Golden Anyon Chains in Quantum Systems

Updated 26 July 2025

Golden anyon chains are one-dimensional quantum lattice models built from Fibonacci anyons that obey non-Abelian fusion rules and showcase robust topological order.
They serve as an analytically tractable platform for studying quantum criticality and integrability, linking fusion tree methods with conformal field theory analysis.
These chains are pivotal in advancing topological quantum computation due to their experimentally motivated design and effective braiding operations.

Golden anyon chains are one-dimensional quantum lattice models in which the local degrees of freedom are Fibonacci (or "golden") anyons, governed by non-Abelian fusion rules. These systems serve as exemplars of non-Abelian anyonic matter, displaying unique topological, algebraic, and quantum critical properties. Their paper is motivated by both foundational questions in topological phases and practical considerations for topological quantum computation, due to the universality of Fibonacci anyons under braiding and their robust topological order. Golden anyon chains provide a minimal, analytically tractable, and experimentally motivated setting to probe a range of phenomena including non-factorizing Hilbert spaces, non-invertible topological symmetries, integrability, crossing into irrational conformal field theories, and behavior under strong disorder and finite-dimensional constraints.

1. Algebraic Structure and Fusion Rules

Golden anyon chains are constructed using Fibonacci anyons, which support two superselection sectors: the trivial particle (𝟙) and the nontrivial anyon (τ). The defining fusion rule,

$\tau \otimes \tau = \mathbf{1} \oplus \tau,$

implements "golden" statistics: fusing two τ yields either the vacuum or a single τ. Distinct from SU(2) spin chains, golden anyon Hilbert spaces do not admit a simple tensor product structure. Instead, many-body basis states are labeled by "fusion trees," where each internal edge encodes the cumulative fusion outcome of its left subtree.

The closure of the Fibonacci category—every fusion outcome remains within {𝟙, τ}—permits consistent application of real-space decimation, under both clean and disordered settings (0807.1123). Basis changes relating different fusion orderings are computed via F-matrices, and braidings are determined by R-matrices with matrix elements involving phases set by the golden ratio, φ = (1 + √5)/2. Algebraically, two-site projection operators onto the 𝟙 channel, after appropriate normalization, satisfy a Temperley–Lieb algebra with parameter φ (0902.3275), bridging the theory with integrable models and quantum groups.

2. Model Hamiltonians: Nearest-Neighbor, Extended, and Disordered Chains

The canonical golden anyon chain is defined by a Hamiltonian favoring the fusion of adjacent τ anyons into a specified channel,

$\mathcal{H}_\mathrm{Fib} = -J \sum_{\langle i, j \rangle} \Pi^1_{ij},$

where $\Pi^1_{ij}$ projects onto the trivial fusion outcome (0902.3275). The sign and strength of J determine the ferromagnetic (FM) or antiferromagnetic (AFM) character.

Hamiltonian representations require explicit construction in the fusion tree basis using F-moves, often leading to intricate nonlocal matrix elements. Extensions studied include:

Three-anyon and longer-range interactions: Terms involving the fusion outcome of three (or more) adjacent anyons, requiring sequential F-moves and potentially nontrivial braiding (0902.3275).
Two-leg ladders: Hilbert spaces formed via "zig-zag" fusion paths, mapping rung and chain interactions onto nearest and next-nearest neighbor terms (0902.3275).
Disordered chains: Random sign and strength in the fusion-preference couplings, leading to random singlet and infinite randomness phases (0807.1123).
Chains with pairing terms: Generalizations to su(2)_k anyon chains, including non-conserving pairing processes that create and annihilate τ pairs, yielding a broader phase diagram and exact zero-energy ground states at projector points (Garjani et al., 2016).

Computationally, the fusion constraints lead to exponential reduction in the Hilbert space dimension, scaling as Fibonacci numbers rather than 2ⁿ, with associated constraints on local Hilbert space structure (Chepiga et al., 2018).

