Some Simple Examples

Steven H. Simon

Published Sep 29, 2023• 0 citations

Abstract

Abstract We explicitly work out the details of some simple examples anyon theories. We start with a fusion ring and using consistency conditions we build F-matrices, then the possible braidings (R-matrices). We consider Z2 fusion rules, Fibonacci fusion rules and Z3 fusion rules as simple cases. We consider Ising fusion rules, obtaining eight possible (modular) unitary braided theories. We then describe several more abelian theories, including the toric code, and general ZN anyons. These can be extended to describe all possible abelian braided theories in terms of a limited number of so-called “prime” theories.

DOI: 10.1093/oso/9780198886723.003.0018

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Some Simple Examples

Steven H. Simon

Published Sep 29, 2023• 0 citations

Publisher

Abstract

DOI: 10.1093/oso/9780198886723.003.0018

Related papers

Anyons and Inherently Complex F-symbols

Matthew Buican, Peter Huston, Jiannis K. Pachos

Anyons in $2+1$ dimensions are not only characterized by exotic braiding statistics but also by intricate fusion properties. Two anyons may fuse into multiple topological charge sectors, and associativity of fusing three anyons to produce a fixed charge sector is governed by $F$ -symbols. While braiding invariants, such as the modular data, are typically complex valued, a complete description of general anyon models requires understanding the arithmetic properties of its fusion associativity data as well. The $F$ -symbols for many of the most common $2+1$ d topological orders, including all Abelian anyon models as well as Fibonacci and Ising anyons, can be made real valued. We show this phenomenon is not universal by exhibiting braided fusion categories whose $F$ -symbols cannot be made real. We call such $F$ -symbols "inherently complex." The examples we study lack a charge-conjugation symmetry and our results are therefore consistent with the converse of a statement proved in a companion work linking real $F$ -symbols in braided fusion categories with the existence of a suitable charge-conjugation symmetry. We analyse the smallest-rank braided fusion categories we know of with inherently complex $F$ -symbols: ${\rm Rep}(\mathbb{Z}_7\rtimes\mathbb{Z}_3)$ and ${\rm Rep}(\mathbb{Z}_5\rtimes\mathbb{Z}_4)$ . Consequently, the corresponding $\mathcal Z({\rm Rep}(\mathbb{Z}_7\rtimes\mathbb{Z}_3))$ and $\mathcal Z({\rm Rep}(\mathbb{Z}_5\rtimes\mathbb{Z}_4))$ anyon models also have inherently complex $F$ -symbols. Our presentation connects these examples with recent results on classifying anyons beyond modular data.Anyons in $2+1$ dimensions are not only characterized by exotic braiding statistics but also by intricate fusion properties. Two anyons may fuse into multiple topological charge sectors, and associativity of fusing three anyons to produce a fixed charge sector is governed by $F$-symbols. While braiding invariants, such as the modular data, are typically complex valued, a complete description of general anyon models requires understanding the arithmetic properties of its fusion associativity data as well. The $F$-symbols for many of the most common $2+1$d topological orders, including all Abelian anyon models as well as Fibonacci and Ising anyons, can be made real valued. We show this phenomenon is not universal by exhibiting braided fusion categories whose $F$-symbols cannot be made real. We call such $F$-symbols "inherently complex." The examples we study lack a charge-conjugation symmetry and our results are therefore consistent with the converse of a statement proved in a companion work linking real $F$-symbols in braided fusion categories with the existence of a suitable charge-conjugation symmetry. We analyse the smallest-rank braided fusion categories we know of with inherently complex $F$-symbols: ${\rm Rep}(\mathbb{Z}_7\rtimes\mathbb{Z}_3)$ and ${\rm Rep}(\mathbb{Z}_5\rtimes\mathbb{Z}_4)$. Consequently, the corresponding $\mathcal Z({\rm Rep}(\mathbb{Z}_7\rtimes\mathbb{Z}_3))$ and $\mathcal Z({\rm Rep}(\mathbb{Z}_5\rtimes\mathbb{Z}_4))$ anyon models also have inherently complex $F$-symbols. Our presentation connects these examples with recent results on classifying anyons beyond modular data.

