauge field similar to the one in Yang–Mills theory, which gives rise to a non-abelian theory. Non-abelian Chern–Simons theories result from the action

^{14–16}

### C. Ising anyon $ SU ( 2 ) 2$ topological quantum computer

^{21}Amplitudes of processes involving Ising anyon are governed by a particular instance of non-abelian Chern–Simons theory. Specifically, the $ SU ( 2 ) k$ theory with

*k*= 2, where the integer coupling constant

*k*in Eq. (14) is also known as the level of the theory. Each level

*k*gives rise to a unique theory pertaining to a particular deformation of the full $ SU ( 2 )$ representation space. The integer

*k*plays the role of a deformation parameter so theories labeled by different

*k*values host different types of anyons with different topological charges. Ising anyons, in particular, have the topological charges $ 1 , \sigma $, and

*ψ*(anyons), which obey the following fusion rules:

*σ*is known as the Ising anyon, and

*ψ*as the

*ψ*-anyon. Level

*k*= 2 non-abelian Chern–Simons theory is believed to bear relevance to fractional quantum Hall fluids with filling fraction $ \nu = 5 / 2$, which is why Majorana fermion quasi-particles have been predicted to emerge in such systems.

^{28–30}They are also believed to be found in topological superconductor–semiconductor nanowires

^{32–39}and chiral

*p*-wave paired Fermi superfluids.

^{23–25,27,80}In a topological quantum computer based on Ising anyons, the fusion product $ \sigma \u2297 \sigma = 1 \u2295 \psi $ constitutes the qubit where the two possible measurement outcomes correspond to $1$ (the vacuum) and

*ψ*(the

*ψ*-anyon). In order to process the information encoded in the fusion product, and thus, to carry out computation, logic gates must be implemented. The special trait possessed by non-abelian anyons that distinguish them from bosons and fermions is that by interchanging them, an implementation of a braid group representation is realized, as opposed to that of an abelian permutation group. These transformations may then be used as logic gates. In Fig. 5, a computational process is illustrated, where a set of Ising anyons are created from the vacuum and then braided according to the quantum circuit one wishes to implement. Finally, the anyons are brought together, or fused, to read out the outcome of the computation. Since any deformation of the braid that does not change the topology leave the transformation invariant, the braids are protected from external noise that otherwise would cause decoherence and the computation to fail. In a one-qubit system, there are three anyons participating in the computation and, thus, two braid matrices implementing the braiding of the first with the second anyon, and the second with the third, respectively. The braid matrices in the Ising anyon model are

However, this model is computationally non-universal, meaning that the braid group generated from *σ*_{1} and *σ*_{2} is finite and is, thus, only capable of implementing a finite set of unitary rotations of the Bloch sphere. In order to be able to implement every logic gate, we need a braid group of infinite order that is generating a topologically dense cover in $ SU ( 2 )$, so that any rotation of the Bloch sphere can be realized. Nevertheless, universality can be achieved by supplementing the set of braids with an additional conventional operation at the cost of sacrificing the complete topological protection.^{81,82} Such a quantum computer is, therefore, to be regarded as a hybrid as arbitrary computational processes will still rely on error correcting protocols to some extent. Next, we shall redirect our attention toward bosonic systems and their quantum double structure in phases with spontaneously broken symmetry.

## III. QUANTUM DOUBLE ANYONS

In Sec. II A, we described how two-dimensional electrodynamics with interactions governed by continuous $ U ( 1 )$ gauge symmetry give rise to an Aharonov–Bohm like phase. The emergence of this phase is due to the punctures caused by the vortex-like excitations (fluxons), which, thus, play the role of flux tubes. We also saw that a modified Gauss's law emerged as a consequence of the additional Chern–Simons term, which lead to a flux-charge attachment, and thus, to fractional statistics. Topological physics can also be realized in the absence of such a term via the Higg's mechanism, since the gauge bosons of the theory may acquire mass, thus resulting in a topological field theory, as the non-topological interactions are rendered short range.

### A. The general picture

#### 1. Discrete gauge theory in (2 + 1)-D

Here, we shall consider bosonic condensates which have undergone spontaneous symmetry breaking from a continuous symmetry group *G* to a finite discrete subgroup *H*, where each subgroup *H* serves as a signature of a low-temperature phase. We refer to this group as the isotropy group of the condensate as it does not alter the physical properties of the system. However, since all cosets of the isotropy group possess the same structure, the full order parameter manifold $M$ is defined by the quotient space $ M = G / H$. Just like in Sec. II B, where it was described how the fluxons can be classified by means of the first homotopy group over the order parameter manifold $ U ( 1 )$, here, the excitations are classified according to $ \pi 1 ( G / H )$. However, if *H* is discrete and *G*/*H* is simply connected, they are connected by an isomorphism so $ \pi 1 ( G / H ) \u2243 H$, which entails that all information about the excitations is encoded in *H*, so that the elements of *H* may be used for labeling the excitations. However, the exact classification is slightly more subtle since charge conservation must be respected. That is, if we regard two fluxons $ h i , h j \u2208 H$ and bring the *h _{i}* fluxon around the

*h*one, it returns in a transformed state $ h i \u2032$. Flux conservation, thus, enforces the condition $ h i \u2032 h j = h j h i$, which further entails that the state must transform under conjugation $ h i \u2032 = h j h i h j \u2212 1$. Conjugation defines an equivalence relation on the group so the fluxons should, therefore, be classified according to the conjugacy class (CC) partitioning, which consequently implies that all particles labeled by an element within the same conjugacy class are indistinguishable. This phenomenon is known as

_{j}*flux metamorphosis,*

^{83,84}and if the residual group

*H*is non-commutative, there are generally more than one element in each conjugacy class which together span a multi-dimensional Hilbert space inhabited by a non-abelian fluxon.

#### 2. Higg's mechanism and symmetry breaking

*g*is the coupling constant and

*T*is a generator of

_{a}*G*. The matter fields transform according to various unitary irreducible representations (UIRs) $ \Gamma i$ of

*H*. Similarly to the $ U ( 1 )$ bundle in Fig. 4, we are now dealing with an

*H*bundle, where

*H*is attached to each point in space, acting on the matter field according to its UIRs. Such fields are called

*chargeons*.

Inheriting the nomenclature adopted in the Aharonov–Bohm experiment, the fluxons play the role of flux tubes while the matter fields correspond to electric charges, or *chargeons*, which are arranged according $ \Gamma i$. If a chargeon labeled by some UIR $ \Gamma i$ encircles a fluxon labeled by some element *h _{j}* the Aharonov–Bohm phase is given by $ \Gamma i ( h j )$. Thus, due to the minimal coupling introduced in the covariant derivative in Eq. (21), we have a notion of charge–flux attachment akin to that in Eq. (10). Such composite objects are called

*dyons*. However, the labeling of the dyons is subtle and requires a more careful analysis. Following Ref. 85, we shall consider the Aharonov–Bohm experiment with two dyons where one of them is hidden between two slits in a plate placed in front of a screen. The flux part of the two dyons is labeled by

*h*and

_{i}*h*, respectively, so if the first dyon goes through the left slit, the flux part of the dyon between the slits transforms according to $ h j \u2192 h i h j h i \u2212 1$. However, if the first dyon goes through the right slit, the second one sitting between the slits is left invariant, i.e., $ h j \u2192 h j$, and consequently, we have asymmetry between the two beams. This is due to the fact that there is a Dirac string connecting the fluxon–anti-fluxon pairs, which is crossed only when passing the fluxon on one of its sides (see Fig. 6). Owing to this correlation, we only have constructive interference if

_{j}*h*and

_{i}*h*commute since the

_{j}*h*will slip through so that $ h j \u2192 h i h j h i \u2212 1 = h j h i h i \u2212 1 = h j$. This implies that the charge attached to

_{i}*h*must transform under UIRs of elements that commute with

_{j}*h*. Such a set of elements always possesses group structure and is known as the centralizer group $ Z ( h j )$ of

_{j}*h*. In conclusion, the adequate labeling of the dyons is given by the conjugacy classes

_{j}*C*partitioning

_{i}*H*(the fluxon part) and the UIRs of the centralizers of the conjugacy classes $ \Gamma j ( Z ( C i ) )$ (the chargeon part). We have now dissected the structure of the quantum double construction, which revealed that fluxon Hilbert spaces are spanned by the elements of the conjugacy class considered. That is, a generic fluxon state is a coherent superposition $ \u2211 i c i | h i \u27e9 \u2208 \u2102 [ H ]$, where

*h*are elements within the same conjugacy class and

_{i}*c*are coefficients belonging to the field $\u2102$. Moreover, the chargeon Hilbert space is a space of functions $ F ( H )$ on

_{i}*H*(the centralizer UIRs), which means that the full quantum double Hilbert space is given by their tensor product $ F ( H ) \u2297 \u2102 [ H ]$. Before we move on and discuss the quantum double algebra in more detail, we shall provide an example of a simple quantum double model based on the group $ \mathbb{Z} N$, which will help us to motivate the various components of the structure.

### B. $ \mathbb{Z} N$ lattice electrodynamics—A simple example

*D*lattice with a continuous group

*G*[ $ U ( 1 )$ for instance] that is spontaneously broken down to $ \mathbb{Z} N$ so that there are

*N*degrees of freedom per site. This is an abelian model where the gauge field is inducing transformations in the group $ \mathbb{Z} N$.

^{86–88}As such, a theory measurements of charges is still possible through the Aharonov–Bohm effect, despite the electric fields vanish globally as the $ U ( 1 )$ symmetry is broken. That is, Gauss's law breaks down globally, since the photons have become massive and decay. Thus, the particles are only interacting via topology. The Hamiltonian can be constructed from two distinct types of four-body interactions given by the “plaquette” operators $ A \u25a1 i = Z k \u2020 Z k + 1 Z l \u2020 Z l + 1$ and the “star” operators $ B + i = X k X k + 1 \u2020 X l \u2020 X l + 1$ (see Fig. 7),

*Z*and

*X*are Weyl matrices defined as

*N*. Note that these matrices reduce to the standard Pauli

*X*and

*Z*if we set

*N*= 2. A nice property of this model is that all terms in the Hamiltonian commute since there are always two overlapping

*X*and

*Z*operators for the adjacent plaquettes and stars, and all other terms commute trivially since they do not overlap and, thus, occupy different blocks in their matrix representations. Consequently, all terms simultaneously diagonalize the Hamiltonian, which further entails that the ground state $ | \Psi 0 \u27e9$ corresponds to the maximum eigenvalue of all plaquette and star operators, i.e., $ A \u25a1 | \Psi 0 \u27e9 = | \Psi 0 \u27e9$ and $ B + | \Psi 0 \u27e9 = | \Psi 0 \u27e9$. The excitations are, thus, created by violating these conditions. For instance, we can create a particle anti-particle pair on two adjacent plaquettes by applying an

*X*operator on the edge they share since $ A \u25a1 ( X k | \Psi 0 \u27e9 ) = \omega \xb1 k X k ( A \u25a1 | \Psi 0 \u27e9 ) = \omega \xb1 k ( X k | \Psi 0 \u27e9 )$ due to the algebra defined in Eq. (24), where ± refers to the eigenvalues corresponding to the two plaquettes, + for the particle and − for the anti-particle. Similarly, we can create pairs at the center of the stars as $ B + ( Z l | \Psi 0 \u27e9 ) = \omega \xb1 l ( Z | \Psi 0 \u27e9 )$. Note that if we consider an arbitrary state $ | l \u27e9$, the

*X*and

*Z*operators act according to

*X*operator is rotating the state like a gauge transformation similar to the chargeon and that

*Z*is pulling out a phase like the fluxon. We can, thus, view plaquettes occupied by fluxons as magnetic flux tubes, which the chargeons interact with via the $ \mathbb{Z} N$ gauge potential. If there are no fluxons present the gauge potential is everywhere flat and, therefore, there is no gauge transformation implemented on the chargeons. Furthermore, the excitations can be moved around by applying string operators, which create and annihilate excitations in a sequence which effectively translates the particle in space. This allows us to perform braiding and fusion with the chargeons and fluxons. In fact, the chargeons live on the Fourier dual to the lattice inhabited by the fluxons, which means that their trajectories should be interpreted in reciprocal space. This insight will help us to interpret the real physical objects realizing the chargeons. A discussion on the interpretation of chargeons in spinor BECs is provided in Sec. III G.