3. Criticality, Integrability, and Conformal Field Theory

Golden anyon chains display a rich spectrum of critical and massive phases, linked to the properties of rational and irrational conformal field theories (CFTs) in the thermodynamic (continuum) limit. Several key results include:

Clean chain criticality: The AFM golden chain exhibits criticality governed by tricritical Ising CFT, with central charge c = 7/10, as established through mapping to hard-boson and quantum dimer models and confirmed by DMRG and CFT analysis (Chepiga et al., 2018). The underlying critical theory is inherited through duality from the special exclusion rules of anyonic fusion and hard-boson occupation.
Integrability and conformal spectra: At specific fine-tuned couplings, golden and generalized anyon chains admit integrable structures. The mapping to restricted solid-on-solid (RSOS) and composite height models enables solution via corner transfer matrix methods. In these cases, conformal spectra are governed by Zₖ parafermionic CFTs (AFM regime) or coset theories such as $\mathrm{su}(2)_1 \times \mathrm{su}(2)_1 \times \mathrm{su}(2)_{k-2}/\mathrm{su}(2)_k$ (FM regime) (1110.0719). For SO(5)_2 anyonic chains, CFTs with extended chiral algebras appear, such as $\mathcal{W}B_2$ (c=8/7) and $\mathcal{W}D_5$ (c=1) (Finch et al., 2014).
Emergence of irrational CFTs: Coupled golden anyon chains with preserved non-invertible (Fibonacci) symmetries yield, via MPS/DMRG numerics, quantum critical phases with central charges 1.15 (N=2 chains) or 1.54 (N=3), operator scaling dimensions inconsistent with rational CFTs, and only the stress tensor as a conserved current, strongly suggesting compact irrational CFTs as emergent descriptions (Antunes et al., 18 Jul 2025). The lattice construction ensures each chain retains its Fibonacci topological symmetry.
Disorder and infinite randomness: The disordered golden chain displays random singlet and mixed infinite-randomness phases, with RG flows tracked by real-space methods. The effective central charge c_eff increases from ln(φ) ≈ 0.481 (singlet) to ≈ 0.702 (mixed), directly violating the monotonicity expected from a conventional c-theorem (0807.1123).

The upshot is a direct quantitative link between anyonic chain Hamiltonians, their topological symmetry content, and the universality class of quantum critical points realized in one dimension.

4. Topological Symmetry, Sectors, and Boundary Phenomena

Non-invertible topological symmetries, encoded by defect lines satisfying the fusion algebra W × W = 1 + W, play a central role in the classification and protection of criticality in golden anyon chains (Antunes et al., 18 Jul 2025, 1103.0159). The following features are salient:

Classification of scaling sectors: Local scaling operators are systematically classified by specifying which topological charge measurements are supported in regions complementary to the anyon chain on arbitrary manifolds (disk, torus, etc.), with fusion multiplicity tensors (e.g., $N^{z_1}_{\overline{y_1}y_2}$ ) controlling equivalence between sectors (1103.0159).
Protection of criticality: For the AFM golden chain on the ring without free charge exchange, translation invariance suppresses all relevant perturbations except a $\sigma$ -type operator with scaling dimension 7/8, and even this is forbidden by symmetry. However, once charge can be exchanged with the environment, additional relevant operators appear, destabilizing the critical phase (1103.0159).
Boundary/field correspondence: Matching field-theoretic boundary conditions to distinct conformal towers is achieved by exploiting the mapping of the chain to dimer or hard-boson models. DMRG studies on varying boundary polarizations establish selection of specific tricritical Ising or Potts conformal towers (Chepiga et al., 2018).

This explicit control of topological sector content not only stabilizes critical phases but also determines edge and impurity physics in open or finite geometries.

