Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0601242 · Physical Review Letters

Probing Non-Abelian Statistics with Quasiparticle Interferometry

Parsa Bonderson, Kirill Shtengel, Johannes K. Slingerland

Published Jul 3, 2006• 109 citations
arXivHTMLPDF

Abstract

We examine interferometric experiments in systems that exhibit non-Abelian braiding statistics, expressing outcomes in terms of the modular S-matrix. In particular, this result applies to fractional quantum Hall interferometry, and we give a detailed treatment of the Read-Rezayi states, providing explicit predictions for the recently observed nu = 12/5 plateau.

DOI
10.1103/physrevlett.97.016401

Related papers

Braiding Non-Abelian Quasiholes in Fractional Quantum Hall States

Yang-Le Wu, Benoit Estienne, Nicolas Regnault +1 more

Quasiholes in certain fractional quantum Hall states are promising candidates for the experimental realization of non-Abelian anyons. They are assumed to be localized excitations, and to display non-Abelian statistics when sufficiently separated, but these properties have not been explicitly demonstrated except for the Moore-Read state. In this work, we apply the newly developed matrix product state technique to examine these exotic excitations. For the Moore-Read and the Z_{3} Read-Rezayi states, we estimate the quasihole radii, and determine the correlation lengths associated with the exponential convergence of the braiding statistics. We provide the first microscopic verification for the Fibonacci nature of the Z_{3} Read-Rezayi quasiholes. We also present evidence for the failure of plasma screening in the nonunitary Gaffnian wave function.

arXiv:1405.1720•Physical Review Letters•Sep 8, 2014•56 citations

Monte Carlo Evaluation of Non-Abelian Statistics

Yaroslav Tserkovnyak, Steven H. Simon

We develop a general framework to (numerically) study adiabatic braiding of quasiholes in fractional quantum Hall systems. Specifically, we investigate the Moore-Read (MR) state at nu=1/2 filling factor, a known candidate for non-Abelian statistics, which appears to actually occur in nature. The non-Abelian statistics of MR quasiholes is demonstrated explicitly for the first time, confirming the results predicted by conformal field theories.

arXiv:cond-mat/0209650•Physical Review Letters•Jan 10, 2003•79 citations

Probing non-Abelian statistics in ν=12/5 quantum Hall state

K. T. Law

The tunneling current and shot noise of the current between two fractional quantum hall (FQH) edges in the $\ensuremath{\nu}=12/5$ FQH state in an electronic Mach--Zehnder interferometer are studied. It is shown that the tunneling current and shot noise can be used to probe the existence of $k=3$ parafermion statistics in the $\ensuremath{\nu}=12/5$ FQH state. More specifically, the dependence of the current on the Aharonov--Bohm flux in the Read--Rezayi state is asymmetric under the change in the sign of the applied voltage. This property is absent in the Abelian--Laughlin states. Moreover, the Fano factor can exceed 12.7 electron charges in the $\ensuremath{\nu}=12/5$ FQH state. This number well exceeds the maximum possible Fano factor in all Laughlin states and the $\ensuremath{\nu}=5/2$ Moore--Read state, which was previously shown to be $e$ and $3.2e$, respectively.

arXiv:0707.3995•Physical Review B•May 12, 2008•25 citations

Conformal Field Theory Approach to Abelian and Non-Abelian Quantum Hall Quasielectrons

T. H. Hansson, Maria Hermanns, Nicolas Regnault +1 more

The quasiparticles in quantum Hall liquids carry fractional charge and obey fractional quantum statistics. Of particular recent interest are those with non-Abelian statistics, since their braiding properties could, in principle, be used for robust coding of quantum information. There is already a good theoretical understanding of quasiholes in both Abelian and non-Abelian quantum Hall states. Here we develop conformal field theory methods that allow for an equally precise description of quasielectrons and explicitly construct two- and four-quasielectron excitations of the non-Abelian Moore-Read state.

arXiv:0810.0636•Physical Review Letters•Apr 23, 2009•38 citations

Effective 1D Language for Fractional Quantum Hall States

John Flavin

In the theory of the fractional quantum Hall effect, much attention is paid to the correspondence between fractional quantum Hall wave functions and conformal blocks in certain rational conformal field theories (CFT). This correspondence is powerful, enabling the calculation of the fractional statistics of a state that would be difficult if not impossible to calculate directly from the wave functions. But it is, in general, conjectural, remaining without microscopic justification in many cases of interest, and involves heavy mathematical machinery. We detail an alternative method to calculate Abelian and non-Abelian fractional statistics, the coherent state method. The method relies on assumptions which are independent of those underlying the CFT correspondence, so it serves as an independent check of results for the statistics where they exist, and an alternative source of results when a CFT is not known. We show how the coherent state method can be used to derive the statistics of several increasingly complicated trial wave functions: ν=1/2 Laughlin, Moore-Read, and k=3 Read-Rezayi. We discuss implications of our method for a possible notion of braiding statistics of the Gaffnian state, a "nonunitary" quantum Hall state for which the ramifications of the CFT correspondence are less well understood. We go on to derive formulas for the counting of zero modes of all these states on the torus.

