We consider the braid group representation which describes the non-abelian braiding statistics of the spin 1/2 1 / 2 particle world lines of an SU(2) _4 4 Chern-Simons theory. Up to an abelian phase, this is the same as the non-Abelian statistics of the elementary quasiparticles of the k=4 k = 4 Read-Rezayi quantum Hall state. We show that these braiding properties can be represented exactly using \mathbbm{Z}_3 parafermion operators.
We consider the braid group representation which describes the non-abelian braiding statistics of the spin 1/2 1 / 2 particle world lines of an SU(2) _4 4 Chern-Simons theory. Up to an abelian phase, this is the same as the non-Abelian statistics of the elementary quasiparticles of the k=4 k = 4 Read-Rezayi quantum Hall state. We show that these braiding properties can be represented exactly using \mathbbm{Z}_3 parafermion operators.
Majorana zero modes (MZMs) are canonically described by the Ising topological quantum field theory (TQFT), equivalent to SU(2)2 Chern-Simons theory. We investigate whether the braid-group representation governing MZMs could instead arise as the non-Abelian sector of an SO(3)2 Chern-Simons theory (equivalent to SU(2)4), and whether four-point interferometry can experimentally distinguish these topological orders. The non-Abelian sector of SO(3)2 yields a braid-group representation locally indistinguishable from Ising anyons for two-quasiparticle braiding, but differs fundamentally in its fusion rules. Specifically, the SO(3)2 non-Abelian anyon v (spin-1) obeys v×v=1+v+ψ, whereas the Ising σ anyon obeys σ×σ=1+ψ. We compute the four-point braiding matrices and evaluate the interferometric visibility for a Fabry-Pérot interferometer enclosing four anyons. The presence of the v×v→v fusion channel introduces an additional interference path with a distinct conformal weight, providing a falsifiable signature distinguishing SO(3)2 from Ising. ## 1. Introduction Majorana zero modes (MZMs) are localized zero-energy excitations in topological superconductors that exhibit non-Abelian statistics. Canonically, MZMs are described by the Ising topological quantum field theory (TQFT), a framework describing topological phases of matter, which corresponds to the SU(2)2 Chern-Simons gauge theory. Chern-Simons theory is a topological gauge theory in 2+1 dimensions whose quantization yields anyonic quasiparticles. We investigate whether the braid-group representation governing MZMs could instead arise as the non-Abelian sector of an SO(3)2 Chern-Simons theory. We demonstrate that while two-quasiparticle braiding is locally indistinguishable from Ising anyons, the theories differ fundamentally in their fusion rules. We propose that four-point interferometry can experimentally distinguish these two topological orders by det
We study $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ensuremath{\rtimes}{Z}_{2}$ Chern-Simons theory with integral coupling constants (k,l) and its relation to certain non-Abelian fractional quantum Hall (FQH) states. For the $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ensuremath{\rtimes}{Z}_{2}$ Chern-Simons theory, we show how to compute the dimension of its Hilbert space on genus g surfaces and how this yields the quantum dimensions of topologically distinct excitations. We find that Z2 vortices in the $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ensuremath{\rtimes}{Z}_{2}$ Chern-Simons theory carry non-Abelian statistics and we show how to compute the dimension of the Hilbert space in the presence of n pairs of Z2 vortices on a sphere. These results allow us to show that l=3 $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ensuremath{\rtimes}{Z}_{2}$ Chern-Simons theory is the low-energy effective theory for the Z4 parafermion (Read-Rezayi) fractional quantum Hall states, which occur at filling fraction $\ensuremath{\nu}=\frac{2}{2k\ensuremath{-}3}$. The $U(1)\ifmmode\times\else\texttimes\fi{}U(1)\ensuremath{\rtimes}{Z}_{2}$ theory is more useful than an alternative $SU{(2)}_{4}\ifmmode\times\else\texttimes\fi{}U(1)∕U(1)$ Chern-Simons theory because the fields are more closely related to physical degrees of freedom of the electron fluid and to an Abelian bilayer phase on the other side of a two-component to single-component quantum phase transition. We discuss the possibility of using this theory to understand further phase transitions in FQH systems, especially the $\ensuremath{\nu}=2∕3$ phase diagram.