*C*of the symmetry group (the fluxon part) and an UIR of the centralizer of this conjugacy class $ \Gamma j ( Z ( C i ) )$. This is the essence of the quantum double structure. The structure is doubled via a Fourier duality, which allows for a unified description of the two excitation spectra. A dyon can be viewed as a fusion product of a fluxon and a chargeon and since the fluxons correspond to the plaquette operators $ A \u25a1$, which project out the flux and the chargeons to the gauge transformations $ B +$, the dyons must be classified according to the UIRs $ \Lambda ( A \u25a1 B + )$. All fusion rules in this model are presented in Eq. (26),

_{i}We can also derive the monodromy by creating string operators that, for instance, are bringing an *e* around an *m*, as in Fig. 8(b). Note that the loop corresponds to the same operation as the product of the plaquettes it encircles. All of the plaquettes return eigenvalue 1 except the one on which the fluxon is residing. That is, $ S loop | \Psi \u27e9 = \u220f \u25a1 A \u25a1 | \Psi \u27e9 = A \u2009\u2009\u22a1 | \Psi \u27e9 = \omega | \Psi \u27e9$. The eigenvalue of a braiding process where an *e* is braided twice with an *m* is, thus, given by *ω*. This is, in fact, a discrete version of Stokes' theorem where the plaquette inhabited by the fluxon carries circulation so that the loop can be smoothly deformed around this plaquette, without affecting the outcome. As shown in Fig. 8(b), the Dirac string connecting *m* and $ m *$ is crossed once independent of the deformation of the loop, which results in a phase *ω* due to the relation between *X* and *Y* in Eq. (24). We may, thus, conclude that topological equivalence is naturally encoded in this model. The plaquette operator $ A \u22a1$ acting on the fluxon can be regarded as a measurement operator that is projecting out the flux. In electromagnetism, this is equivalent to $ \u2207 \xd7 A = B$ and the action of the vertex operator, thus, represents something akin to Gauss's law $ \u2207 \xb7 \u2207 \varphi = \u2212 \rho \epsilon 0$. Generally, the flux is measured by *Z* operators while the *X* operators correspond to gauge transformations. Hence, the dyons, which can be regarded as the elementary objects of the quantum double structure, must be arranged according to the UIRs implementing the action of a flux measurement followed by a gauge transformation.

#### 1. Chargeon–fluxon duality

*N*'th root of unity such that $ ( \Gamma n ( 1 ) ) N = 1 \u2009 \u2009 \u2200 n$. This fixes the number of UIRs to

*N*and the maps are given by $ \Gamma n ( m ) = e i ( 2 m n \pi / N ) = \omega n m$, where $ m \u2208 \mathbb{Z} N$. These UIRs are exactly the elements representing the fluxons, which means that we can interpret the fluxons and chargeons as duals of one another. The fluxons, thus, perceive the chargeons as flux tubes in the same way the chargeons experience the fluxons as flux tubes, to which they couple via the gauge field. Furthermore, for the special case

*N*= 2, we have that the operators

*X*and

*Z*are hermitian (the standard Pauli matrices), that is $ X = X \u2020$ and $ Z = Z \u2020$, which implies that they must square to the identity so that all excitations are their own anti-particles. Hence, in addition to the $ e \u2194 m$ symmetry, we also have $ e \u2194 e *$ and $ m \u2194 m *$ symmetry in the $ \mathbb{Z} 2$ model. In Sec. III E 1, we shall see an explicit example of a physical system realizing this particular structure.

### C. The quantum double construction

*m*) and chargeons (

*e*), and together they form a composite object, the dyon, which can be classified according to UIRs $ \Pi ( P h g )$ implementing a gauge transformation

*g*followed by a flux measurement

*P*. These elements constitute the building blocks of the quantum double of a group. In the example provided in Sec. III B, the quantum double is given by $ D ( \mathbb{Z} N ) = { P h i g j} i , j = 0 N \u2212 1$. Now, due to flux metamorphosis and the property $ P h P h \u2032 = P h \delta h , h \u2032$ of the projection operators, we enforce the following multiplication rule:

_{h}*C*(the fluxons) and the UIRs of the corresponding centralizers $\Gamma $, with basis $ | h i C , u j \Gamma \u27e9$, where $ i = 0 , 1 , \u2026 , | C |$ and $ j = 0 , 1 , \u2026 , dim ( \Gamma )$. The action of the image of $ \Pi ( P h g )$ can be thought of as performing a gauge transformation

*g*subsequently followed by a flux measurement

*P*which yields

_{h}^{41,84}

*a*is defined via $ h k = g h i g \u2212 1$, where $ a i \u2208 H$ is the element that maps the conjugacy class representative element $ h * C \u2208 C$ to $ h i C = a i h * C a i \u2212 1 \u2208 C$. For a more detailed discussion, the reader may consult Appendix B 2 or, e.g., Refs. 41 and 84. Now, if we tensor two quantum double representations (dyons), we find the fusion product of the two particles by decomposing the reducible tensored representation into its irreducibles subblocks. For instance, consider particle

_{k}*a*and

*b*with fusion channel

*c*,

*C*is brought around a distant flux

*A*, then it returns as $ A C A \u2212 1$ so that $ C \u2212 1 A C A \u2212 1 \u2260 I$ when it is fused with the anti-flux $ C \u2212 1$. These quantum numbers are labeling delocalized charges that are not attached to the positions of the fluxes and are often referred to as

*Cheshire charges*(inspired by the Cheshire cat in

*Alice in Wonderland*who suddenly vanished while leaving the reminiscent of its grin behind). The integer $ N a b c$ can be calculated by applying the projection operator that is mapping the reducible tensored representation $ \Pi a \u2297 \Pi b$ onto the irreducible orthogonal Π

_{c}blocks. That way one can disassemble the vector space corresponding to $ \Pi a \u2297 \Pi b$ into its irreducibles where the multiplicity of each irreducible is given by

^{41}

^{,}

^{89}

^{,}

*Shur's lemma*

^{90}which states that if a complex valued matrix

*A*commutes with the image of some UIR $\Gamma $, then the matrix

*A*must be represented as a complex matrix that is proportional to the identity in the representation $\Gamma $, i.e., $ Rep ( A ) = e i \theta \Gamma ( A ) 1 \Gamma ( A )$. Given some anyon model, we may compute the phases corresponding to each of the particles in the model and then store them in a matrix by stacking them on the diagonal. The resulting matrix is known as the modular T-matrix, which together with the S-matrix discussed above, span the modular group $ SL ( N , F )$ of conformal transformations which encodes the conformal structure of anyon models. Note that this fits into the quantum double construction as the flux

*h*of a dyon, per definition, commutes with its centralizer

*Z*(

*h*). Thus, considering some UIR $\Gamma $ of

*Z*(

*h*), we may establish a generalized spin-statistics connection

^{91,92}$ e i 2 \pi s = e i 2 \pi \theta \Gamma ( h )$. This relation can be understood from a graphical perspective by considering the twisting of the world lines of two anyons as they are braided around one another. Here, it is useful to think of the world lines as ribbons (see Fig. 9), as the twisting becomes more apparent. Figure 9 is a pictorial illustration of the topological equivalence between the double interchange $ R 2$ of

*a*and

*b*[this is exactly the monodromy illustrated in Figs. 4 and 8(b)] and the twisting of their world lines. From this, we can straightforwardly deduce the monodromy matrix elements as

*s*are the topological spins of the particles. We shall return to this in Secs. III E and III F and calculate the spins explicitly for various excitations in spinor BECs.

_{i}### D. Vortex chromodynamics picture

^{93}by defining the superfluid velocity $ v s = ( \u210f / m ) \u2207 \theta $ as the electric field and the mass density of particles $ \rho s = m | \Psi | 2$ as the magnetic field, jointly forming a Maxwellian field strength tensor $ F \mu \nu $. Alternatively, the phase change $ \u2202 t \theta $ may be assigned to the role of magnetic field to avoid nonzero vacuum magnetization.

^{94}Building on this idea, it is conceivable that a superfluid comprised of spinful particles may reproduce a generic Yang–Mills theory.