5. Numerical and Analytical Methodologies

Analytical and computational methods tailored to the fusion-constrained Hilbert space include:

Tensor network approaches: Anyonic matrix product states (MPS) and density matrix renormalization group (DMRG) algorithms natively incorporate non-Abelian fusion and charge structure, using hybrid normalization conventions and explicit fusing/splitting tensors. DMRG yields ground state energies for the golden chain matching or surpassing traditional matrix product methods, and is readily generalized to periodic topologies (Pfeifer et al., 2015).
Entanglement renormalization/MERA: Anyonic MERA, using charge-preserving isometries and unitary moves constructed from F-transforms and diagrammatic calculus, provides variational descriptions of ground states and algebraic two-point correlators, with computational scaling optimized via symmetry blocking. The ansatz also naturally encodes non-Abelian statistics and is successful at reproducing exactly known ground state structures at integrable points (1006.2478).
Valence-bond Monte Carlo: VBMC adapted to nonorthogonal valence-bond bases with nontrivial quantum dimension d simulates energy and entanglement properties of "temperley–lieb" anyon chain models, with direct applications to the golden chain and broader fractional quantum Hall contexts (1008.1819).
Exact solutions via Bethe ansatz and integrable mappings: Bethe equations are derived for critical models and integrable lines, using RSOS/composite height model techniques or interaction-round-a-face (IRF) algebraic frameworks. Conformal data and modular properties are extracted from finite-size scaling and comparison with known CFT characters (1110.0719, 1211.4449, Finch et al., 2014).

These tools enable not just calculation of spectra or ground state energy, but full access to correlation, entanglement, and topological properties sensitive to the underlying anyonic algebra.

6. Quantum Information and Topological Quantum Computation

Golden anyons are central to topological quantum computation because braiding Fibonacci anyons realizes a set of gates dense in SU(2). Braiding operations (implemented via sequences of F- and R-moves) on three-τ anyon logical qubits enact non-Clifford gates: $\sigma_2\,\overline{|0\rangle} = \phi^{-1}\,e^{4\pi i/5}\,\overline{|0\rangle} + \phi^{-1/2}\,e^{-3\pi i/5}\,\overline{|1\rangle},$ with phases and amplitudes fixed by the golden ratio (Minev et al., 18 Jun 2024). Experimental realization through the dynamic string-net preparation (DSNP) protocol on superconducting qubits has achieved anyon creation, braiding, and charge measurement with average accuracy of 94%, and demonstrated extraction of the golden ratio in fusion probabilities at 98% accuracy (Minev et al., 18 Jun 2024).

Beyond quantum information, the amplitude of a string-net configuration in the Fibonacci string-net condensate encodes, up to normalization, the chromatic polynomial of the dual graph evaluated at $\phi+2$ , a #P-hard classical invariant. Thus, sampling these states realizes classical computationally intractable tasks directly in quantum many-body physics (Minev et al., 18 Jun 2024).

7. Chiral Topological Order, Higher Dimensions, and Novel Phases

Recent advances generalize golden anyon chains to 2+1D lattice models enforcing exact modular tensor category (MTC) symmetry on the lattice, which generates robust chiral topological phases with protected edge states (Ueda et al., 5 Aug 2024). Here, a 2+1D Hamiltonian with MTC input for Fibonacci anyons supports gapless edge modes, with entanglement spectra matching those of chiral CFTs (central charges near 0.7–0.8 for Fibonacci, 0.5 for Ising). This approach does not rely on commuting projectors or solvability and highlights the essential role of non-invertible categorical symmetry in stabilizing topological order and sharply defined edge physics.

Additionally, coupling multiple golden chains in a symmetry-preserving manner leads to emergent irrational CFTs with discrete spectra but lacking extra conserved currents, broadening the landscape of attainable quantum critical theories in 1+1D (Antunes et al., 18 Jul 2025).

In summary, golden anyon chains constitute a highly tractable, physically motivated setting for investigating non-Abelian lattice topological phases, conformal quantum criticality—including both rational and irrational CFTs—robustness to disorder, edge phenomena, and the interplay of non-invertible symmetries, and they offer a rich platform for experimental realization of quantum computational and classically-intractable measurement tasks.