A method of construction of harmonic functions for the system of anyons on a plane is given. Spectra of angular momentum are derived. Anyonic statistics implies a restriction of the range of angular momentum which is stronger in comparison to the bosonic and fermion case.

Journal of Mathematical PhysicsMar 1, 20012 citations

Non-Abelian quasi-particles in electronic systems

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arXiv:1611.05848 · SciPost Physics

The Conformal Spectrum of Non-Abelian Anyons

Nima Doroud, David Tong, Carl Turner

Published Apr 30, 2018• 7 citations

arXiv HTML PDF

Abstract

DOI: 10.21468/scipostphys.4.4.022

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Ever since the proposal of Kitaev for decoherence-free quantum computing based on non-Abelian anyons physical realizations of these exotic particles have been investigated extensively. Starting from one-dimensional models of interacting fermions with different symmetries the emergence and condensation of $\mathfrak{su}(2)_{N_f}$ , $\mathfrak{su}(3)_{N_f}$ and $\mathfrak{so}(5)_{N_f}$ anyons is studied in the framework of integrable perturbed WZNW models. For sufficiently small temperatures and fields non-Abelian anyons residing on massive solitonic excitations are identified by their quantum dimension. By tuning the external fields the density of anyons can be increased continuously to study the effect of interactions between them. For each model the conformal field theories describing the various collective states of the interacting anyons are determined and a phase diagram for the anyonic modes is proposed.Ever since the proposal of Kitaev for decoherence-free quantum computing based on non-Abelian anyons physical realizations of these exotic particles have been investigated extensively. Starting from one-dimensional models of interacting fermions with different symmetries the emergence and condensation of $\mathfrak{su}(2)_{N_f}$, $\mathfrak{su}(3)_{N_f}$ and $\mathfrak{so}(5)_{N_f}$ anyons is studied in the framework of integrable perturbed WZNW models. For sufficiently small temperatures and fields non-Abelian anyons residing on massive solitonic excitations are identified by their quantum dimension. By tuning the external fields the density of anyons can be increased continuously to study the effect of interactions between them. For each model the conformal field theories describing the various collective states of the interacting anyons are determined and a phase diagram for the anyonic modes is proposed.

Jan 1, 20180 citations

θ

θ= 0, π

correspond to bosons and fermions respectively; at intermediate values we say that we have fractional statistics. In two dimensions,

θ

describes the phase acquired by the wave function as two anyons wind around one another counterclockwise. It generates a shift in the allowed values for the relative angular momentum. Composites of localized electric charge and magnetic flux associated with an abelian U(1) gauge group realize this behavior. More complex charge-flux constructions can involve non-abelian and product groups acting on a spectrum of allowed charges and fluxes, giving rise to nonabelian and mutual statistics. Interchanges of non-abelian anyons implement unitary transformations of the wave function within an emergent space of internal states. Anyons of all kinds are described by quantum field theories that include Chern--Simons terms. The crossings of one-dimensional anyons on a ring are uni-directional, such that a fractional phase

θ

acquired upon interchange gives rise to fractional shifts in the relative momenta between the anyons. The quasiparticle excitations of fractional quantum Hall states have long been predicted to include anyons. Recently the anyon behavior predicted for quasiparticles in the

ν= 1/3

fractional quantum Hall state has been observed both in scattering and in interferometric experiments. Excitations within designed systems, notably including superconducting circuits, can exhibit anyon behavior. Such systems are being developed for possible use in quantum information processing.

\mathfrak{su}(2)_{N_f}

\mathfrak{su}(3)_{N_f}

and

\mathfrak{so}(5)_{N_f}

anyons is studied in the framework of integrable perturbed WZNW models. For sufficiently small temperatures and fields non-Abelian anyons residing on massive solitonic excitations are identified by their quantum dimension. By tuning the external fields the density of anyons can be increased continuously to study the effect of interactions between them. For each model the conformal field theories describing the various collective states of the interacting anyons are determined and a phase diagram for the anyonic modes is proposed.