1D Non-Abelian Anyons

Updated 19 August 2025

One-dimensional non-Abelian anyons are quasiparticle excitations with non-commutative fusion and braiding operations, enabling topologically protected state spaces.
They emerge in systems like topological superconductors, quantum spin chains, and optical lattices, providing potential routes for fault-tolerant quantum computation.
Their behavior is modeled using modified braid and traid groups, with fusion rules and quantum dimensions that are critical for both theoretical insights and experimental validation.

One-dimensional non-Abelian anyons are quasiparticle excitations or collective degrees of freedom in one-dimensional quantum systems whose exchange or fusion processes are described by non-commuting (matrix-valued) operations rather than simple phase factors. Unlike their two-dimensional counterparts, the topological and algebraic structures underpinning non-Abelian statistics in 1D are subtle, requiring careful distinction between physical motion, internal Hilbert space structure, fusion rules, and how topological protection is implemented. Such anyons emerge in a variety of physical systems, including engineered topological superconductors, quantum spin chains, ladders, condensed matter heterostructures, and cold atom implementations with nontrivial multi-body interactions. Their paper is motivated in part by the prospects for fault-tolerant quantum computation and for creating novel topologically ordered quantum phases in reduced dimensionality.

1. Fundamental Properties and Topological Structure

Non-Abelian anyons in 1D are characterized by two core features: (i) the degeneracy of their multi-anyon Hilbert space increases exponentially with particle number, governed by the quantum dimension $d_x > 1$ of the underlying anyon type (Rowell et al., 2015), and (ii) their braiding or fusion acts via non-commuting matrices implementing a representation of a non-Abelian group (often, but not always, the braid group or a variant adapted to the 1D context such as the traid group) (Harshman et al., 2018, Harshman et al., 2021).

The fusion rules of these anyons encode the possible outcomes of bringing two or more together. For example:

Ising anyons: $\sigma \times \sigma = I + \psi$ , with $I$ the vacuum and $\psi$ a fermion;
Fibonacci anyons: $\tau \times \tau = I + \tau$ ;
Drinfeld double (e.g., $D(S_3)$ ) models: $G \times G = A + B + G$ (Byles et al., 6 Aug 2024).

The degeneracy of the ground (or multi-anyon) state space is quantified as $\dim V_{x,n,a} \sim d_x^n$ for $n$ anyons of type $x$ with total fusion channel $a$ (Rowell et al., 2015). Non-Abelian statistics is implied by this degeneracy, since no unitary action that is solely a phase can mix an exponentially growing set of fusion states.

In 1D, the absence of planar braiding necessitates alternative constructs. Theories often rely on variant exchange groups such as the traid group (for systems with hard-core three-body constraints) (Harshman et al., 2018), or on representations of the symmetric group modified by orbifold configuration space structures that preserve path ambiguities at collision points (Harshman et al., 2021). For lattice models and spin chains, the effective non-Abelian structure manifests in the fusion tree of the Hilbert space, which is equivalent to the Bratteli diagram of the associated fusion category (Greiter et al., 2019).

2. Physical Realizations and Model Systems

Non-Abelian anyons in one dimension arise across various platforms:

a. Kitaev Chain and Majorana Zero Modes:

The Kitaev model realizes a topological superconducting phase with spatially separated Majorana zero modes at the boundaries. These modes are self-conjugate ( $\gamma = \gamma^\dagger$ ) and satisfy $\gamma^2 = 1$ . Exchanging two Majorana modes is implemented by the unitary operation

$U_{ij} = \exp\left( \frac{\pi}{4} \gamma_j \gamma_i \right),$

which, for collections of more than two modes, acts non-commutatively in the degenerate ground-state space (Rao, 2016, Masaki et al., 2023).

b. Spin Chains and Ladder Models:

Critical spin chains and ladders built from deformations of $SU(2)$ (e.g., $SU(2)_k$ quantum-group chains) support spinon excitations obeying non-Abelian fusion rules (Poilblanc et al., 2011, Soni et al., 2015, Greiter et al., 2019). For instance, in $k=3$ (Fibonacci) models, the fusion algebra matches that of Fibonacci anyons. The fractionalization of magnon-like excitations leads to unique universality classes, with critical chains described by conformal field theories of central charge $c = 1/2$ (Ising), $c = 7/10$ (tricritical Ising), or $c = 4/5$ (3-state Potts), depending on model parameters (Poilblanc et al., 2011, Poilblanc et al., 2012).

c. Quantum Wires and Proximitized Systems:

One-dimensional wires with spin-orbit coupling, Zeeman fields, and strong interactions can support phases analogous to those in the fractional quantum Hall effect, with fractionalized excitations and, upon proximity-induced superconductivity, "fractional Majorana" zero modes with non-Abelian fusion and braiding rules (Oreg et al., 2013).

d. Ultracold Atomic and Optical Lattice Systems:

Strongly correlated gases in 1D optical traps with engineered three-body "hard-core" interactions can realize the traid group and corresponding non-Abelian exchange statistics (Harshman et al., 2018, Harshman et al., 2021).

e. Quantum Double and Lattice Qudit Models:

Minimal qudit lattices, such as the $d=6$ qudit lattice for $D(S_3)$ , offer direct anyon encoding and manipulation via braiding and fusion operations implemented through sequences of lattice operators, culminating in explicit measurement of non-Abelian statistics with minimal resource overhead (Byles et al., 6 Aug 2024).

3. Mathematical Structures: Fusion, Braiding, and Exchange Algebra

The algebraic framework is defined by fusion rules, $F$ -matrices (associativity isomorphisms), and $R$ -matrices (braiding isomorphisms). Non-Abelianity is exhibited by the non-commutativity of fusion and braiding operations: $[R^{GG}, F^{G}_{GGG}] \neq 0$ as shown for $D(S_3)$ anyons (Byles et al., 6 Aug 2024). In concrete models, these matrices are tabulated and satisfy pentagon and hexagon identities.