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Andrew Stewart Caldin

FINDING: Non-Abelian anyon fusion algebra exhibits self-similarity under braid group representations, enabling 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 $|i-j|>1$ . - Hecke algebra relations: $(\sigma_i - q^{1/2})(\sigma_i + q^{-1/2}) = 0$ , with $q = e^{2\pi i/k}$ for SU(2)_k anyons. - Fusion rules: $\phi_a \times \phi_b = \sum_c N_{ab}^c \phi_c$ , where $N_{ab}^c$ are non-negative integers (fusion coefficients). - Fibonacci anyon fusion: $\tau \times \tau = 1 + \tau$ , with quantum dimension $d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618$ . CONNECTION: - Fibonacci anyon quantum dimension $d_\tau = \phi = 1.618$ (golden ratio). - Braid matrix eigenvalues for Fibonacci anyons: $e^{\pm 3\pi i/5}$ , related to golden ratio via $ 2\cos(3\pi/5) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Non-Abelian anyon fusion algebra exhibits self-similarity under braid group representations, enabling 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 $. - Hecke algebra relations: $ (\sigma_i - q^{1/2})(\sigma_i + q^{-1/2}) = 0 $, with $ q = e^{2\pi i/k} $ for SU(2)_k anyons. - Fusion rules: $ \phi_a \times \phi_b = \sum_c N_{ab}^c \phi_c $, where $ N_{ab}^c $ are non-negative integers (fusion coefficients). - Fibonacci anyon fusion: $ \tau \times \tau = 1 + \tau $, with quantum dimension $ d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618 $. CONNECTION: - Fibonacci anyon quantum dimension $ d_\tau = \phi = 1.618 $ (golden ratio). - Braid matrix eigenvalues for Fibonacci anyons: $ e^{\pm 3\pi i/5} $, related to golden ratio via \( 2\cos(3\pi/5) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Jul 21, 20260 citations

Anyons in

2+1

dimensions are not only characterized by exotic braiding statistics but also by intricate fusion properties. Two anyons may fuse into multiple topological charge sectors, and associativity of fusing three anyons to produce a fixed charge sector is governed by

F

-symbols. While braiding invariants, such as the modular data, are typically complex valued, a complete description of general anyon models requires understanding the arithmetic properties of its fusion associativity data as well. The

F

-symbols for many of the most common

2+1

d topological orders, including all Abelian anyon models as well as Fibonacci and Ising anyons, can be made real valued. We show this phenomenon is not universal by exhibiting braided fusion categories whose

F

-symbols cannot be made real. We call such

F

-symbols "inherently complex." The examples we study lack a charge-conjugation symmetry and our results are therefore consistent with the converse of a statement proved in a companion work linking real

F

-symbols in braided fusion categories with the existence of a suitable charge-conjugation symmetry. We analyse the smallest-rank braided fusion categories we know of with inherently complex

F

-symbols:

{\rm Rep}(\mathbb{Z}_7\rtimes\mathbb{Z}_3)

and

{\rm Rep}(\mathbb{Z}_5\rtimes\mathbb{Z}_4)

. Consequently, the corresponding

\mathcal Z({\rm Rep}(\mathbb{Z}_7\rtimes\mathbb{Z}_3))

and

\mathcal Z({\rm Rep}(\mathbb{Z}_5\rtimes\mathbb{Z}_4))

anyon models also have inherently complex

F

-symbols. Our presentation connects these examples with recent results on classifying anyons beyond modular data.

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

FINDING: Non-Abelian anyon fusion algebra exhibits self-similarity under braid group representations, enabling 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

. - Hecke algebra relations:

(\sigma_i - q^{1/2})(\sigma_i + q^{-1/2}) = 0

, with

q = e^{2\pi i/k}

for SU(2)_k anyons. - Fusion rules:

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

, where

N_{ab}^c

are non-negative integers (fusion coefficients). - Fibonacci anyon fusion:

\tau \times \tau = 1 + \tau

, with quantum dimension

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

. CONNECTION: - Fibonacci anyon quantum dimension

d_\tau = \phi = 1.618

(golden ratio). - Braid matrix eigenvalues for Fibonacci anyons:

e^{\pm 3\pi i/5}

, related to golden ratio via \( 2\cos(3\pi/5) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com