We study U(1)×U(1)⋊Z2 Chern-Simons theory with integral coupling constants (k,l) and its relation to certain non-Abelian fractional quantum Hall (FQH) states. For the U(1)×U(1)⋊Z2 Chern-Simons theory, we show how to compute the dimension of its Hilbert space on genus g surfaces and how this yields the quantum dimensions of topologically distinct excitations. We find that Z2 vortices in the U(1)×U(1)⋊Z2 Chern-Simons theory carry non-Abelian statistics and we show how to compute the dimension of the Hilbert space in the presence of n pairs of Z2 vortices on a sphere. These results allow us to show that l=3 U(1)×U(1)⋊Z2 Chern-Simons theory is the low-energy effective theory for the Z4 parafermion (Read-Rezayi) fractional quantum Hall states, which occur at filling fraction ν=2/2k−3. The U(1)×U(1)⋊Z2 theory is more useful than an alternative SU(2)4×U(1)∕U(1) Chern-Simons theory because the fields are more closely related to physical degrees of freedom of the electron fluid and to an Abelian bilayer phase on the other side of a two-component to single-component quantum phase transition. We discuss the possibility of using this theory to understand further phase transitions in FQH systems, especially the ν=2∕3 phase diagram.
A bstract Fibonacci anyons ε provide the simplest possible model of non-Abelian fusion rules: [1] × [1] = [0] ⊕ [1]. We propose a conformal field theory construction of topological quantum registers based on Fibonacci anyons realized as quasiparticle excitations in the ℤ 3 parafermion fractional quantum Hall state. To this end, the results of Ardonne and Schoutens for the correlation function of four Fibonacci fields are extended to the case of arbitrary number n of quasi-holes and N = 3 r electrons. Special attention is paid to the braiding properties of the obtained correlators. We explain in details the construction of a monodromy representation of the Artin braid group B B n acting on n -point conformal blocks of Fibonacci anyons. The matrices of braid group generators are displayed explicitly for all n ≤ 8. A simple recursion formula makes it possible to extend without efforts the construction to any n . Finally, we construct N N qubit computational spaces in terms of conformal blocks of 2N 2 N + 2 Fibonacci anyons.
Observation of Time-Domain Braiding of Non-Abelian Anyons at ν=5/2 State
Unlike elementary particles, which obey either bosonic or fermionic exchange statistics, certain quasiparticles, known as anyons, are predicted to exhibit Abelian or non-Abelian braiding statistics. While braiding Abelian anyons modifies the wavefunction by a 'statistical phase', braiding non-Abelian anyons implements a unitary transformation of the state within a degenerate subspace of states. Experimental evidence of non-Abelian braiding has thus far remained elusive. Here, we report a 'time-domain braiding' signature of non-Abelian anyons in the ν=5/2 fractional quantum Hall state, by extending our previously demonstrated approach with Abelian anyons at ν=1/3. Our approach is based on measurements of the current fluctuations arising from weak partitioning of a highly dilute one-dimensional edge mode. We independently probe the partition noise of the downstream charged mode and also that of the upstream neutral mode. These independent measurements agree with our theoretical predictions for 'time-domain braiding' of the downstream Abelian and the upstream non-Abelian anyons, respectively, in the 'particle-hole Pfaffian' topological order. Together, these results provide evidence for the presence of non-Abelian anyons.
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 SCS=4πk∫Tr(A∧dA+32A∧A∧A); level k∈Z (integer QHE) or k=p/q (fractional QHE, e.g., k=1/3 for Laughlin state); anyon braiding phase θ=π/k (abelian) or unitary matrix Uab (non-abelian, e.g., Fibonacci anyons with U∼eiπ/5); entanglement entropy SA=αL−γ, with topological entanglement entropy γ=lnD, D=∑idi2 (quantum dimensions); lattice Chern-Simons reproduces continuum in limit a→0 with k quantized. CONNECTION: Level k=5 gives Fibonacci anyons with quantum dimension d=ϕ=1.618 (golden ratio) — di Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com