Chern-Simons Theory and Non-Abelian Anyons in Fractional Quantum Hall States — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 12, 2026• 0 citations

Abstract

FINDING: Chern-Simons theory at level $k$ governs fractional quantum Hall anyons; braiding produces non-abelian phases encoded in the Jones polynomial; lattice realizations confirm topological invariance; a novel CKM angle measurement is unrelated to topological physics. MATH: Chern-Simons action $S_{CS} = \frac{k}{4\pi}\int \text{Tr}(A\wedge dA + \frac{2}{3}A\wedge A\wedge A)$; level $k \in \mathbb{Z}$ (integer QHE) or $k = p/q$ (fractional QHE, e.g., $k=1/3$ for Laughlin state); anyon braiding phase $\theta = \pi/k$ (abelian) or unitary matrix $U_{ab}$ (non-abelian, e.g., Fibonacci anyons with $U \sim e^{i\pi/5}$); entanglement entropy $S_A = \alpha L - \gamma$, with topological entanglement entropy $\gamma = \ln \sqrt{D}$, $D = \sum_i d_i^2$ (quantum dimensions); lattice Chern-Simons reproduces continuum in limit $a \to 0$ with $k$ quantized. CONNECTION: Level $k=5$ gives Fibonacci anyons with quantum dimension $d = \phi = 1.618$ (golden ratio) — di Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.23309728

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Chern-Simons Theory and Non-Abelian Anyons in Fractional Quantum Hall States — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 12, 2026• 0 citations

Publisher

Abstract

FINDING: Chern-Simons theory at level (k) governs fractional quantum Hall anyons; braiding produces non-abelian phases encoded in the Jones polynomial; lattice realizations confirm topological invariance; a novel CKM angle measurement is unrelated to topological physics. MATH: Chern-Simons action (S_{CS} = \frac{k}{4\pi}\int \text{Tr}(A\wedge dA + \frac{2}{3}A\wedge A\wedge A)); level (k \in \mathbb{Z}) (integer QHE) or (k = p/q) (fractional QHE, e.g., (k=1/3) for Laughlin state); anyon braiding phase (\theta = \pi/k) (abelian) or unitary matrix (U_{ab}) (non-abelian, e.g., Fibonacci anyons with (U \sim e^{i\pi/5})); entanglement entropy (S_A = \alpha L - \gamma), with topological entanglement entropy (\gamma = \ln \sqrt{D}), (D = \sum_i d_i^2) (quantum dimensions); lattice Chern-Simons reproduces continuum in limit (a \to 0) with (k) quantized. CONNECTION: Level (k=5) gives Fibonacci anyons with quantum dimension (d = \phi = 1.618) (golden ratio) — di Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.23309728

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FINDING: Anyon braiding forms the computational substrate for topological quantum computation, with non-Abelian statistics encoded in braid group representations rather than local qubit states. | MATH: Braid group $B_n$ generators $\sigma_i$ satisfy $\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}$ (Yang–Baxter equation); non-Abelian anyons carry unitary representations $U(\sigma_i)$ with $U(\sigma_i)U(\sigma_j)=U(\sigma_j)U(\sigma_i)$ for $|i-j|\geq 2$ . Fibonacci anyons: fusion rule $\tau \times \tau = 1 + \tau$ , Hilbert space dimension grows as Fibonacci sequence $F_n$ , golden ratio $\phi = (1+\sqrt{5})/2 \approx 1.618$ appears as asymptotic dimension growth rate. Topological protection arises from degeneracy of ground states on genus- $g$ surfaces: dimension $= k^g$ for level- $k$ Chern–Simons theory. | CONNECTION: The Fibonacci anyon fusion space dimension ratio $F_{n+1}/F_n \to 1.618$ — direct geometric harmony link. Braid group relations ar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Anyon braiding forms the computational substrate for topological quantum computation, with non-Abelian statistics encoded in braid group representations rather than local qubit states. | MATH: Braid group $B_n$ generators $\sigma_i$ satisfy $\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}$ (Yang–Baxter equation); non-Abelian anyons carry unitary representations $U(\sigma_i)$ with $U(\sigma_i)U(\sigma_j)=U(\sigma_j)U(\sigma_i)$ for $|i-j|\geq 2$. Fibonacci anyons: fusion rule $\tau \times \tau = 1 + \tau$, Hilbert space dimension grows as Fibonacci sequence $F_n$, golden ratio $\phi = (1+\sqrt{5})/2 \approx 1.618$ appears as asymptotic dimension growth rate. Topological protection arises from degeneracy of ground states on genus-$g$ surfaces: dimension $= k^g$ for level-$k$ Chern–Simons theory. | CONNECTION: The Fibonacci anyon fusion space dimension ratio $F_{n+1}/F_n \to 1.618$ — direct geometric harmony link. Braid group relations ar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Andrew Stewart Caldin

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Andrew Stewart Caldin

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Sep 8, 20260 citations

B_n

generators

\sigma_i

satisfy

\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}

(Yang–Baxter equation); non-Abelian anyons carry unitary representations

U(\sigma_i)

with

U(\sigma_i)U(\sigma_j)=U(\sigma_j)U(\sigma_i)

for

|i-j|\geq 2

. Fibonacci anyons: fusion rule

\tau \times \tau = 1 + \tau

, Hilbert space dimension grows as Fibonacci sequence

F_n

, golden ratio

\phi = (1+\sqrt{5})/2 \approx 1.618

appears as asymptotic dimension growth rate. Topological protection arises from degeneracy of ground states on genus-

g

surfaces: dimension

= k^g

for level-

k

Chern–Simons theory. | CONNECTION: The Fibonacci anyon fusion space dimension ratio

F_{n+1}/F_n \to 1.618

— direct geometric harmony link. Braid group relations ar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

B

acts on degenerate ground-state manifold with unitary representation

\rho(B)

, where

\rho(B_1)\rho(B_2) \neq \rho(B_2)\rho(B_1)

for non-abelian anyons. Exchange phase

\theta

generalizes Fermi (

\theta=\pi

) and Bose (

\theta=0

) to

\theta = 2\pi s

with

s

fractional (e.g.,

s=1/3, 1/5

). Fibonacci anyon fusion rules:

\tau \times \tau = 1 + \tau

, with quantum dimension

d_\tau = \varphi = (1+\sqrt{5})/2 \approx 1.618

. | CONNECTION: **Strong.** The Fibonacci anyon quantum dimension is exactly the golden ratio

\varphi = 1.618

. Braid group

B_n

representations relate to Temperley–Lieb algebra with l Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

B_n

generators

\sigma_i

satisfy

\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}

and

\sigma_i \sigma_j = \sigma_j \sigma_i

for

|i-j|>1

. Anyonic exchange yields unitary matrices

U(\sigma_i)

with non-commuting phases

e^{i\theta}

, where

\theta

depends on topological charge (e.g., Ising anyons:

\theta = \pi/8

for

\sigma

particles). Fibonacci anyons: dimension

d = \phi = (1+\sqrt{5})/2 \approx 1.618

, with fusion rules

\tau \times \tau = 1 + \tau

. Majorana zero modes:

\gamma_i^\dagger = \gamma_i

\{\gamma_i, \gamma_j\} = 2\delta_{ij}

, parity

P = i^k \prod \gamma_i

. | CONNECTION: Fibonacci anyon dimension

d = \phi = 1.618

— direct golden ratio e Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com