^{95}Indeed, it was demonstrated in Refs. 96–98 that certain aspects of quantum chromodynamics (QCD), such as quark confinement, can be simulated in BECs with spin. Other forms of gauge theories, such as that involving an emergent Ising gauge field,

^{99}have also been studied using BECs comprised of spinful particles. By adopting a mean-field description (see Appendix E), a Yang–Mills field strength tensor may be constructed by applying the Madelung transformation $ \Psi a ( r , t ) = | \psi a ( r , t ) | e i \theta a ( r , t )$ to the spinor components. This allows one to separate the phase and the density which can be interpreted as Yang–Mills field strength and gravity, respectively. However, if we consider a static phase, the full field strength $ F \mu \nu a$ is completely defined by the spin current (phase gradient)

^{47}

^{,}

*a*denotes the components of the spin. In this picture, the particles can be thought of as color charged “quarks” which are interacting via a generalized electric field—the “strong” force. However, if we instead consider a viewpoint in which one particle is a charged analog quark, and a second particle is a generalized flux tube, then the interpretation of Eq. (35) is that it corresponds to a non-abelian gauge field $ A \mu a$, such that instead $ F \mu \nu a = \u2202 \mu A \nu a \u2212 \u2202 \nu A \mu a + i g [ A \mu a , A \nu a ]$, where

*g*is the coupling. Since the superfluid velocity field arises due to the non-vanishing of the phase gradient $ \u2207 \theta a$, the Aharonov–Bohm phase acquired as the vortex is encircled is given by the contour integral of $ A \mu a = [ v s ] a$. Furthermore, the vorticity is given by $ \omega = \u2207 \xd7 v s$ allowing it to be interpreted as the magnetic part of the field strength. If the vorticity, or field strength, is only non-vanishing inside of the vortex core the resulting interaction is topological. In the Gross–Pitaevskii equation (GPE)

^{47}which governs the dynamics of a BEC, the covariant derivative encoding the interactions mediated by $ A \mu a = [ v s ] a$ is obtained by transforming to a frame co-moving with the vortex. The Laplacian in the kinetic term can then be expressed as $ \u2207 2 \u2192 ( \u2207 + \kappa a [ v s ] a ) 2 ,$ so the charge

*g*corresponds to the generalized circulation quantum

*κ*. This term is responsible for the interactions within the quantum double as it encodes the response of the vortex due to the motion of the surrounding atoms. Note also that this Laplacian is taking on exactly the same form as that giving you the kinetic energy of an electron coupled to a magnetic field in, e.g., the quantum Hall effects. In analogy with QCD, we may, therefore, say that the particles of the theory carry color charge which is preserved under the interactions mediated by $ A \mu a$. This quantum number can, thus, be identified as the topological spin, or generalized quantum of circulation, of the particles defined in Subsection III C. The vortices represent defects, which carry a non-zero field strength to which the chargeons couple topologically via $ A \mu a$ (or the reversed if implementing the S-matrix duality which is to be discussed in Secs. III E and III F). A generic dyonic state can, in analogy with Gell-Mann's eightfold way, thus, be regarded as a superposition in an

^{a}*N*-tuplet color space, where $ N = | C | \xd7 dim ( \Gamma )$. Interestingly, as demonstrated in Ref. 100, by choosing the density profile of the system in a specific way, the gravity-like force arising from the quantum pressure can be implemented topologically. In particular, arbitrary topological phase rotations can be achieved which may be employed as additional quantum logic gates. Hence, such a vortex QCD plus gravity may result in a universal platform for topological quantum computation.

### E. Anyons in spin-0 and spin-1 BECs

In a spinor BEC, the order parameter representing the phase under consideration is not a scalar but a spinor. The order parameter wave function of a spin-*F* condensate belongs to the complex vector space $ H = \u2102 2 S + 1$, which is isomorphic to the real space $ \mathbb{R} 4 S + 2$, i.e., $ H = \u2102 2 S + 1 \u2243 \mathbb{R} 4 S + 2$, where *S* is the spin. This entails that the order parameter is a map onto a surface isomorphic to a sphere, that is, it maps onto the manifold $ M \u2243 S 4 S + 2 \u2212 1 = S 4 S + 1$. Consequently, we can conclude that the full symmetry group of the system is $ SO S ( 3 )$ accompanied by phase invariance, i.e., $ SO S ( 3 ) \xd7 U ( 1 )$. Here, we consider two types of condensates with spin-0 and spin-1 degrees of freedom, respectively, which are both governed by abelian theories. In the *S* = 0 case, we know that the corresponding wave function is a scalar $ \Psi ( r , \theta ) = | \Psi ( r , \theta ) | e i \theta $. Moreover, if we break the $ U ( 1 )$ symmetry, we know that each subgroup is inheriting the commutative property, more precisely each subgroup must be isomorphic to a $ \mathbb{Z} N$ group, thus giving rise to a quantum double akin to the lattice model discussed in Sec. III B. In the spin-1 case, we have three low-temperature inert states $ ( 1 , 0 , 0 ) T , ( 0 , 0 , 1 ) T$, and $ ( 0 , 1 , 0 ) T$, which correspond to the two ferromagnetic states and the polar state, respectively. The symmetry groups of these states are given by rotations about the axes parallel to the states so that $ SO S ( 3 ) \xd7 U ( 1 )$ reduces to $ SO ( 2 ) S z \xd7 U ( 1 )$ for the ferromagnetic states and $ D \u221e \xd7 U ( 1 )$ for the polar state. Here, $ D \u221e$ denotes the infinite dihedral group which also includes reflection symmetry, as opposed to $ SO ( 2 ) S z$. As for discrete symmetries in these phases, we can conclude that we only have one corresponding to the reflection part of the $ D \u221e$ symmetry, which can be represented by the cyclic group *C*_{2}.

A convenient way to deduce the symmetries is to consider a graphical representations of the spinorial states, such as the spherical harmonics or the Majorana star representation.^{47} Note that since these groups are abelian, spin-0 or a spin-1 condensates are not able to support non-abelian anyons as their natural excitations. This fact can be established from the analysis outlined in Sec. II B, where we discussed how the topological excitations in planar systems can be classified according to the first homotopy group $ \pi 1 ( G / H )$ of the coset space *G*/*H* generated by the subgroup *H*.

#### 1. The quantum double $ D ( C 2 )$ of C_{2}

*I*and one

*π*-rotation

*r*, which are the two conjugacy classes of the group, and consequently the centralizer has two UIRs, the 1

*D*trivial representation $ \Lambda sym$ and the 1

*D*asymmetric representation $ \Lambda asym$. The character table for this model is provided in Table I. We, thus, conclude that there are four abelian anyons in the model corresponding to the UIRs of the quantum double element $ P h g$. In particular, we have a vacuum sector $1$, one fluxon

*m*, one chargeon

*e*and one dyon

*ε*. The fusion rules can, thus, be summarized as

. | I
. | r
. |
---|---|---|

$ \Lambda sym$ | 1 | 1 |

$ \Lambda asym$ | 1 | −1 |

. | I
. | r
. |
---|---|---|

$ \Lambda sym$ | 1 | 1 |

$ \Lambda asym$ | 1 | −1 |

*N*= 2 instance of the model discussed in Sec. III B. As for braiding, we can conclude that the fluxon

*m*as well as the chargeon

*e*are both bosons. This fact is evident from the character Table I, from which we can also deduce that the dyon

*ε*is a fermion. Moreover, while the bosons have trivial self-braiding, winding the chargeon around he fluxon (or the fluxon around the chargeon) yields a factor −1, which can be derived from the generalized spin-statistics theorem in Eq. (34). The topological spins

*s*can be deduced from the character table as $ \chi / d = e i 2 \pi s$, where

*χ*is the character and

*d*is the dimension of the corresponding UIR. This relationship can be derived from the definition of the Wilson loop defined in Eq. (11), which is nothing but a measurement of the monodromy illustrated in Fig. 4, which is equivalent to the character values of the representation under consideration. Reading off the character table, we may establish that $ s 1 = s m = s e = 0$ and $ s \epsilon = 1 / 2$ as expected for bosons and fermions. We next compute the S-matrix of the model which we briefly encountered in Sec. III C, as well as the T-matrix, which together generate the modular group $ SL ( 2 , \mathbb{Z} )$. Applying Eq. (33) yields the matrix

*i*and

*j*, where the indices run through the charges $ 1 , e , m , \epsilon $. Moreover, the T-matrix is given by stacking the twist factors $ e i 2 \pi s$ of the anyons on the diagonal which yields the T-matrix

#### 2. Particle-vortex duality in $ D ( C 2 )$

Interestingly, as shown in Ref. 101, the modular S-matrix can be interpreted as a generalization of the quantum Fourier transform based on the representation theory of the quantum double. In fact, this is exactly the transformation that is implementing the chargeon–fluxon duality discussed in Sec. III B, where the chargeons live on the direct lattice and the fluxons live on the reciprocal lattice. To illustrate this explicitly, all we have to show is that the set of characters corresponding to the chargeon transform, under the action of *S*, into the set of characters corresponding to the fluxon, since these sets form an orthonormal basis for their respective Hilbert space. We first need the representations labeling the two anyons which can be worked out by virtue of Eq. (29), which defines the quantum double action. Using $ \Pi e = \Pi I asym$ to denote the representation corresponding to the chargeon *e* and $ \Pi m = \Pi r sym$ to denote that of the fluxon *m*, then the image of their 1*D* quantum double representations, given the action defined in Eq. (29), is given by Table II.

. | $ P I I$ . | $ P I r$ . | $ P r I$ . | $ P r r$ . |
---|---|---|---|---|

$ \Pi I asym ( P h g )$ | 1 | −1 | 0 | 0 |

$ \Pi r sym ( P h g )$ | 0 | 0 | 1 | 1 |

. | $ P I I$ . | $ P I r$ . | $ P r I$ . | $ P r r$ . |
---|---|---|---|---|

$ \Pi I asym ( P h g )$ | 1 | −1 | 0 | 0 |

$ \Pi r sym ( P h g )$ | 0 | 0 | 1 | 1 |

Since these representations are one-dimensional, they yield the characters directly since the characters are computed as the trace of the elements in the representation image. In light of this, we may denote by $ \chi e = ( 1 , \u2212 1 , 0 , 0 ) T$ the set of characters for the chargeon and by $ \chi m = ( 0 , 0 , 1 , 1 ) T$ the set of characters for the fluxon. It can then be shown that indeed $ S \chi e = \chi m$, which proves that *S* is implementing a duality between the two anyons. Moreover, if we work out the characters corresponding to the dyon *ε*, we have that $ \chi \epsilon = ( 0 , 0 , 1 , \u2212 1 ) T$, and if we act with *S* we find that the state is left invariant, i.e., $ S \chi \epsilon = \chi \epsilon $, as expected since *ε* is a fluxon–chargeon composite so it must be left untouched if we swap the constituent anyons $ e \u2194 m$. This duality is exactly the same as the one discussed in Sec. III B 1 for *N* = 2. Here, we took a different route via the representation theory and the S-matrix and arrived at the same result, without any direct knowledge about the structure of the Hamiltonian, other than its symmetries. Next, we shall consider spin-2 systems which exhibit phases with non-abelian symmetry groups that may be capable of topological quantum computation.

### F. Anyons in spin-2 BECs

Unlike the spin-0 and spin-1 condensates, the spin-2, and in fact any spin $ F \u2265 2$, exhibit phases whose order parameters are invariant under non-abelian groups. The order parameters can be worked out from the mean-field theory described in Appendix E. In the unbroken phase, such order parameters have rotational $ SO ( 3 )$ symmetry as well as phase invariance $ U ( 1 )$, which results in the full symmetry $ SO ( 3 ) \xd7 U ( 1 )$, but if we consider instead the simply connected special unitary representation $ SU ( 2 )$, the group structure is given by $ G = SU ( 2 ) \xd7 U ( 1 )$. If this symmetry is broken to a subgroup $ H \u2282 G$, the full order parameter manifold is given by the coset space formed by taking the quotient *G*/*H*, i.e., $ M = SU ( 2 ) \xd7 U ( 1 ) / H$. Here, we will direct our attention toward two particular ground state phases, the binary tetrahedral phase and the biaxial-nematic phase. The particle content of the emerging anyon models can be derived by extending the analysis carried out in the abelian case, but as we shall see, these structures are much more complex which gives rise to vast particle spectra. The topological spins of the particles are, again, derived from the character theory in the same way as in Sec. III E.

