While it is well known that three dimensional quantum many-body systems can support nontrivial braiding statistics between particlelike and looplike excitations, or between two looplike excitations, we argue that a more fundamental quantity is the statistical phase associated with braiding one loop α around another loop β, while both are linked to a third loop γ. We study this three-loop braiding in the context of (Z(N))(K) gauge theories which are obtained by gauging a gapped, short-range entangled lattice boson model with (Z(N))(K) symmetry. We find that different short-range entangled bosonic states with the same (Z(N))(K) symmetry (i.e., different symmetry-protected topological phases) can be distinguished by their three-loop braiding statistics.While it is well known that three dimensional quantum many-body systems can support nontrivial braiding statistics between particlelike and looplike excitations, or between two looplike excitations, we argue that a more fundamental quantity is the statistical phase associated with braiding one loop α around another loop β, while both are linked to a third loop γ. We study this three-loop braiding in the context of (Z(N))(K) gauge theories which are obtained by gauging a gapped, short-range entangled lattice boson model with (Z(N))(K) symmetry. We find that different short-range entangled bosonic states with the same (Z(N))(K) symmetry (i.e., different symmetry-protected topological phases) can be distinguished by their three-loop braiding statistics.

arXiv:1403.7437Physical Review LettersAug 19, 2014193 citations

Fibonacci Anyons Enable Universal Quantum Computation via Hexagonal Symmetry — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Fibonacci anyons realize non-Abelian braiding statistics with quantum dimension φ, enabling universal topological quantum computation via the A₂ root system's hexagonal symmetry. MATH: - Quantum dimension of Fibonacci anyon: $d = \phi = \frac{1+\sqrt{5}}{2} \approx 1.618$ - Fusion rule: $\tau \times \tau = 1 + \tau$ (where τ is the Fibonacci anyon) - Braiding matrices satisfy the Yang–Baxter equation and produce the Fibonacci representation of the braid group $B_n$ - A₂ root system: 6 roots at 60° angles, hexagonal lattice, Coxeter number 3, Weyl group order 6 CONNECTION: - The quantum dimension φ is exactly the golden ratio, linking anyon statistics to the Fibonacci sequence and pentagonal/quasicrystalline symmetry. - The A₂ root system's hexagonal lattice (crystallographic symmetry) underlies the fusion and braiding structure of Fibonacci anyons; the anyon's fusion algebra corresponds to the representation theory of the quantum group $ U_q(sl_2) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Fibonacci anyons realize non-Abelian braiding statistics with quantum dimension φ, enabling universal topological quantum computation via the A₂ root system's hexagonal symmetry. MATH: - Quantum dimension of Fibonacci anyon: \( d = \phi = \frac{1+\sqrt{5}}{2} \approx 1.618 $ - Fusion rule: $ \tau \times \tau = 1 + \tau $ (where τ is the Fibonacci anyon) - Braiding matrices satisfy the Yang–Baxter equation and produce the Fibonacci representation of the braid group $ B_n $ - A₂ root system: 6 roots at 60° angles, hexagonal lattice, Coxeter number 3, Weyl group order 6 CONNECTION: - The quantum dimension φ is exactly the golden ratio, linking anyon statistics to the Fibonacci sequence and pentagonal/quasicrystalline symmetry. - The A₂ root system's hexagonal lattice (crystallographic symmetry) underlies the fusion and braiding structure of Fibonacci anyons; the anyon's fusion algebra corresponds to the representation theory of the quantum group \( U_q(sl_2) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Jul 25, 20260 citations

Braid Statistics in Three-Dimensional Local Quantum Theory

Jürg Martin Fröhlich, Fabrizio Gabbiani, P. A. Marchetti

NATO ASI series. Series B : PhysicsJan 1, 199025 citations

d = \phi = \frac{1+\sqrt{5}}{2} \approx 1.618

- Fusion rule:

\tau \times \tau = 1 + \tau

(where τ is the Fibonacci anyon) - Braiding matrices satisfy the Yang–Baxter equation and produce the Fibonacci representation of the braid group

B_n

- A₂ root system: 6 roots at 60° angles, hexagonal lattice, Coxeter number 3, Weyl group order 6 CONNECTION: - The quantum dimension φ is exactly the golden ratio, linking anyon statistics to the Fibonacci sequence and pentagonal/quasicrystalline symmetry. - The A₂ root system's hexagonal lattice (crystallographic symmetry) underlies the fusion and braiding structure of Fibonacci anyons; the anyon's fusion algebra corresponds to the representation theory of the quantum group \( U_q(sl_2) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com