One-dimensional constraints alter the possible exchange operations: particles cannot physically overtake each other, leading to the requirement that anyon's internal Hilbert space must be manipulated by operations associated with fusion diagrams or by effective braiding in junction/network topology (Maciążek et al., 2020).

The traid group $T_n$ is defined by generators $t_i$ satisfying $t_i^2 = 1$ and $t_i t_j = t_j t_i$ for $|i - j| > 1$ , but lacking the standard Yang-Baxter (braid) relations. Its representations (abelian and non-abelian) give rise to novel anyon types in 1D not reducible to bosonic, fermionic, or standard braid group categories (Harshman et al., 2018, Harshman et al., 2021).

4. Fractionalization, Excitation Spectrum, and Entanglement

The low-energy excitation spectrum of 1D non-Abelian anyon models universally exhibits fractionalization:

Charge sector: Described by a hard-core boson (or fermion) band structure, dispersing according to effective hopping models.
Neutral/Anyonic sector: Encoded by quantum-number "fusion labels" (the fusion tree), governed by the fusion algebra and the associated CFT. For Ising anyons: $c=1/2$ neutral mode; for Fibonacci: $c=7/10$ (tricritical Ising) or $4/5$ (3-state Potts) (Poilblanc et al., 2012, Poilblanc et al., 2011, Soni et al., 2015).

The full spectrum is synthesized as

$E_{p,m} = E_{\text{charge}}(p, k_m) + E_{\text{anyon}}(m),$

where $k_m$ parameterizes the neutral (fusion) state (Poilblanc et al., 2011, Poilblanc et al., 2012).

Entanglement studies further confirm this structure. The entanglement spectrum and derived entanglement entropy in these systems display universal features, with block structure and eigenvalue ratios determined by the underlying quantum dimensions and fusion multiplicities: $\lambda_c = d_c / (d_{a_L} d_{a_R}),$ where $d_c$ is the quantum dimension of fusion channel $c$ (Cornfeld et al., 2018). In practice, measuring the entanglement spectrum in, e.g., Majorana wires enables direct access to the quantum dimension and hence a probe of non-Abelian statistics.

5. Interfaces, Networks, and Hybrid Phases

Interfacing distinct 1D non-Abelian phases (e.g., Ising vs. Fibonacci) or constructing complex wire networks introduces new physics:

Interfacial CFTs: At the boundary between Moore–Read (Ising) and NASS (Fibonacci) quantum Hall states, the interface supports neutral modes described by a $c=7/10$ CFT (tricritical Ising), reconciling the mismatch in quantum statistics (e.g., in the allowed fusion channels and charge assignments) (0810.1955).
Wire Networks and Network Connectedness: The braiding relations and the representational content of non-Abelian anyon braiding depend strongly on the network topology. In modular (2-connected) networks, moving a particle between modules can change its effective braiding statistics, while 3-connected networks recover uniform 2D braid group properties (Maciążek et al., 2020).
Chiral Phases and Topological Transitions: Exactly solvable chains (e.g., Drinfeld double $D(D_3)$ models) display phase diagrams partitioned by level crossings, some exhibiting chiral ground states with nonzero ground-state momentum and first-order phase transitions associated with changes in the fusion-braiding structure (Finch et al., 2010).

6. Quantum Computation and Information Encoding

Non-Abelian anyons in 1D can encode qubits and implement quantum gates:

Majorana-based (Ising anyon) qubits: Exchange (physical or effective) of Majorana modes implements Clifford gates topologically. For universality, supplementary gates are required (Masaki et al., 2023).
Fibonacci anyons: Braiding is dense in $SU(2)$ , permitting universal quantum computation by topological means if implemented in a suitable 1D or quasi-1D system (0810.1955).
Dense Coding and Measurement: Minimal qudit systems, such as a two-qudit d=6 lattice, suffice to fully determine non-Abelian braiding and fusion matrices, and can be used for quantum error correction schemes (Byles et al., 6 Aug 2024).
Robustness: Topologically protected degeneracy of the encoded space, as enforced by fusion and braiding algebra, underpins fault tolerance (Rowell et al., 2015).

7. Open Challenges and Theoretical Directions

While the non-Abelian statistics of 1D anyons are well-defined in multiple model settings, several conceptual and practical questions remain:

The full classification of non-Abelian exchange groups (e.g., affine traid/fraid groups) and their representations is incomplete (Harshman et al., 2021).
In strictly 1D systems, physical implementation of braiding requires engineered junctions or effective adiabatic protocols, since particles cannot truly traverse each other (Maciążek et al., 2020).
The distinction between topological (configuration-space) and dynamical (model-dependent) origins for fractional statistics is especially acute in one dimension; whereas abelian phase statistics are typically dynamical, non-Abelian statistics in 1D models often inherit a strictly topological origin, particularly when multi-body coincidences (e.g., in traid models) are excluded, producing non-simply connected spaces (Harshman et al., 2021, Harshman et al., 2018).
The robustness of topological protection against disorder and nonideal perturbations is context-dependent (e.g., fractional helical liquids are fragile to disorder (Oreg et al., 2013)).

Potential applications span not only quantum computation but also the paper of collective behavior in cold-atom and solid-state contexts, as well as theoretical insights into topological phase transitions.

One-dimensional non-Abelian anyons thus provide a rich meeting ground of algebra, topology, and condensed matter physics. Through fusion and braiding in constrained geometries, they realize robust, topologically protected internal state spaces suitable both for fundamental exploration and for practical applications in robust quantum technologies.