#### 1. Binary tetrahedral phase and its quantum double $ D ( T * )$

^{47}The tetrahedral group comprises the symmetries of a tetrahedron and has 12 elements in total formed by π $ / 3$-rotations and reflections. The binary representation, however, has 24 elements due to the two-to-one double cover as each element $ t \u2208 T$ is mapped onto $ t , t \xaf \u2208 T *$, i.e., $ f : \u2009 \u2009 t \u2192 { t , t \xaf}$. The structure of $ T *$ can be represented as $ { a , b , c | a 3 = b 3 = ( a b ) 2 = \u2212 1 ,}$,

^{102}and since $ T * \u2282 SU ( 2 )$, we may pick the elements $ a = 1 / 2 ( 1 + i \sigma x +\u2009 i \sigma y + i \sigma z )$ and $ b = 1 / 2 ( 1 + i \sigma x + i \sigma y \u2212 i \sigma z )$ as generators. It is straightforward to show that these satisfy the relationships between the generators of the group. Unpacking the structure, we first note that they both generate disjoint six-cycles $ \mathbb{Z} 6$, thus, comprising in total 12 distinct elements individually, which further implies that they must form another 12 distinct elements when combined, since there are 24 elements in the group in total. Noting that $ a b = i \sigma x$ and $ b a = i \sigma y$, and that $ i \sigma y i \sigma x = i \sigma z$ and $ a 4 = \u2212 a$ (and consequently $ a 5 = \u2212 a 2$ where the same relations hold for

*b*), we can conclude that all of the 24 elements can be generated from $ \xb1 a = \xb1 \sigma \u0303$ (or

*b*) and its product with $ \xb1 i \sigma i$ for $ i \u2208 { x , y , z}$. In particular, we have the following partitioning into conjugacy classes (CC):

^{46}

*i*= 4, 5, 6), due to Burnsides's theorem

^{90}in the theory of finite groups. Furthermore, we must have a faithful two-dimensional complex representation since the binary tetrahedron is naturally embedded in $ \u2102 2$. We can construct yet another two two-dimensional UIRs by taking the tensor product of it with the two non-trivial one-dimensional UIRs, thus resulting in a total of three inequivalent two-dimensional representations which we shall denote by $ \Lambda i T *$ (

*i*= 1, 2, 3). Now, as a collorary of Burnside's theorem, the order of the group must be equal to the sum of the UIR dimensions squared. Thus far, we have $ 3 \xd7 1 2 + 3 \xd7 2 2 = 15$, which implies that there must exist one three-dimensional representation since $ 24 \u2212 15 = 3$. In order to find this representation, we may exploit the fact $ SU ( 2 )$ is the double cover of $ SO ( 3 )$, and hence, there must exist a two-to-one map between $ T *$ and

*T. T*has, owing its embedding, a natural three-dimensional faithful representation $ \Lambda 7 T *$, which is simply permuting the corners of the tetrahedron in real $ \mathbb{R} 3$ space, so by composing this map with the double covering map, we find a three dimensional UIR for $ T *$.

Instead of writing all of these matrices out explicitly as we did in the abelian *C*_{2} case, we are simply providing the corresponding character tables as all essential information can be extracted from there. The characters of the one-dimensional UIRs are found trivially since the traces of them are the same as the group elements themselves. As for the two-dimensional representations, we pick one element from each conjugacy class in Eq. (41) which we multiply by each of the one-dimensional UIRs and then take the trace, since the trace is invariant under conjugacy due to its invariance under cyclic permutations. Finally, the characters for the three-dimensional UIR can be found simply by defining a map *f* that sends each Pauli matrix *σ _{i}* in Eq. (41) to the corresponding three-dimensional Lorentz rotation matrix

*R*, since the Lorentz rotations $ SO ( 3 )$ and $ SU ( 2 )$ adhere to the same algebra. The characters for all of these seven representations can be found in Table III. Again, the topological spins of the anyons in the spectrum can be deduced from the character table via the relation $ e i 2 \pi s = \chi / d$, where

_{i}*s*is the spin,

*χ*is the character value and

*d*is the dimension. Reading off the first and the second column, corresponding to fluxons $ CC 1 ( T * )$ and $ CC 2 ( T * )$, we note that all characters are integers which means that they are all bosons. Next, we turn to the $ CC 3 ( T * )$ conjugacy class which has $ \mathbb{Z} 4$ as centralizer group. This can be concluded by noting that all elements in this set are pure rotations around the

*x*,

*y*, and

*z*axes, respectively. Hence, the only elements commuting with a representative from this conjugacy class is a rotation about the same axis. Considering the group structure, there can only be 4 of those (generated by the element itself), which leads us to the conclusion that the centralizer is $ \mathbb{Z} 4$. Moreover, each element in $ \mathbb{Z} 4$ is its own conjugacy class since it is an abelian group, and consequently, as per Burnside's theorem, there must be four UIRs $ \Gamma i$ ( $ i = 1 , 2 , 3 , 4$), which can be found simply by permuting the trivial representation which send all elements to 1, by rotations of $ \pi / 2$. We, thus, obtain the character table presented in Table IV.

. | $ CC 1 ( T * )$ . | $ CC 2 ( T * )$ . | $ CC 3 ( T * )$ . | $ CC 4 ( T * )$ . | $ CC 5 ( T * )$ . | $ CC 6 ( T * )$ . | $ CC 7 ( T * )$ . |
---|---|---|---|---|---|---|---|

$ \Lambda 1 T *$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

$ \Lambda 2 T *$ | 1 | 1 | 1 | ω | ω | $ \omega *$ | $ \omega *$ |

$ \Lambda 3 T *$ | 1 | 1 | 1 | $ \omega *$ | $ \omega *$ | ω | ω |

$ \Lambda 4 T *$ | 2 | −2 | 0 | −1 | 1 | 1 | −1 |

$ \Lambda 5 T *$ | 2 | −2 | 0 | $ \u2212 \omega *$ | $ \omega *$ | ω | $ \u2212 \omega $ |

$ \Lambda 6 T *$ | 2 | −2 | 0 | $ \u2212 \omega $ | ω | $ \omega *$ | $ \u2212 \omega *$ |

$ \Lambda 7 T *$ | 3 | 3 | −1 | 0 | 0 | 0 | 0 |

. | $ CC 1 ( T * )$ . | $ CC 2 ( T * )$ . | $ CC 3 ( T * )$ . | $ CC 4 ( T * )$ . | $ CC 5 ( T * )$ . | $ CC 6 ( T * )$ . | $ CC 7 ( T * )$ . |
---|---|---|---|---|---|---|---|

$ \Lambda 1 T *$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

$ \Lambda 2 T *$ | 1 | 1 | 1 | ω | ω | $ \omega *$ | $ \omega *$ |

$ \Lambda 3 T *$ | 1 | 1 | 1 | $ \omega *$ | $ \omega *$ | ω | ω |

$ \Lambda 4 T *$ | 2 | −2 | 0 | −1 | 1 | 1 | −1 |

$ \Lambda 5 T *$ | 2 | −2 | 0 | $ \u2212 \omega *$ | $ \omega *$ | ω | $ \u2212 \omega $ |

$ \Lambda 6 T *$ | 2 | −2 | 0 | $ \u2212 \omega $ | ω | $ \omega *$ | $ \u2212 \omega *$ |

$ \Lambda 7 T *$ | 3 | 3 | −1 | 0 | 0 | 0 | 0 |

. | 1 . | x
. | x^{2}
. | x^{3}
. |
---|---|---|---|---|

$ \Gamma 1 \mathbb{Z} 4$ | 1 | 1 | 1 | 1 |

$ \Gamma 2 \mathbb{Z} 4$ | 1 | i | −1 | −i |

$ \Gamma 3 \mathbb{Z} 4$ | 1 | −1 | 1 | −1 |

$ \Gamma 4 \mathbb{Z} 4$ | 1 | −i | −1 | i |

. | 1 . | x
. | x^{2}
. | x^{3}
. |
---|---|---|---|---|

$ \Gamma 1 \mathbb{Z} 4$ | 1 | 1 | 1 | 1 |

$ \Gamma 2 \mathbb{Z} 4$ | 1 | i | −1 | −i |

$ \Gamma 3 \mathbb{Z} 4$ | 1 | −1 | 1 | −1 |

$ \Gamma 4 \mathbb{Z} 4$ | 1 | −i | −1 | i |

The self-statistics, and hence the spin, can be read off from the second column in Table IV which reveals that the $ ( CC 3 , \Gamma 1 \mathbb{Z} 4 )$ dyon is a boson, the $ ( CC 3 , \Gamma 2 \mathbb{Z} 4 )$ is a spin- $ 1 / 4$ particle, $ ( CC 3 , \Gamma 3 \mathbb{Z} 4 )$ is a fermion and the $ ( CC 3 , \Gamma 4 \mathbb{Z} 4 )$ must be a spin- $ 3 / 4$ particle. Finally, by the same argument we applied to deduce the $ \mathbb{Z} 4$ centralizer, we can establish that the centralizer of the remaining conjugacy classes $ CC 4 ( T * ) , CC 5 ( T * ) , \u2009 CC 6 ( T * )$, and $ CC 7 ( T * )$ must be the six-cycle $ \mathbb{Z} 6$ since $ \sigma \u0303$ and $ \sigma \u0303 \sigma i$ (where $ i \u2208 { x , y , z}$) have order 6. Again, owing to the abelian structure of $ \mathbb{Z} 6$, it must have six UIRs whose characters are provided in Table V.

