arXiv:2009.12709 · Complex Analysis and Operator Theory

Exchange and Exclusion in the Non-abelian Anyon Gas

Douglas Lundholm, Viktor Qvarfordt

Published Oct 1, 2026• 9 citations

Abstract

Abstract We review and develop the many-body spectral theory of ideal anyons, i.e. identical quantum particles in the plane whose exchange rules are governed by unitary representations of the braid group on N strands. Allowing for arbitrary rank (dependent on N ) and non-abelian representations, and letting $$N \rightarrow \infty $$ N → ∞ , this defines the ideal non-abelian many-anyon gas. We compute exchange operators and phases for a common and wide class of representations defined by fusion algebras, including the Fibonacci and Ising anyon models. Furthermore, we extend methods of statistical repulsion (Poincaré and Hardy inequalities) and a local exclusion principle (also implying a Lieb–Thirring inequality) developed for abelian anyons to arbitrary geometric anyon models, i.e. arbitrary sequences of unitary representations of the braid group, for which two-anyon exchange is nontrivial.

DOI: 10.1007/s11785-026-02050-z

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arXiv:2009.12709 · Complex Analysis and Operator Theory

Exchange and Exclusion in the Non-abelian Anyon Gas

Douglas Lundholm, Viktor Qvarfordt

Published Oct 1, 2026• 9 citations

arXiv HTML PDF

Abstract

N \rightarrow \infty

N → ∞ , this defines the ideal non-abelian many-anyon gas. We compute exchange operators and phases for a common and wide class of representations defined by fusion algebras, including the Fibonacci and Ising anyon models. Furthermore, we extend methods of statistical repulsion (Poincaré and Hardy inequalities) and a local exclusion principle (also implying a Lieb–Thirring inequality) developed for abelian anyons to arbitrary geometric anyon models, i.e. arbitrary sequences of unitary representations of the braid group, for which two-anyon exchange is nontrivial.

DOI: 10.1007/s11785-026-02050-z

Related papers

Non-Abelian Anyons: Statistical Repulsion and Topological Quantum Computation

Viktor Qvarfordt

As opposed to classical mechanics, quantum mechanical particles can be truly identical and lead to new and interesting phenomena. Identical particles can be of different types, determined by their exchange symmetry, which in turn gives rise to statistical repulsion. The exchange symmetry is given by a representation of the exchange group; the fundamental group of the configuration space of identical particles. In three dimensions the exchange group is the permutation group and there are only two types of identical particles; bosons and fermions. While any number of bosons can be at the same place or in the same state, fermions repel each other. In two dimensions the exchange group is the braid group and essentially any exchange symmetry is allowed, such particles are called anyons. Abelian anyons are described by abelian representations of the exchange group and can be seen as giving a continuous interpolation between bosons and fermions. Non-abelian anyons are much more complex and their statistical repulsion is yet largely unexplored. We use the framework of modular tensor categories to show how the statistical repulsion of non-abelian anyons depends on the exchange symmetry. The Fibonacci anyon model is studied, for which explicit results are obtained. We also show how Fibonacci anyons can be used to implement topological quantum computation, providing topologically stable quantum information encoded in the state of non-abelian anyons that can be manipulated via the non-abelian exchange symmetry.As opposed to classical mechanics, quantum mechanical particles can be truly identical and lead to new and interesting phenomena. Identical particles can be of different types, determined by their exchange symmetry, which in turn gives rise to statistical repulsion. The exchange symmetry is given by a representation of the exchange group; the fundamental group of the configuration space of identical particles. In three dimensions the exchange group is the permutation group and there are only two types of identical particles; bosons and fermions. While any number of bosons can be at the same place or in the same state, fermions repel each other. In two dimensions the exchange group is the braid group and essentially any exchange symmetry is allowed, such particles are called anyons. Abelian anyons are described by abelian representations of the exchange group and can be seen as giving a continuous interpolation between bosons and fermions. Non-abelian anyons are much more complex and their statistical repulsion is yet largely unexplored. We use the framework of modular tensor categories to show how the statistical repulsion of non-abelian anyons depends on the exchange symmetry. The Fibonacci anyon model is studied, for which explicit results are obtained. We also show how Fibonacci anyons can be used to implement topological quantum computation, providing topologically stable quantum information encoded in the state of non-abelian anyons that can be manipulated via the non-abelian exchange symmetry.