•Jan 1, 2012•0 citations

Quasiparticle tunneling in the Moore-Read fractional quantum Hall state

Hua Chen, Zi-Xiang Hu, Kun Yang +2 more

In fractional quantum Hall systems, quasiparticles of fractional charge can tunnel between the edges at a quantum point contact. Such tunneling (or backscattering) processes contribute to charge transport and provide information on both the charge and statistics of the quasiparticles involved. Here, we study quasiparticle tunneling in the Moore-Read state, in which quasiparticles of charges $e/4$ (non-Abelian) and $e/2$ (Abelian) may coexist and both contribute to edge transport. On a disk geometry, we calculate the matrix elements for $e/2$ and $e/4$ quasiholes to tunnel through the bulk of the Moore-Read state, in an attempt to understand their relative importance. We find that the tunneling amplitude for charge $e/2$ quasihole is exponentially smaller than that for charge $e/4$ quasihole, and the ratio between them can be (partially) attributed to their charge difference. We find that including long-range Coulomb interaction only has a weak effect on the ratio. We discuss briefly the relevance of these results to recent tunneling and interferometry experiments at filling factor $\ensuremath{\nu}=5/2$.

arXiv:0905.3607•Physical Review B•Dec 3, 2009•17 citations
We examine interferometric experiments in systems that exhibit non-Abelian braiding statistics, expressing outcomes in terms of the modular S-matrix. In particular, this result applies to fractional quantum Hall interferometry, and we give a detailed treatment of the Read-Rezayi states, providing explicit predictions for the recently observed nu = 12/5 plateau.
Quasiholes in certain fractional quantum Hall states are promising candidates for the experimental realization of non-Abelian anyons. They are assumed to be localized excitations, and to display non-Abelian statistics when sufficiently separated, but these properties have not been explicitly demonstrated except for the Moore-Read state. In this work, we apply the newly developed matrix product state technique to examine these exotic excitations. For the Moore-Read and the Z_{3} Read-Rezayi states, we estimate the quasihole radii, and determine the correlation lengths associated with the exponential convergence of the braiding statistics. We provide the first microscopic verification for the Fibonacci nature of the Z_{3} Read-Rezayi quasiholes. We also present evidence for the failure of plasma screening in the nonunitary Gaffnian wave function.
The tunneling current and shot noise of the current between two fractional quantum hall (FQH) edges in the $\ensuremath{\nu}=12/5$ FQH state in an electronic Mach--Zehnder interferometer are studied. It is shown that the tunneling current and shot noise can be used to probe the existence of k=3k=3k=3 parafermion statistics in the $\ensuremath{\nu}=12/5$ FQH state. More specifically, the dependence of the current on the Aharonov--Bohm flux in the Read--Rezayi state is asymmetric under the change in the sign of the applied voltage. This property is absent in the Abelian--Laughlin states. Moreover, the Fano factor can exceed 12.7 electron charges in the $\ensuremath{\nu}=12/5$ FQH state. This number well exceeds the maximum possible Fano factor in all Laughlin states and the $\ensuremath{\nu}=5/2$ Moore--Read state, which was previously shown to be eee and 3.2e3.2e3.2e, respectively.
The quasiparticles in quantum Hall liquids carry fractional charge and obey fractional quantum statistics. Of particular recent interest are those with non-Abelian statistics, since their braiding properties could, in principle, be used for robust coding of quantum information. There is already a good theoretical understanding of quasiholes in both Abelian and non-Abelian quantum Hall states. Here we develop conformal field theory methods that allow for an equally precise description of quasielectrons and explicitly construct two- and four-quasielectron excitations of the non-Abelian Moore-Read state.
In the theory of the fractional quantum Hall effect, much attention is paid to the correspondence between fractional quantum Hall wave functions and conformal blocks in certain rational conformal field theories (CFT). This correspondence is powerful, enabling the calculation of the fractional statistics of a state that would be difficult if not impossible to calculate directly from the wave functions. But it is, in general, conjectural, remaining without microscopic justification in many cases of interest, and involves heavy mathematical machinery. We detail an alternative method to calculate Abelian and non-Abelian fractional statistics, the coherent state method. The method relies on assumptions which are independent of those underlying the CFT correspondence, so it serves as an independent check of results for the statistics where they exist, and an alternative source of results when a CFT is not known. We show how the coherent state method can be used to derive the statistics of several increasingly complicated trial wave functions: ν=1/2 Laughlin, Moore-Read, and k=3 Read-Rezayi. We discuss implications of our method for a possible notion of braiding statistics of the Gaffnian state, a "nonunitary" quantum Hall state for which the ramifications of the CFT correspondence are less well understood. We go on to derive formulas for the counting of zero modes of all these states on the torus.
In fractional quantum Hall systems, quasiparticles of fractional charge can tunnel between the edges at a quantum point contact. Such tunneling (or backscattering) processes contribute to charge transport and provide information on both the charge and statistics of the quasiparticles involved. Here, we study quasiparticle tunneling in the Moore-Read state, in which quasiparticles of charges e/4e/4e/4 (non-Abelian) and e/2e/2e/2 (Abelian) may coexist and both contribute to edge transport. On a disk geometry, we calculate the matrix elements for e/2e/2e/2 and e/4e/4e/4 quasiholes to tunnel through the bulk of the Moore-Read state, in an attempt to understand their relative importance. We find that the tunneling amplitude for charge e/2e/2e/2 quasihole is exponentially smaller than that for charge e/4e/4e/4 quasihole, and the ratio between them can be (partially) attributed to their charge difference. We find that including long-range Coulomb interaction only has a weak effect on the ratio. We discuss briefly the relevance of these results to recent tunneling and interferometry experiments at filling factor $\ensuremath{\nu}=5/2$.