. | 1 . | y
. | y^{2}
. | y^{3}
. | y^{4}
. | y^{5}
. |
---|---|---|---|---|---|---|

$ \Gamma 1 \mathbb{Z} 6$ | 1 | 1 | 1 | 1 | 1 | 1 |

$ \Gamma 2 \mathbb{Z} 6$ | 1 | $\phi $ | $ \phi 3$ | −1 | $ \phi 4$ | $ \phi 5$ |

$ \Gamma 3 \mathbb{Z} 6$ | 1 | $ \phi 2$ | $ \phi 4$ | 1 | $ \phi 2$ | $ \phi 4$ |

$ \Gamma 4 \mathbb{Z} 6$ | 1 | −1 | 1 | −1 | 1 | −1 |

$ \Gamma 5 \mathbb{Z} 6$ | 1 | $ \phi 4$ | $ \phi 2$ | 1 | $ \phi 2$ | $ \phi 4$ |

$ \Gamma 6 \mathbb{Z} 6$ | 1 | $ \phi 5$ | $ \phi 4$ | −1 | $ \phi 4$ | $\phi $ |

. | 1 . | y
. | y^{2}
. | y^{3}
. | y^{4}
. | y^{5}
. |
---|---|---|---|---|---|---|

$ \Gamma 1 \mathbb{Z} 6$ | 1 | 1 | 1 | 1 | 1 | 1 |

$ \Gamma 2 \mathbb{Z} 6$ | 1 | $\phi $ | $ \phi 3$ | −1 | $ \phi 4$ | $ \phi 5$ |

$ \Gamma 3 \mathbb{Z} 6$ | 1 | $ \phi 2$ | $ \phi 4$ | 1 | $ \phi 2$ | $ \phi 4$ |

$ \Gamma 4 \mathbb{Z} 6$ | 1 | −1 | 1 | −1 | 1 | −1 |

$ \Gamma 5 \mathbb{Z} 6$ | 1 | $ \phi 4$ | $ \phi 2$ | 1 | $ \phi 2$ | $ \phi 4$ |

$ \Gamma 6 \mathbb{Z} 6$ | 1 | $ \phi 5$ | $ \phi 4$ | −1 | $ \phi 4$ | $\phi $ |

Even here we find some interesting dyonic particles with fractional spin $ s = ( 1 / 6 ) , ( 1 / 3 ) , ( 2 / 3 ) , ( 5 / 6 )$, which can be deduced from the second column. We have now worked out the entire particle spectrum of the quantum double of $ T *$ which is comprised of one vacuum, six pure fluxons, six pure chargeons, and 29 dyons, thus amounting to a total of 42 distinguishable particles. The fusion rules of these anyons can be obtained by first computing the S-matrix according to Eq. (33) and then employing, e.g., the Verlinde equation in Eq. (32). However, computing all such combinations would be a monstrous task so we will not present these here. The underlying principle is the same though as in the much simpler $ D ( C 2 )$ anyon model. Interestingly, as already pointed out in Sec. III C, fusing a fluxon with its anti-partner may result in a particle with Cheshire charge. This possibility stems from the reducibility of the tensored representation space of the two fluxons. If the resulting space has invariant subspaces, these subspaces correspond to multiplets, each of which is labeled by an invariant charge quantum number.

#### 2. Biaxial nematic phase and its quantum double $ D ( D 4 * )$

*D*

_{4}whose structure can be represented as $ D 4 * = { a , b | a 8 = 1 , b = a 2 , b a b \u2212 1 = a \u2212 1}$.

^{103}In words, we have an eight-cycle

*a*about a primary axis, say the z-axis, and a four-cycle

*b*which reverses

*a*under conjugation. In the $ SU ( 2 )$ representation, we may pick $ R z ( 2 \pi / 8 ) = 1 / 2 ( 1 + i \sigma z )$ as the $ \mathbb{Z} 8$ generator, and one can easily verify that either $ i \sigma x$ or $ i \sigma y$ works as an $ SU ( 2 )$ representation of

*b*, so we may pick for instance $ i \sigma x$. Note that $ i \sigma x i \sigma z = i \sigma y$, meaning that all elements can be expressed as products of $ i \sigma x , i \sigma y$, and $ \sigma \u0303 = 1 / 2 ( 1 + i \sigma z )$. Moreover, since $ D 4 *$ has sixteen elements in total, and the $ \mathbb{Z} 8$ subgroup generated by $ 1 / 2 ( 1 + i \sigma z )$ has eight elements, the remaining eight can straightforwardly be found by acting with $ \xb1 i \sigma x$ and $ \xb1 i \sigma y$ on $ \sigma \u0303$. These elements can be partitioned into seven conjugacy classes according to

^{46}

The conjugacy class structure and the centralizer UIRs of this group can be found by applying the same set of arguments as in the $ T *$ case, so we are simply jumping straight to the character tables here. Again, the abelian fluxons $ CC 1 ( D 4 * )$ and $ CC 2 ( D 4 * )$ have the entire group $ D 4 *$ as centralizer whose character table is provided in Table VI. The dyons with flux corresponding to $ CC 1$ and $ CC 2$ are consequently all bosons since all of the characters in the first and second column are integers. The centralizer of $ CC 3 ( D 4 * )$ and $ CC 7 ( D 4 * )$ is given by $ \mathbb{Z} 4$, which coincides with the centralizer of $ CC 3 ( T * )$, whose character table is already provided in Table IV. Finally, the centralizer for the remaining conjugacy classes $ CC 4 ( D 4 * ) , \u2009 CC 5 ( D 4 * )$, and $ CC 6 ( D 4 * )$ are given by the cyclic abelian group $ \mathbb{Z} 8$ whose character table is provided in Table VII.

. | $ CC 1 ( D 4 * )$ . | $ CC 2 ( D 4 * )$ . | $ CC 3 ( D 4 * )$ . | $ CC 4 ( D 4 * )$ . | $ CC 5 ( D 4 * )$ . | $ CC 6 ( D 4 * )$ . | $ CC 7 ( D 4 * )$ . |
---|---|---|---|---|---|---|---|

$ \Lambda 1 D 4 *$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

$ \Lambda 2 D 4 *$ | 1 | 1 | −1 | 1 | 1 | 1 | −1 |

$ \Lambda 3 D 4 *$ | 1 | 1 | 1 | 1 | −1 | −1 | −1 |

$ \Lambda 4 D 4 *$ | 1 | 1 | −1 | 1 | −1 | −1 | 1 |

$ \Lambda 5 D 4 *$ | 2 | −2 | 0 | 0 | −2 | 0 | 0 |

$ \Lambda 6 D 4 *$ | 4 | −4 | 0 | 0 | $ 2 2$ | $ \u2212 2 2$ | 0 |

$ \Lambda 7 D 4 *$ | 4 | −4 | 0 | 0 | $ \u2212 2 2$ | $ 2 2$ | 0 |

. | $ CC 1 ( D 4 * )$ . | $ CC 2 ( D 4 * )$ . | $ CC 3 ( D 4 * )$ . | $ CC 4 ( D 4 * )$ . | $ CC 5 ( D 4 * )$ . | $ CC 6 ( D 4 * )$ . | $ CC 7 ( D 4 * )$ . |
---|---|---|---|---|---|---|---|

$ \Lambda 1 D 4 *$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

$ \Lambda 2 D 4 *$ | 1 | 1 | −1 | 1 | 1 | 1 | −1 |

$ \Lambda 3 D 4 *$ | 1 | 1 | 1 | 1 | −1 | −1 | −1 |

$ \Lambda 4 D 4 *$ | 1 | 1 | −1 | 1 | −1 | −1 | 1 |

$ \Lambda 5 D 4 *$ | 2 | −2 | 0 | 0 | −2 | 0 | 0 |

$ \Lambda 6 D 4 *$ | 4 | −4 | 0 | 0 | $ 2 2$ | $ \u2212 2 2$ | 0 |

$ \Lambda 7 D 4 *$ | 4 | −4 | 0 | 0 | $ \u2212 2 2$ | $ 2 2$ | 0 |

. | 1 . | z
. | z^{2}
. | z^{3}
. | z^{4}
. | z^{5}
. | z^{6}
. | z^{7}
. |
---|---|---|---|---|---|---|---|---|

$ \Gamma 1 \mathbb{Z} 8$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

$ \Gamma 2 \mathbb{Z} 8$ | 1 | −1 | 1 | −1 | 1 | −1 | 1 | −1 |

$ \Gamma 3 \mathbb{Z} 8$ | 1 | i | −1 | −i | 1 | i | −1 | −i |

$ \Gamma 4 \mathbb{Z} 8$ | 1 | −i | −1 | i | 1 | −i | −1 | i |

$ \Gamma 5 \mathbb{Z} 8$ | 1 | θ | $ \theta 2$ | $ \theta 3$ | $ \theta 4$ | $ \theta 5$ | $ \theta 6$ | $ \theta 7$ |

$ \Gamma 6 \mathbb{Z} 8$ | 1 | $ \theta 7$ | $ \theta 6$ | $ \theta 5$ | $ \theta 4$ | $ \theta 3$ | $ \theta 2$ | θ |

$ \Gamma 7 \mathbb{Z} 8$ | 1 | $ \theta 3$ | $ \theta 6$ | θ | $ \theta 4$ | $ \theta 7$ | $ \theta 2$ | $ \theta 5$ |

$ \Gamma 8 \mathbb{Z} 8$ | 1 | $ \theta 5$ | $ \theta 2$ | $ \theta 7$ | $ \theta 4$ | θ | $ \theta 6$ | $ \theta 3$ |

. | 1 . | z
. | z^{2}
. | z^{3}
. | z^{4}
. | z^{5}
. | z^{6}
. | z^{7}
. |
---|---|---|---|---|---|---|---|---|

$ \Gamma 1 \mathbb{Z} 8$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

$ \Gamma 2 \mathbb{Z} 8$ | 1 | −1 | 1 | −1 | 1 | −1 | 1 | −1 |

$ \Gamma 3 \mathbb{Z} 8$ | 1 | i | −1 | −i | 1 | i | −1 | −i |

$ \Gamma 4 \mathbb{Z} 8$ | 1 | −i | −1 | i | 1 | −i | −1 | i |

$ \Gamma 5 \mathbb{Z} 8$ | 1 | θ | $ \theta 2$ | $ \theta 3$ | $ \theta 4$ | $ \theta 5$ | $ \theta 6$ | $ \theta 7$ |

$ \Gamma 6 \mathbb{Z} 8$ | 1 | $ \theta 7$ | $ \theta 6$ | $ \theta 5$ | $ \theta 4$ | $ \theta 3$ | $ \theta 2$ | θ |

$ \Gamma 7 \mathbb{Z} 8$ | 1 | $ \theta 3$ | $ \theta 6$ | θ | $ \theta 4$ | $ \theta 7$ | $ \theta 2$ | $ \theta 5$ |

$ \Gamma 8 \mathbb{Z} 8$ | 1 | $ \theta 5$ | $ \theta 2$ | $ \theta 7$ | $ \theta 4$ | θ | $ \theta 6$ | $ \theta 3$ |

We deduce from these characters non-abelian dyons with fractional topological spins $ s = ( 1 / 8 ) , ( 1 / 4 ) , ( 3 / 8 ) , ( 5 / 8 ) , ( 3 / 4 ) , ( 7 / 8 )$.

#### 3. A note on the quaternionic phase and its quantum double $ D ( Q 8 * )$

In a separate work,^{104} devoted entirely to the $ D ( Q 8 * )$ model, we analyzed the feasibility of employing the quaternionic phase of a spin-2 BEC as a TQC platform. This phase can be obtained from the $ D 4 *$ one by reducing its rotational symmetry to a four-cycle. For the sake of completeness, we are here outlining some key aspect of this model. One of the primary challenges of developing a theoretical model of a TQC based on the non-abelian phases considered in this work, is that the dimensionality of the fusion spaces are generally larger than two. This is problematic since we only need a two-level system to form a qubit, meaning that the redundant particles in the fusion outcomes will generally absorb amplitude and consequently cause information leakage.^{82,105} It is, therefore, of interest to find a low-temperature symmetry group whose quantum double contains fusion products that form two-level states. The quaternionic phase offers such a model if one consider a closed subset of the particles in the theory.

