arXiv:1508.04793 · Physical Review A

Degeneracy and non-Abelian statistics

Eric C. Rowell, Zhenghan Wang

Published Mar 7, 2016• 72 citations

Abstract

A non-Abelian anyon can only occur in the presence of ground-state degeneracy in the plane. It is conceivable that for some strange anyon with quantum dimension $>1$ that the resulting representations of all $n$-strand braid groups ${B}_{n}$ are overall phases, even though the ground-state manifolds for $n$ such anyons in the plane are in general Hilbert spaces of dimensions $>1$. We observe that degeneracy is all that is needed: For an anyon with quantum dimension $>1$ the non-Abelian statistics cannot all be overall phases on the degeneracy ground-state manifold. Therefore, degeneracy implies non-Abelian statistics, which justifies defining a non-Abelian anyon as one with quantum dimension $>1$. Since non-Abelian statistics presumes degeneracy, degeneracy is more fundamental than non-Abelian statistics.

DOI: 10.1103/physreva.93.030102

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arXiv:1508.04793 · Physical Review A

Degeneracy and non-Abelian statistics

Eric C. Rowell, Zhenghan Wang

Published Mar 7, 2016• 72 citations

arXiv HTML PDF

Abstract

A non-Abelian anyon can only occur in the presence of ground-state degeneracy in the plane. It is conceivable that for some strange anyon with quantum dimension

>1

that the resulting representations of all

n

-strand braid groups

{B}_{n}

are overall phases, even though the ground-state manifolds for

n

such anyons in the plane are in general Hilbert spaces of dimensions

>1

. We observe that degeneracy is all that is needed: For an anyon with quantum dimension

>1

the non-Abelian statistics cannot all be overall phases on the degeneracy ground-state manifold. Therefore, degeneracy implies non-Abelian statistics, which justifies defining a non-Abelian anyon as one with quantum dimension

>1

. Since non-Abelian statistics presumes degeneracy, degeneracy is more fundamental than non-Abelian statistics.

DOI: 10.1103/physreva.93.030102

Related papers

Non-Abelian Anyons: Theoretical Promise, Experimental Gap — E8 Intelligence Research

Andrew Stewart Caldin

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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.

Jan 1, 20173 citations

A non-Abelian anyon can only occur in the presence of ground-state degeneracy in the plane. It is conceivable that for some strange anyon with quantum dimension

>1

that the resulting representations of all

n

-strand braid groups

{B}_{n}

are overall phases, even though the ground-state manifolds for

n

such anyons in the plane are in general Hilbert spaces of dimensions

>1

. We observe that degeneracy is all that is needed: For an anyon with quantum dimension

>1

>1

. Since non-Abelian statistics presumes degeneracy, degeneracy is more fundamental than non-Abelian statistics.

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}

(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