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:1504.06620 · Physical Review B

Matrix product state representation of non-Abelian quasiholes

Yang-Le Wu, Benoit Estienne, Nicolas Regnault, Bogdan Andrei Bernevig

Published Jul 9, 2015• 21 citations
arXivHTMLPDF

Abstract

We provide a detailed explanation of the formalism necessary to construct matrix product states for non-Abelian quasiholes in fractional quantum Hall model states. Our construction yields an efficient representation of the wave functions with conformal-block normalization and monodromy, and complements the matrix product state representation of fractional quantum Hall ground states.

DOI
10.1103/physrevb.92.045109

Related papers

Matrix product state description of Halperin states

Valentin Crépel, Benoit Estienne, Bogdan Andrei Bernevig +2 more

Many fractional quantum Hall states can be expressed as a correlator of a given conformal field theory used to describe their edge physics. As a consequence, these states admit an economical representation as an exact matrix product state (MPS) that was extensively studied for the systems without any spin or any other internal degrees of freedom. In that case, the correlators are built from a single electronic operator, which is primary with respect to the underlying conformal field theory. We generalize this construction to the archetype of Abelian multicomponent fractional quantum Hall wave functions, the Halperin states. These can be written as conformal blocks involving multiple electronic operators and we explicitly derive their exact MPS representation. In particular, we deal with the caveat of the full wave-function symmetry and show that any additional SU(2) symmetry is preserved by the natural MPS truncation scheme provided by the conformal dimension. We use our method to characterize the topological order of the Halperin states by extracting the topological entanglement entropy. We also evaluate their bulk correlation lengths, which are compared to plasma analogy arguments.

arXiv:1802.05278•Physical review. B./Physical review. B•Apr 23, 2018•22 citations

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

Quasihole condensates in quantum Hall liquids

Juha Suorsa, S. Viefers, T. H. Hansson

We develop a formalism to describe quasihole condensates in quantum Hall liquids and thereby extend the conformal field theory approach to the full hierarchy of spin-polarized Abelian states and to several classes of non-Abelian hierarchical states. Most previously proposed spin-polarized quantum Hall wave functions appear as special cases. In this paper we explain the physical motivations for the approach, and exemplify it by explicitly constructing the level-two quasihole condensate state at filling fraction 2/3, and the two level-three states at 5/13 and 5/7 which are built from combinations of quasielectron and quasihole condensates.

arXiv:1004.3657•Physical Review B•Jun 16, 2011•23 citations

Condensing Non-Abelian Quasiparticles

Maria Hermanns

A most interesting feature of certain fractional quantum Hall states is that their quasiparticles obey non-Abelian fractional statistics. So far, candidate non-Abelian wave functions have been constructed from conformal blocks in cleverly chosen conformal field theories. In this work we present a hierarchy scheme by which we can construct daughter states by condensing non-Abelian quasiparticles (as opposed to quasiholes) in a parent state, and show that the daughters have a non-Abelian statistics that differs from the parent. In particular, we discuss the daughter of the bosonic, spin-polarized Moore-Read state at nu=4/3 as an explicit example.

arXiv:0906.2073•Physical Review Letters•Feb 5, 2010•30 citations

Non-Abelian fusion and braiding in many-body parton states

Koyena Bose

Fractional quantum Hall (FQH) states host fractionally charged anyons with exotic exchange statistics. Of particular interest are FQH phases supporting non-Abelian anyons, which can encode topologically protected quantum information. In this work, we construct quasihole bases for a broad family of non-Abelian FQH states using parton wave functions, which reproduces the fusion-space dimensionality expected from their underlying conformal field theory, consistent with level-rank duality across the parton family. As an application, we numerically compute braiding matrices for representative parton states for large systems, providing a general framework for diagnosing non-Abelian characteristics in candidate FQH states.