A natural question that may arise is why we do not want to utilize all of the particles in a fusion outcome to represent a generic qudit. The issue with this idea, as thoroughly discussed in Ref. 104, is that one would need to have access to a group whose algebra is of a higher dimension. For instance, the braid group of a four-level qudit still only have two generators *σ*_{1} and *σ*_{2}, which is insufficient to span an $ SU ( 4 )$ Bloch sphere. Instead, by considering two qubits, which combined form a four-level system, the braid group has five generators, which may, thus, make universality more attainable. For instance, as already touched upon in Subsection III D and demonstrated in Ref. 100, a topological gravity may be simulated via the quantum pressure in the fluid, which results in additional topologically protected phase rotations which one may supplement any non-universal braid set with. Only one such additional phase gate would be required in order to make the $ D ( Q 8 * )$ universal. Another advantage of this model is that all fusion rules are multiplicity free. As already argued in Ref. 106, the presence of multiplicities is complicating the calculation of the Clebsch–Gordan coefficients of the theory. This is also apparent in our derivation presented in Appendix D since the number of terms on the right hand side in Eq. (D10) depends on the multiplicity. Only in the multiplicity-free case is an explicit solution accessible.

### G. Symmetry-based normal mode analysis

As indicated by the Fourier duality connecting the fluxons and the chargeons, if one of the two is considered to be localized in real space, its dual must be localized in the reciprocal space, with respect to a generalized quantum Fourier transform (the S-matrix). While it is, as per homotopy theory, well established that the fluxons must map onto quantized vortices, what the physical incarnations of chargeons are is less clear. The purpose of this section is, therefore, to offer an interpretation of the chargeon degrees of freedom. We find that, by means of a symmetry analysis, these normal modes correspond to spin-rotations and spin-waves or *magnons*. These systems also have regular phonon excitations but those are naturally associated with the breaking of translational symmetry. In general, a system in equilibrium exhibits some sort of symmetry that, if broken, excites modes in the system. These solutions transform according to the various UIRs of the, possibly broken, symmetry group, and as such, furnish a basis for the vector spaces the matrices in images of the UIRs act on. Guided by these considerations, we may now proceed and work out the normal modes in a spinor BEC.

#### 1. Tetrahedral phase—An example

*T*instead of its binary representation $ T *$ previously considered, since this group has a simpler geometric structure which will facilitate the discussion and analysis. To be more explicit in our formulation, we consider the Majorana star representation of the order parameter in which the spin nodes are located on the corners of a tetrahedron (see Fig. 10). The problem of finding the modes of motion of such a system can be cast into the form of an eigenvalue problem

*α*and

*β*(Euler angles), and consequently, the system must have 4 × 2 = 8 degrees of freedom in total. The resulting vector space constructed from the space of node labels and the vectors describing the displacements of the nodes must consequently have a tensor structure $ V = V nodes \u2297 V direction$. This eight-dimensional vector space has a basis of the form $ \theta \xaf = ( \alpha 1 , \beta 1 , \alpha 2 , \beta 2 , \alpha 3 , \beta 3 , \alpha 4 , \beta 4 ) T$, where

*α*and

_{i}*β*are the angular coordinates relative to the equilibrium position of each node. Due to the uniqueness of the solution to Eq. (48), given some initial state vector $ \theta i \xaf$, we know that the evolution of the system is completely determined by these values at time

_{i}*t*= 0. Furthermore, since the eigenvalue spectrum of $ A \u0302$ is invariant under

*T*, we may consider the action of the image of the full eight-dimensional representation on

*V*. Generally, this representation is, owing to the tensor structure of

*V*, reducible, which implies that it can be constructed from a set of irreducible orthogonal subblocks

*m*is the multiplicity of block

_{i}*i*. Since the various eigenmodes $ \theta \xaf i$ of

*A*are orthogonal and inhabit distinct blocks of

*V*, they must transform according to the UIRs of

*T*. Hence, the action of $ \Gamma ( T )$ can be decomposed as

*V*cannot be altered by $ \Gamma ( T )$, that is to say

_{i}What we learn from this is that there is one unique eigenvalue $ \omega i 2$ attached to each UIR, and that this eigenvalue must be the same for $ \theta \xaf i \u2032$ and $ \theta \xaf i$. Moreover, the dimension of the subspace *V _{i}*, and hence the degeneracy of $ \omega i 2$, must be equal to the dimension of the UIR $ \Gamma i$. We have now reduced the problem of finding the normal modes of the system to finding the various irreducible representations of the underlying symmetry group. Remarkably, the number of distinct eigenvalues and their degeneracies can be completely deduced from the representation theory, without any consideration of the physical parameters; the entire measurable spectrum of possible eigenvalues is classified by the symmetry of the system. Note that this is in complete agreement with the quantum double particle labeling. The fluxons are point particles labeled by the conjugacy classes of the group, and the chargeons are labeled by the various UIRs (or the reverse if we act with the S-matrix and consider the system in reciprocal space).

*T*as it contains all of the pertinent information required in order to work out the normal modes. A tetrahedron is constructed from four triangular faces, thus comprising 4 × 3 = 12 rotational symmetries in total. It has four conjugacy classes corresponding to the identity

*I*, all $ 2 \pi / 3$-rotations $ R ( 2 \pi / 3 )$, all $ 4 \pi / 3$-rotations $ R ( 4 \pi / 3 )$, in addition to

*π*-rotations $ R ( \pi )$. Let us now deduce the UIRs. First, we always have a trivial one-dimensional representation $ \Lambda 1$ sending all elements to unity. Then, we must also have another one-dimensional representation $ \Lambda 2$ sending all of the $ 2 \pi / 3$-rotations to $ e i ( 2 \pi / 3 )$. We can further permute this representation giving us a map $ \Lambda 3$ that sends all of the $ 2 \pi / 3$-rotations to $ e i ( 4 \pi / 3 )$. Again, using the corollary to Burnside's theorem which states that the order of a finite group is equal to the sum of the representation dimensions squared, we can conclude that there must also exist a representation of dimension $ 12 \u2212 1 2 \u2212 1 2 \u2212 1 2 = 3$. This is, of course, the three-dimensional representation acting on the $ \mathbb{R} 3$ space in which the tetrahedron is embedded. By virtue of the above-mentioned considerations, in addition to the orthogonality condition enforced on the rows in a character table, we may construct the complete character table of the group which is presented in Table VIII. Directing our attention to the fist column of Table VIII reveals that the characters add up to six, which means that the combined dimension of the UIRs is six since these characters correspond to the traces of the identity element. This means that there must be some multiplicities

*m*involved since the full representation $ \Gamma ( T )$ is eight-dimensional. All these multiplicities can be found by projecting the UIRs onto the full representation through the following equation:

_{i}*χ*and $ \chi full$ are the characters of the

_{i}*i*'th UIR and the full representation, respectively, and $ t \u2208 T$. This is a somewhat cumbersome, yet straight forward, calculation to perform so we will not carry it out here as the normal modes can be found without any knowledge of these multiplicities. As already noted in the previous cases, the number of UIRs must be the same as the number of conjugacy classes, that is four, and consequently we must have four normal modes in total. This can be understood intuitively by considering the geometry of the tetrahedron. The three nodes can either rotate about the three axes or the nodes can oscillate in an out-of-phase fashion. Moreover, we see from the character table that the $ \Lambda 3$ UIR must be three-dimensional since the character corresponding to the identity is three. From this, we can infer that the corresponding mode must be three-fold degenerate as these degenerate modes span the irreducible subspace of $ \Lambda 3$. Again, we can confirm this result through some simple geometrical considerations. The tetrahedron has four vertices in total so if one node is pairing up with one other node, the remaining two nodes are forced to pair up as well, but since the first node has three choices of partner, and this choice automatically fixes the choices of the remaining nodes, we can only have three inequivalent configurations of pair-vibrations. We have now, guided by the symmetry of the system, completely determined the normal mode spectrum as well as the degeneracies without any knowledge of the physical parameter values. We found three non-degenerate spin-rotation modes and one spin-wave mode with three-fold degeneracy corresponding to the out-of-phase oscillations of the nodes. The strategy deployed here is completely generic and can as such be applied to systems exhibiting any type of symmetry, to obtain the normal mode spectra.

. | I
. | $ R ( 2 \pi / 3 )$ . | $ R ( 4 \pi / 3 )$ . | $ R ( \pi )$ . |
---|---|---|---|---|

$ \Lambda 1$ | 1 | 1 | 1 | 1 |

$ \Lambda 2$ | 1 | $ e i \pi 3$ | $ e i 2 \pi 3$ | 1 |

$ \Lambda 3$ | 1 | $ e i 2 \pi 3$ | $ e i \pi 3$ | 1 |

$ \Lambda 4$ | 3 | 0 | 0 | −1 |

. | I
. | $ R ( 2 \pi / 3 )$ . | $ R ( 4 \pi / 3 )$ . | $ R ( \pi )$ . |
---|---|---|---|---|

$ \Lambda 1$ | 1 | 1 | 1 | 1 |

$ \Lambda 2$ | 1 | $ e i \pi 3$ | $ e i 2 \pi 3$ | 1 |

$ \Lambda 3$ | 1 | $ e i 2 \pi 3$ | $ e i \pi 3$ | 1 |

$ \Lambda 4$ | 3 | 0 | 0 | −1 |

## IV. VORTEX ANYONS IN SUPERFLUID GASES OF COLD ATOMS

To draw a connection between the theoretical concepts and experiments, we briefly mention two closely related physical systems, superfluid Fermi gases and Bose–Einstein condensates, which may be able to host non-abelian anyons inside the cores of quantized vortices. We will first discuss abelian vortices, routinely created and observed in experiments, and how to manipulate them before contemplating their respective non-abelian extensions yet to be realized in the laboratory.

### A. Quantized vortices in cold atomic superfluids

*m*is the mass of a bosonic atom, while for a simple superfluid Fermi gas $ m * = 2 m$ due to Cooper pairing of fermionic atoms of mass

*m*. The circulation of the superflow is quantized according to

Scalar vortices were created and observed in Bose–Einstein condensates in 1999^{107} and in superfluid Fermi gas in 2005.^{108} Since then, they have been routinely observed in numerous cold atom laboratories using a variety of techniques. Further details about vortex experiments in cold atom superfluids may be found for instance in Refs. 109–111.

In both bosonic and fermionic cases, the vortices, if left to equilbrate, will typically arrange into a regular Wigner-crystalline triangular vortex lattice due to the repulsive 2D-Coulomb-like interaction between the vortices.