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Andrew Stewart Caldin

FINDING: Anyon braiding provides non-Abelian statistics for topological quantum computation, with braid group representations forming the computational basis. | 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 braid matrices $U(\sigma_i)$ with $U(\sigma_i)U(\sigma_{i+1})U(\sigma_i) = U(\sigma_{i+1})U(\sigma_i)U(\sigma_{i+1})$ ; Fibonacci anyons have fusion rule $\tau \times \tau = 1 + \tau$ , yielding quantum dimension $\phi = (1+\sqrt{5})/2 \approx 1.618$ (golden ratio); topological protection arises from ground-state degeneracy on genus- $g$ surfaces: $\dim \mathcal{H} = 4^g$ for toric code, or $2^g$ for Ising anyons. | CONNECTION: Fibonacci anyons' quantum dimension is *exactly* the golden ratio $\phi = 1.618$ , and its inverse $\phi^{-1} = 0.618$ appears in braid fusion probabilities; the Yang–Baxter equation is a braided analogue of th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Anyon braiding provides non-Abelian statistics for topological quantum computation, with braid group representations forming the computational basis. | 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 braid matrices $U(\sigma_i)$ with $U(\sigma_i)U(\sigma_{i+1})U(\sigma_i) = U(\sigma_{i+1})U(\sigma_i)U(\sigma_{i+1})$; Fibonacci anyons have fusion rule $\tau \times \tau = 1 + \tau$, yielding quantum dimension $\phi = (1+\sqrt{5})/2 \approx 1.618$ (golden ratio); topological protection arises from ground-state degeneracy on genus-$g$ surfaces: $\dim \mathcal{H} = 4^g$ for toric code, or $2^g$ for Ising anyons. | CONNECTION: Fibonacci anyons' quantum dimension is *exactly* the golden ratio $\phi = 1.618$, and its inverse $\phi^{-1} = 0.618$ appears in braid fusion probabilities; the Yang–Baxter equation is a braided analogue of th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Non-Abelian Anyon Fusion Mirrors Braid Groups for Fault-Tolerant Quantum Computing — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Non-Abelian anyon fusion algebras exhibit self-similarity that mirrors the braid group structure, enabling fault-tolerant topological quantum computation via braiding operations. MATH: - Braid group $B_n$ 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 $|i-j|>1$ . - Fusion algebra: $\phi_a \times \phi_b = \sum_c N_{ab}^c \phi_c$ , where $N_{ab}^c$ are non-negative integers (fusion coefficients). For non-Abelian anyons, $\sum_c N_{ab}^c > 1$ for some $a,b$ . - Fibonacci anyon model: single nontrivial particle $\tau$ with fusion rule $\tau \times \tau = 1 + \tau$ , leading to quantum dimension $d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618$ . - Braiding matrices yield phases $e^{i\theta}$ with $\theta = 2\pi/5, 4\pi/5$ (related to golden angle $2\pi/\phi^2 \approx 137.5^\circ$ ). CONNECTION: - Fibonacci anyon quant Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Non-Abelian anyon fusion algebras exhibit self-similarity that mirrors the braid group structure, enabling fault-tolerant topological quantum computation via braiding operations. MATH: - Braid group $ 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 $. - Fusion algebra: $ \phi_a \times \phi_b = \sum_c N_{ab}^c \phi_c $, where $ N_{ab}^c $ are non-negative integers (fusion coefficients). For non-Abelian anyons, $ \sum_c N_{ab}^c > 1 $ for some $ a,b $. - Fibonacci anyon model: single nontrivial particle $ \tau $ with fusion rule $ \tau \times \tau = 1 + \tau $, leading to quantum dimension $ d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618 $. - Braiding matrices yield phases $ e^{i\theta} $ with $ \theta = 2\pi/5, 4\pi/5 $ (related to golden angle $ 2\pi/\phi^2 \approx 137.5^\circ $). CONNECTION: - Fibonacci anyon quant Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Jul 20, 20260 citations

N \rightarrow \infty

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

FINDING: Anyon braiding provides non-Abelian statistics for topological quantum computation, with braid group representations forming the computational basis. | 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 braid matrices

U(\sigma_i)

with

U(\sigma_i)U(\sigma_{i+1})U(\sigma_i) = U(\sigma_{i+1})U(\sigma_i)U(\sigma_{i+1})

; Fibonacci anyons have fusion rule

\tau \times \tau = 1 + \tau

, yielding quantum dimension

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

(golden ratio); topological protection arises from ground-state degeneracy on genus-

g

surfaces:

\dim \mathcal{H} = 4^g

for toric code, or

2^g

for Ising anyons. | CONNECTION: Fibonacci anyons' quantum dimension is *exactly* the golden ratio

\phi = 1.618

, and its inverse

\phi^{-1} = 0.618

appears in braid fusion probabilities; the Yang–Baxter equation is a braided analogue of th 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

. - Fusion algebra:

\phi_a \times \phi_b = \sum_c N_{ab}^c \phi_c

, where

N_{ab}^c

are non-negative integers (fusion coefficients). For non-Abelian anyons,

\sum_c N_{ab}^c > 1

for some

a,b

. - Fibonacci anyon model: single nontrivial particle

\tau

with fusion rule

\tau \times \tau = 1 + \tau

, leading to quantum dimension

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

. - Braiding matrices yield phases

e^{i\theta}

with

\theta = 2\pi/5, 4\pi/5

(related to golden angle

2\pi/\phi^2 \approx 137.5^\circ

). CONNECTION: - Fibonacci anyon quant Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com