arXiv:2601.16819•Jan 23, 2026•0 citations

Properties of Non-Abelian Fractional Quantum Hall States at Filling ν=k/r

Bogdan Andrei Bernevig, F. D. M. Haldane

We compute the physical properties of non-Abelian fractional quantum Hall (FQH) states described by Jack polynomials at general filling nu = k/r. For r = 2, these states are the Zk Read-Rezayi parafermions, whereas for r > 2 they represent new FQH states. The r = k+1 states, multiplied by a Vandermonde determinant, are a non-Abelian alternative construction of states at fermionic filling 2/5,3/7,4/9,.... We obtain the thermal Hall coefficient, the quantum dimensions, the electron scaling exponent, and the non-Abelian quasihole propagator. The properties of the r > 2 Jack polynomials indicate they are correlators of fields of nonunitary conformal field theories (CFT), but the CFT-FQH connection fails when invoked to compute physical properties such as the quasihole propagator. The quasihole wave function, written as a coherent state representation of Jack polynomials, has an identical structure for all non-Abelian states.

arXiv:0803.2882•Physical Review Letters•Dec 10, 2008•104 citations
We provide a detailed explanation of the formalism necessary to construct matrix product states for non-Abelian quasiholes in fractional quantum Hall model states. Our construction yields an efficient representation of the wave functions with conformal-block normalization and monodromy, and complements the matrix product state representation of fractional quantum Hall ground states.
Many fractional quantum Hall states can be expressed as a correlator of a given conformal field theory used to describe their edge physics. As a consequence, these states admit an economical representation as an exact matrix product state (MPS) that was extensively studied for the systems without any spin or any other internal degrees of freedom. In that case, the correlators are built from a single electronic operator, which is primary with respect to the underlying conformal field theory. We generalize this construction to the archetype of Abelian multicomponent fractional quantum Hall wave functions, the Halperin states. These can be written as conformal blocks involving multiple electronic operators and we explicitly derive their exact MPS representation. In particular, we deal with the caveat of the full wave-function symmetry and show that any additional SU(2) symmetry is preserved by the natural MPS truncation scheme provided by the conformal dimension. We use our method to characterize the topological order of the Halperin states by extracting the topological entanglement entropy. We also evaluate their bulk correlation lengths, which are compared to plasma analogy arguments.
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.
We develop a formalism to describe quasihole condensates in quantum Hall liquids and thereby extend the conformal field theory approach to the full hierarchy of spin-polarized Abelian states and to several classes of non-Abelian hierarchical states. Most previously proposed spin-polarized quantum Hall wave functions appear as special cases. In this paper we explain the physical motivations for the approach, and exemplify it by explicitly constructing the level-two quasihole condensate state at filling fraction 2/3, and the two level-three states at 5/13 and 5/7 which are built from combinations of quasielectron and quasihole condensates.
A most interesting feature of certain fractional quantum Hall states is that their quasiparticles obey non-Abelian fractional statistics. So far, candidate non-Abelian wave functions have been constructed from conformal blocks in cleverly chosen conformal field theories. In this work we present a hierarchy scheme by which we can construct daughter states by condensing non-Abelian quasiparticles (as opposed to quasiholes) in a parent state, and show that the daughters have a non-Abelian statistics that differs from the parent. In particular, we discuss the daughter of the bosonic, spin-polarized Moore-Read state at nu=4/3 as an explicit example.
Fractional quantum Hall (FQH) states host fractionally charged anyons with exotic exchange statistics. Of particular interest are FQH phases supporting non-Abelian anyons, which can encode topologically protected quantum information. In this work, we construct quasihole bases for a broad family of non-Abelian FQH states using parton wave functions, which reproduces the fusion-space dimensionality expected from their underlying conformal field theory, consistent with level-rank duality across the parton family. As an application, we numerically compute braiding matrices for representative parton states for large systems, providing a general framework for diagnosing non-Abelian characteristics in candidate FQH states.
We compute the physical properties of non-Abelian fractional quantum Hall (FQH) states described by Jack polynomials at general filling nu = k/r. For r = 2, these states are the Zk Read-Rezayi parafermions, whereas for r > 2 they represent new FQH states. The r = k+1 states, multiplied by a Vandermonde determinant, are a non-Abelian alternative construction of states at fermionic filling 2/5,3/7,4/9,.... We obtain the thermal Hall coefficient, the quantum dimensions, the electron scaling exponent, and the non-Abelian quasihole propagator. The properties of the r > 2 Jack polynomials indicate they are correlators of fields of nonunitary conformal field theories (CFT), but the CFT-FQH connection fails when invoked to compute physical properties such as the quasihole propagator. The quasihole wave function, written as a coherent state representation of Jack polynomials, has an identical structure for all non-Abelian states.