### B. Non-abelian vortices in a superfluid Fermi gas

Here, we briefly outline how the non-abelian anyons may emerge in the vortex cores of chiral p-wave paired superfluid Fermi gas. This situation is generic and believed to be relevant to other topological fermionic superfluids such as in liquid ^{3}He and certain Type II superconductors.^{13,23,26,27,112–114}

*u*,

_{q}*v*, and energies

_{q}*E*are described by the Bogoliubov–deGennes equation

_{q}*E*, there is another, negative energy −

*E*, eigenstate obtained via the transformation $ ( u \u2212 E , v \u2212 E ) T = ( v E * , u E * ) T$. This further implies that $ \gamma E \u2020 = \gamma \u2212 E$, that is, creating a quasi-particle with energy

*E*is equivalent to annihilating a quasi-hole with energy −

*E*.

^{115}Under such circumstances, the system possesses an exact zero energy solution

*E*= 0 with the quasi-particle probability density peaking inside the vortex. This yields the defining property of a Majorana quasi-particle

_{q}*γ*

_{1}and

*γ*

_{2}, one can construct “complex” Dirac fermion operators $ c = 1 / 2 ( \gamma 1 + i \gamma 2 )$ and $ c \u2020 = 1 / 2 ( \gamma 1 \u2212 i \gamma 2 )$ characterized by the usual Fermi statistics. Taking a product of these relations yields

*σ*maps onto

*γ*, $1$ corresponds to the vacuum $ | 0 \u27e9$, and

*ψ*corresponds to a regular fermion $ | 1 \u27e9$. The number operator $ c \u2020 c$ may take two possible values 0, 1 corresponding to respective states $ | 0 \u27e9$ and $ | 1 \u27e9$ resulting in the two values for the Majorana parity $ i \gamma 1 \gamma 2 = \xb1 1$. More rigorous analysis leads to the conclusion that the Majorana zero mode vortices indeed correspond to realizations of the $ SU ( 2 ) 2$ Ising anyons discussed in Sec. II C.

^{11}

From theoretical perspective, vortices must be well separated to avoid quasi-particle tunneling between vortex cores, which would cause energy splitting of the Majorana pairs. From an experimental perspective, the greatest unresolved challenge is to realize a suitable topological superfluid such as a spinless p-wave paired superfluid phase of a Fermi gas. Ultracold Fermi gases near a p-wave Feshbach resonance, have been studied experimentally^{116–123} and offer a potential route to realizing chiral p-wave superfluidity, however, they suffer from inelastic losses which are enhanced near the resonance.^{124} Lower-dimensional gases,^{125–129} optical lattices^{130} and even nonequilibrium^{131} systems provide promising opportunities for studying Fermi gases with resonantly enhanced p-wave interactions and remain the subject of ongoing research.

### C. Non-abelian vortices in a superfluid Bose gas

^{57,132}of

*F*= 2 spinor Bose–Einstein condensates.

^{47,133,134}A canonical transformation of bosonic fields to Bogoliubov quasi-particle basis

*F*, the quasi-particle eigenmodes have $ 2 F + 1$ components such that for

*F*= 2 they are five component spinors. The BEC itself is an exact Nambu–Goldstone zero mode satisfying

*E*= 0 and $ u = v *$, where the energy is measured with respect to the chemical potential. It is, therefore, convenient to discuss the ground state topology in terms of the ground state order parameter $ \Psi = [ u \u2212 2 , u \u2212 1 , u 0 , u 1 , u 2 ] T$, where the subscripts refer to the five spin-projections of the hyperfine spin. For instance, in the cyclic ground state phase, the state respects the symmetry of the non-abelian binary tetrahedral group and, therefore, the topological vortex excitations (fluxons) are determined by the conjugacy classes of the group with the quantum double construction leading to the additional chargeon and dyon degrees of freedom as clarified in Sec. III F.

In contrast to the fermionic case where the role of the Majorana vortices is merely to trap and host the core localized quasi-particles, which define the qubit of the non-abelian fusion algebra, in the quantum double construction of spinor BECs, both the carrier vortices (fluxons) and the core localized quasi-particles (chargeons) participate in the fusion process. As such, the Majorana zero modes may be viewed as the counterparts of pure chargeons of the quantum double.

*F*= 2 BEC, realized by the quaternion phase,

^{104}leads to fusion rules of higher complexity in comparison to the Ising anyon model, such as

_{x}may yield either a vacuum or three different types of pure chargeons. The physical interpretation of this fusion rule that defines a 4-qudit is, thus, that the two vortices annihilate leaving behind one of the four possible chargeons, including the trivial one. This arises due to the fact that the order parameter around the vortex core is now a five component spinor such that there may exist multiple distinct ways spin currents and superflow mass currents can be combined along a path encircling a vortex core that nevertheless result in the same topological vortex invariant. This is to be contrasted with a simple vortex in a scalar BEC where the only degree of freedom along the path around the vortex is a single number, the phase S(r) of the order parameter, which uniquely defines the topology of the vortex flux.

### D. Experiment considerations regarding creation, fusion, and braiding

It is useful to first consider the most abundant laboratory system of a simple scalar BEC with a quantized abelian vortex fluxon. Such vortices are readily produced in research laboratories via a variety of methods,^{109–111} including the so-called “chopsticks” method,^{135} which enables deterministic production of vortices and control over their positions, thereby conceptually facilitating pair creation, braiding, and fusion protocols for vortex anyons. Recently, this method has been utilized to develop a programable vortex collider in a superfluid Fermi gas.^{136} In principle, these techniques can be extended to larger numbers of vortices using optical tweezer arrays, as being deployed for storing Rydberg atom qubits in neutral atom quantum computers.^{137–139}

As the vortices are moved around during braiding, one potential concern is the effect of phonon excitations. Although phonons cannot change the topology of isolated vortices, it is in principle possible for a high-energy phonon to split into a vortex–antivortex pair via a Sauter–Schwinger-like pair creation process and such vortices could then accidentally become braided as the computation is carried out leading to quasi-particle poisoning of the computation. The probability of such processes should, however, be negligible at low enough temperatures achievable in the experiments. In the spinor BECs, it is also conceivable that an absorption of a phonon by the vortex might affect the atomic probability distribution in the different hyperfine spin levels of the core localized modes. This could potentially alter the amplitudes of the quantum double fusion rules. However, we are not aware of any works that would have addressed this issue.

Regarding the readout of the fusion outcome, the greatest experimental challenge is to observe the vortex core localized quasi-particles, which has not been achieved to date even in the case of a scalar BEC. It was only recently discovered^{140} that even in a scalar BEC a vortex core hosts a zero energy BdG quasi-particle mode known as a kelvon.^{141,142} Such a kelvon can be regarded as a quantum mechanical equivalent of a classical Kelvin wave.^{143} In a 3D scalar BEC, kelvons have been indirectly observed^{144} as they distort the shape of a vortex line into a helix. However, in 2D, it is not clear how to experimentally resolve the zero energy kelvon, which can be viewed as a quantum depleted non-condensate atom that is trapped by the vortex core and as such is inseparable from the vortex fluxon. It may, thus, be tempting to interpret the kelvon as the charge of a quantum double anyon. Indeed, from our analysis we found that all pure fluxons are in fact bosons, while the dyons generally have fractional charge. Therefore, the non-abelian anyons in a spinor BEC are always charge–flux composites. We may, therefore, conjecture that the chargeon part of a dyon corresponds to the spin mode of the kelvon attached to the vortex. This view is supported by numerical experiments carried out in Ref. 57, where spin waves were observed after a fusion event involving two vortices. Even if a vortex is annihilated by an anti-vortex, they may still have a charge that survives the fusion process. In a realistic model of TQC based on vortices in a spinor BEC, one would then build the qubits from vortices and their kelvons.

Thus, the technologies to create, braid, and fuse vortex anyons have been demonstrated in the case of abelian vortices. There are two remaining major problems that need to be resolved in order to achieve a complete experimental demonstration of a non-abelian vortex based topological qubit in a cold atom superfluid. The first is the stabilization of a ground state whose order parameter would support non-abelian vortices as its natural excitations. For instance, it is not presently known if the non-abelian cyclic, biaxial nematic, or the quaternion phases can be stabilized as ground states.^{133} The second, which has already been discussed, is the development of experimental techniques that allow probing the kelvon and to measure its charge, a task that is yet to be realized even for vortices in existing quasi-two-dimensional scalar BECs.

## V. DISCUSSION

We have aimed at providing a self-contained account of the emergent low-temperature quantum double structure in spinor BECs, a connection that is not well covered in the literature to date. It is well understood that spinor BECs may support non-abelian anyons in the form of quantized vortices (fluxons). However, as emphasized in this paper, this is just one side of the story. The existence of a particle-vortex duality, whose structure is made exact within the quantum double framework, implies that the description of vortices can be mapped onto a dual side, which describes a different type of particle (chargeons). We have shown that the chargeons belonging to the dual side correspond to spin waves and spin rotation modes. As illustrated, despite being non-abelian, the fluxons only have fractional mutual statistics, whereas their self-statistics is bosonic. The same is true for the chargeons since their algebraic structure is, owing to the duality, equivalent. However, the quantum double structure also accommodate charge–flux composites which, as demonstrated, may indeed have fractional self-statistics.

The strong interest in non-abelian anyons is primarily due to their potential to realize fault-tolerant quantum information processing. Up until now, the main focus has been on fermionic systems, and in particular systems that may host Majorana fermion quasi-particles. Spinor BECs, on the other hand, appear to have been overlooked in this context, which prompted us to carry out this work with the aim of shedding light on their potential for TQC. Theoretically, as we discussed in Sec. III F 3, one of the main hurdles in the development of a spinor BEC-based TQC platform is that the fusion spaces generally are large due to the high dimensionality. To construct a qubit, only two states are needed, which implies that information leakage^{82,105} may occur into the states not belonging to the computational space. Another issue is the possibility of multiplicities in the fusion rules, which complicates severely the calculation of the Clebsch–Gordan coefficients^{106} and, thus, the braid group matrices. Similar issues are, of course, also present in the $ SU ( 2 ) k$ anyon models but mainly for larger values of *k*. To reduce the chance of having to face such problems, it is useful to consider smaller groups since that will restrict the dimensionality of the Hilbert spaces spanned by the anyons of the theory. For instance, the quaternionic subgroup $ Q 8$, which was considered in a separate work,^{104} can be obtained by breaking further the rotational part of the $ D 4 *$ subgroup to a four-cycle. For this group, it is possible to define multiplicity free qubits that are not accompanied by additional non-computational states that generally cause leakage.

Finally, although a scalable fault tolerant topological quantum computer based on non-abelian vortex anyons remains out of reach, a neutral atom superfluid prepared in a non-abelian ground state could certainly facilitate a beautiful demonstration of the fundamental principles of topological quantum computation utilizing a few topologically protected logical qubits.

## ACKNOWLEDGMENTS

This research was supported by the Australian Research Council Future Fellowship FT180100020, and was funded by the Australian Government.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Emil Génetay Johansen:** Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). **Chris Vale:** Supervision (supporting); Validation (equal); Writing – review & editing (equal). **Tapio Simula:** Conceptualization (equal); Formal analysis (supporting); Funding acquisition (lead); Investigation (supporting); Methodology (equal); Supervision (lead); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.

