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 $\mathcal{B}$ 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 $\mathcal{N}$ N qubit computational spaces in terms of conformal blocks of $2\mathcal{N}$ 2 N + 2 Fibonacci anyons.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 $$ \mathcal{B} $$ 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 $$ \mathcal{N} $$ N qubit computational spaces in terms of conformal blocks of $$ 2\mathcal{N} $$ 2 N + 2 Fibonacci anyons.

arXiv:2404.01778Journal of High Energy PhysicsAug 12, 20243 citations

Interacting anyons in one dimension: The Fibonacci chain

Andreas W. W. Ludwig, Adrian E. Feiguin, Simon Trebst +4 more

Bulletin of the American Physical SocietyMar 6, 20070 citations

Quantum computation with Fibonacci anyons

Claire Levaillant

Journal of Material Science & EngineeringNov 16, 20160 citations

Effective models of doped quantum ladders of non-Abelian anyons

Medha Soni, Matthias Troyer, Didier Poilblanc

Quantum spin models have been studied extensively in one and higher dimensions. Furthermore, these systems have been doped with holes to study $t\text{\ensuremath{-}}J$ models of $\mathrm{SU}(2)$ spin-1/2. Their anyonic counterparts can be built from non-Abelian anyons, such as Fibonacci anyons described by $\mathrm{SU}{(2)}_{3}$ theories, which are quantum deformations of the $\mathrm{SU}(2)$ algebra. Inspired by the physics of $\mathrm{SU}(2)$ spins, several works have explored ladders of Fibonacci anyons and also one-dimensional (1D) $t\text{\ensuremath{-}}J$ models. Here, we aim to explore the combined effects of extended dimensionality and doping by studying ladders composed of coupled chains of interacting itinerant Fibonacci anyons. We show analytically that in the limit of strong rung couplings these models can be mapped onto effective 1D models. These effective models can either be gapped models of hole pairs, or gapless models described by $t\text{\ensuremath{-}}J$ (or modified $t\text{\ensuremath{-}}J\text{\ensuremath{-}}V$) chains of Fibonacci anyons, whose spectrum exhibits a fractionalization into charge and anyon degrees of freedom. The charge degrees of freedom are described by the hardcore boson spectra while the anyon sector is given by a chain of localized interacting anyons. By using exact diagonalizations for two-leg and three-leg ladders, we show that indeed the doped ladders show exactly the same behavior as that of $t\text{\ensuremath{-}}J$ chains. In the strong ferromagnetic rung limit, we can obtain a new model that hosts two different kinds of Fibonacci particles, which we denote as the heavy $\ensuremath{\tau}$'s and light $\ensuremath{\tau}$'s. These two particle types carry the same (non-Abelian) topological charge but different (Abelian) electric charges. Once again, we map the two-dimensional ladder onto an effective chain carrying these heavy and light $\ensuremath{\tau}$'s. We perform a finite size scaling analysis to show the appearance of gapless modes for certain anyon densities, whereas a topological gapped phase is suggested for another density regime.Quantum spin models have been studied extensively in one and higher dimensions. Furthermore, these systems have been doped with holes to study $t\text{\ensuremath{-}}J$ models of $\mathrm{SU}(2)$ spin-1/2. Their anyonic counterparts can be built from non-Abelian anyons, such as Fibonacci anyons described by $\mathrm{SU}{(2)}_{3}$ theories, which are quantum deformations of the $\mathrm{SU}(2)$ algebra. Inspired by the physics of $\mathrm{SU}(2)$ spins, several works have explored ladders of Fibonacci anyons and also one-dimensional (1D) $t\text{\ensuremath{-}}J$ models. Here, we aim to explore the combined effects of extended dimensionality and doping by studying ladders composed of coupled chains of interacting itinerant Fibonacci anyons. We show analytically that in the limit of strong rung couplings these models can be mapped onto effective 1D models. These effective models can either be gapped models of hole pairs, or gapless models described by $t\text{\ensuremath{-}}J$ (or modified $t\text{\ensuremath{-}}J\text{\ensuremath{-}}V$) chains of Fibonacci anyons, whose spectrum exhibits a fractionalization into charge and anyon degrees of freedom. The charge degrees of freedom are described by the hardcore boson spectra while the anyon sector is given by a chain of localized interacting anyons. By using exact diagonalizations for two-leg and three-leg ladders, we show that indeed the doped ladders show exactly the same behavior as that of $t\text{\ensuremath{-}}J$ chains. In the strong ferromagnetic rung limit, we can obtain a new model that hosts two different kinds of Fibonacci particles, which we denote as the heavy $\ensuremath{\tau}$'s and light $\ensuremath{\tau}$'s. These two particle types carry the same (non-Abelian) topological charge but different (Abelian) electric charges. Once again, we map the two-dimensional ladder onto an effective chain carrying these heavy and light $\ensuremath{\tau}$'s. We perform a finite size scaling analysis to show the appearance of gapless modes for certain anyon densities, whereas a topological gapped phase is suggested for another density regime.

arXiv:1508.04160Physical review. B./Physical review. BJan 21, 20168 citations

\mathcal{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

\mathcal{N}

N qubit computational spaces in terms of conformal blocks of

2\mathcal{N}

2 N + 2 Fibonacci anyons.

Quantum spin models have been studied extensively in one and higher dimensions. Furthermore, these systems have been doped with holes to study $t\text{\ensuremath{-}}J$ models of

\mathrm{SU}(2)

spin-1/2. Their anyonic counterparts can be built from non-Abelian anyons, such as Fibonacci anyons described by

\mathrm{SU}{(2)}_{3}

theories, which are quantum deformations of the

\mathrm{SU}(2)

algebra. Inspired by the physics of

\mathrm{SU}(2)

spins, several works have explored ladders of Fibonacci anyons and also one-dimensional (1D) $t\text{\ensuremath{-}}J$ models. Here, we aim to explore the combined effects of extended dimensionality and doping by studying ladders composed of coupled chains of interacting itinerant Fibonacci anyons. We show analytically that in the limit of strong rung couplings these models can be mapped onto effective 1D models. These effective models can either be gapped models of hole pairs, or gapless models described by $t\text{\ensuremath{-}}J$ (or modified $t\text{\ensuremath{-}}J\text{\ensuremath{-}}V$) chains of Fibonacci anyons, whose spectrum exhibits a fractionalization into charge and anyon degrees of freedom. The charge degrees of freedom are described by the hardcore boson spectra while the anyon sector is given by a chain of localized interacting anyons. By using exact diagonalizations for two-leg and three-leg ladders, we show that indeed the doped ladders show exactly the same behavior as that of $t\text{\ensuremath{-}}J$ chains. In the strong ferromagnetic rung limit, we can obtain a new model that hosts two different kinds of Fibonacci particles, which we denote as the heavy $\ensuremath{\tau}$'s and light $\ensuremath{\tau}$'s. These two particle types carry the same (non-Abelian) topological charge but different (Abelian) electric charges. Once again, we map the two-dimensional ladder onto an effective chain carrying these heavy and light $\ensuremath{\tau}$'s. We perform a finite size scaling analysis to show the appearance of gapless modes for certain anyon densities, whereas a topological gapped phase is suggested for another density regime.