### APPENDIX A: THE FUNDAMENTAL GROUP

*X*of some topology, and let $ x 0 \u2208 X$ be an arbitrary point in

*X*. Let

*f*

_{1}and

*f*

_{2}be two loop functions

*f*

_{1}start at

*x*

_{0}and

*f*

_{2}end at

*x*

_{0}, such that $ f 1 ( 1 ) = f 2 ( 0 ) = x 0$. Then, one may compose

*f*

_{1}and

*f*

_{2}according to the parametrization

### APPENDIX B: REPRESENTATION THEORY AND CHARACTERS

##### 1. Representation theory of a group G

*G*of the system necessarily commutes with

*H*so that the quantum numbers are invariant under the action of

*G*. Thus, in order to study this action we need to find a set of matrices that implement the elements of

*G*acting on $H$. The map $\Gamma $ from

*G*to this set of matrices

*G*(homomorphic), i.e.,

*N*. If the matrices in the image of $\Gamma $ are block diagonal, they may be decomposed as a direct sum into matrices that are not block diagonal. Hence, the representations that map onto matrices that cannot be decomposed in this way are called irreducible representations. In quantum physics, we are, since normalization must be preserved, primarily concerned with unitary irreducible transformations so in this context $ \Gamma ( a ) \u2208 U ( N )$ for all $ a \u2208 G$, where $\Gamma $ is a unitary irreducible representation (UIR).

##### 2. Representation theory of the quantum double $ D ( H )$ of H

*a*, all of which belong to the same equivalence class, which maps $ h * CC$ to

_{i}*h*under conjugation. The quantum double action, which performs a gauge transformation followed by a flux measurement, is implemented as

_{i}*H*and their centralizers $ Z ( C C i )$, which are defined as the subgroups of elements that are commuting with $ CC i$, and pick one arbitrary element $ h i * CC$ from each conjugacy class $ CC i$ as the representative element and work out the UIRs $ \Gamma j$ of each $ Z ( C C i )$. Then, find the set of elements $ { a j} j = 1 | C C i |$ such that $ h k = a k h * CC i a k \u2212 1$ for each $ h k \u2208 C C i$. These elements belong to the various cosets formed by $ H / Z ( C C i )$. Next, explicitly define a basis to work with for the Hilbert space spanned by the states $ | h i CC , u j \Gamma \u27e9$. The dimension

*d*of this Hilbert space equals $ d = dim ( \Gamma ) \xd7 | C C |$ so any set $ { v \u2192 k} k = 1 d$ of orthonormal vectors $ v \u2192 k \u2208 \u2102 d$ constitutes an orthonormal basis. Given such a basis, the action $ \Pi CC \Gamma ( P h g ) | h i CC , u j \Gamma \u27e9$ is implemented as $ \pi ( h , g h i CC g \u2212 1 ) \Gamma ( g * )$ on the space spanned by $ { v \u2192 k} k = 1 d$, where $ \pi ( h , g h i CC g \u2212 1 )$ is a

*d*-dimensional matrix projecting $ g h i CC g \u2212 1$ onto

*h*and $ \Gamma ( g * )$ is a

*d*-dimensional matrix implementing the gauge transformation $ g *$. Note that in the case when

*H*is non-abelian, it may be possible to further decompose $ g *$ since the group has more than one generator. For instance, in the case of a dihedral group

*D*, an arbitrary element is a combination of a rotation

_{n}*r*and a reflection

*t*, such that $ \Gamma ( g * ) = \Gamma ( t m ) \Gamma ( r n )$, where $ m \u2208 { 0 , 1}$ and $ n \u2208 \mathbb{Z} n$. Splitting the action up in this way makes it easier to construct the quantum double matrices since one can simply multiply the projection with the gauge transformation, whose matrix representations are easier to find.

##### 3. Characters

*a*in the representation $\Gamma $,

^{90}states that the number of irreducible representations of a group equals the number of conjugacy classes in the group. Thus, the characters of the conjugacy classes in the various UIRs can be summarized in a table $ \chi \Gamma i , C C j$, where the rows correspond to the UIRs $ \Gamma i$ and the columns to the conjugacy classes $ C C j$. An important feature of the rows $ \chi \xaf \Gamma m$ in such a table is that they are orthonormal with respect to the inner product

### APPENDIX C: HOPF ALGEBRAS—A UNIFIED FRAMEWORK

##### 1. Coproduct (fusion)

*g*on the individual Hilbert spaces and projects out the total flux

*h*, so that the topological charges are conserved globally under fusion. This action satisfies the important coassociativity property under composition $\u25cb$

##### 2. Braiding

*R*must be an element of $ D ( H ) \u2297 D ( H )$, acting on the flux-charge Hilbert space. More specifically, the operator

*R*implements a gauge transformation

*g*on the first dyon by the flux of the second dyon such that

*R*operators

*R*

_{1}and

*R*

_{2}which must satisfy the so-called quasi-triangularity conditions

^{148}cannot be satisfied if they are not enforced. It is straight forward to check that indeed $ R 1 R 2 R 1 = R 2 R 1 R 2$, which is required for any algebraically consistent anyon model. Thus, given a two-dyon representation, the action on the corresponding Hilbert space $ V j i \u2297 V j j$ is given by a matrix $ R j i , j j j k$,

*j*and

_{i}*j*are labeling the generalized angular momenta of the dyons that are being interchanged and

_{j}*j*that of their fusion channel. Furthermore, $ \sigma m i , m q m j , m p$ are the components of the of the permutation operator, which simply corresponds to de-coupling followed by a re-coupling, where the two dyons are swapped

_{k}*m*is the generalized topological magnetization. The elements denoted by the brackets are nothing but the Clebch–Gordan coefficients for which in Appendix D we shall derive an analytical expression. Now, the braid matrices

_{i}*σ*

_{1}and

*σ*

_{2}at single qubit level are given by $ \sigma 1 = R$ and $ \sigma 2 = F R F \u2212 1$, where $F$ is the change of basis operator.

^{10,149}The action of $F$ is simply changing the order of fusion such that if $ | ( j i j j ) j k ; j l \u27e9$ represents the states corresponding to the fusion channels

*j*, formed when fusing

_{l}*j*,

_{i}*j*and

_{j}*j*, then $ F | ( j i j j ) j k ; j l \u27e9 = | j i ( j j j k ) ; j l \u27e9$, where the anyons within the parenthesis are fused first. The matrix elements of this operator can also be expressed as a series of de-couplings and re-couplings given the Clebsch–Gordan coefficients of the theory, such that

_{k}##### 3. Counit (vacuum)

*ε*which is an algebra morphism

##### 4. Antipod (anti-particles)

### APPENDIX D: DERIVATION OF THE QUANTUM DOUBLE CLEBSCH–GORDAN COEFFICIENTS

*n*denotes the multiplicity and let us define a projection operator $ P \u0302 m k m l j k$ that is projecting out the $ m k \u2032$ s component of the tensored basis

*P*is the residual symmetry group and $ d j k$ is the dimension of the representation

_{h}. H*j*. Moreover, $ \Lambda j i \u2297 j j$ represents the reducible tensored representation and $ \Lambda j k$ the irreducible component we wish to project onto. Now, in order compute the action of this operator on the tensored state, we need to expand the quantum double element with the coproduct

_{k}*n*= 1 to provide an analytical formula for the case with no multiplicities. Finally, by letting

*m*=

_{i}*m*,

_{p}*m*=

_{j}*m*, and

_{p}*m*=

_{k}*m*, and taking the square root, the Clebsch–Gordan coefficients can be expressed as

_{l}### APPENDIX E: MEAN-FIELD THEORY FOR SPIN-*S* BECS

^{47}

^{,}

*p*,

*q*are external magnetic fields coupling linearly and quadratically, respectively, to the wavefunction via the z-component of the spin-

*F*matrix $ [ f z ] m m \u2032$ and

*c*(

_{i}*i*= 1, 2) are couplings to the spin density $ F \u2192 = \u2211 m , m \u2032 = \u2212 S S \psi m * [ f a ] m m \u2032 \psi m \u2032$ and spin-singlet pair amplitude

*A*(

*r*). The Hamiltonian describing the various phases can, thus, be obtained by tuning the parameters to the desired values.

^{47,57}Here, particles belonging to different states of spin participate in the current, thus, resulting in a multi-component velocity

*a*denotes the components if the spin, which serves as a non-abelian gauge field in contrast to the simple abelian phase gradient $ v s = ( \u210f / 2 m ) \u2207 \theta $ of a scalar BEC. When computing the monodromy of such an object the gauge field $ A \mu a$ implements a non-abelian transformation from which the particles spin can be deduces. In particular, in a low-energy phase with residual symmetry group $ H \u2282 SO ( 3 )$, the first homotopy group is isomorphic to the subgroup itself

^{41}$ \pi 1 ( U ( 1 ) \xd7 SO ( 3 ) / H ) \u2243 H$, which is why the non-abelian vortices are labeled by the conjugacy classes of

*H*.

### APPENDIX F: FOURIER ANALYSIS OVER ARBITRARY GROUPS AND THE S-MATRIX

*G*be a finite group and let Λ be a representation of

*G*, then the Fourier transform $ F \u0303 \Lambda $ of a function $ \psi \u2208 L 2 ( G )$, with respect to Λ, is given by the inner product projection

*g*in the image of Λ, since the trace is trivial. As a concrete example, let us consider a system with discrete

*N*-fold cyclic translational symmetry

*T*, generated by

_{a}*a*, such as in a one-dimensional crystal with periodic boundary conditions. A representation Λ in this case is a map $ \Lambda : \u2009 \u2009 T a \u2192 U ( 1 )$, which implies that $ \chi \Lambda ( g ) = \Lambda ( g ) \u2208 U ( 1 )$, which is nothing but the conventional discrete Fourier transform where the set $ { e i ( 2 \pi n / N )} n = 0 N \u2212 1$ forms an orthornormal basis for the Hilbert space. Thus, given a system with a Hamiltonian which is invariant under

*G*, the set of characters $ { \chi \Lambda i ( g )} g \u2208 G$ with respect to a representation Λ

_{i}of

*G*, constitutes a basis vector corresponding to the subspace $ H i \u2282 H$, where $ H = \u2295 i H i$ is the full Hilbert space, and

*i*denotes the invariant label (quantum number) of the subspace $ H i$. The sets of characters are always orthonormal and have a size equivalent to $ dim ( L 2 ( G ) )$ meaning that they always furnish a basis for the Hilbert space of the system.

##### 1. Modular S-matrix

*H*. The matrices in the image of a representation Π of $ D ( H )$ act on a Hilbert space $H$ spanned by the sets of characters corresponding to those matrices. Here, $ H = \u2102 [ H ] \u2297 F [ H ]$, where $ \u2102 [ H ]$ is spanned by the pure fluxon states and $ F [ H ]$ by the pure chargeon states. The Fourier transform relating these two vector spaces is defined by the modular S-matrix

^{89}

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