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:hep-th/9310181 · Physical Review Letters

Classification of quantum Hall universality classes by W1+∞ symmetry

Andrea Cappelli A, Carlo A. Trugenberger, Guillermo R. Zemba

Published Mar 21, 1994• 63 citations
arXivHTMLPDF

Abstract

We show how two-dimensional incompressible quantum fluids and their excitations can be viewed as ${\mathit{W}}_{1+\mathrm{\ensuremath{\infty}}}$ edge conformal field theories, thereby providing an algebraic characterization of incompressibility. The Kac-Radul representation theory of the ${\mathit{W}}_{1+\mathrm{\ensuremath{\infty}}}$ algebra leads then to a purely algebraic complete classification of hierarchical quantum Hall states, which encompasses all measured fractions. Spin-polarized electrons in single-layer devices can only have Abelian anyon excitations.

DOI
10.1103/physrevlett.72.1902

Related papers

Numerical study of hierarchical quantum Hall edge states in the disk geometry

Andrea Cappelli A, Carlos Méndez, J. Simonin +1 more

We present a detailed analysis of the exact numerical spectrum of up to ten interacting electrons in the first Landau level on the disk geometry. We study the edge excitations of the hierarchical plateaus and check the predictions of two relevant conformal field theories: the multicomponent Abelian theory and the ${W}_{1+\ensuremath{\infty}}$ minimal theory of the incompressible fluids. We introduce two criteria for identifying the edge excitations within the low-lying states: the plot of their density profiles and the study of their overlaps with the Jain wave functions in a meaningful basis. We find that the exact bulk and edge excitations are very well reproduced by the Jain states; these, in turn, can be described by the multicomponent Abelian conformal theory. Most notably, we observe that the edge excitations form subfamilies of the low-lying states with a definite pattern, which is explained by the ${W}_{1+\ensuremath{\infty}}$ minimal conformal theory. Actually, the two conformal theories are related by a projection mechanism whose effects are observed in the spectrum. Therefore, the edge excitations of the hierarchical Hall states are consistently described by the ${W}_{1+\ensuremath{\infty}}$ minimal theory, within the finite-size limitations.

arXiv:cond-mat/9806238•Physical review. B, Condensed matter•Dec 15, 1998•19 citations

Bulk-boundary correspondence in the quantum Hall effect

Andrea Cappelli A, Lorenzo Maffi

Abstract We present a detailed microscopic study of edge excitations for n filled Landau levels. We show that the higher-level wavefunctions possess a non-trivial radial dependence that should be integrated over for properly defining the edge conformal field theory. This analysis let’s us clarify the role of the electron orbital spin s in the edge theory and to discuss its universality, thus providing a further instance of the bulk-boundary correspondence. We find that the values s i for each level, parameterize a Casimir effect or chemical potential shift that could be experimentally observed. These results are generalized to fractional and hierarchical fillings by exploiting the W-infinity symmetry of incompressible Hall fluids.

arXiv:1801.03759•Journal of Physics A Mathematical and Theoretical•Jul 3, 2018•13 citations

Study of Quantum Hall Effect and Its Relation to Conformal Field Theory

Shima Bahrampour Lomer, M. N. Najafi, Hosein Mohammadzadeh

The quantum Hall effect is one of the most challenging problems in condensed matter physics in which many rare physical phenomena like fractional charges and generally the fractional statistics are involved. In the integer quantum Hall effect the role of the disorder is more than others. The important point in describing these systems is that their bulk states should be gapful and in this sense the system should be incompressible. For describing the electron condensates in an arbitrary filling fraction with the odd denominator one needs the topological U(1) Chern-Simons effective field theory. Our motivation for using these field theories is the emergence of quantized conductivity. This effective field theory also has the ability of constructing the hierarchical fractional condensates with various filling factors. The excitations of these theories have fractional statistics which are investigated in this review. The non-Abelian Chern-Simons for quantum Hall systems will be also investigated. The interesting point is that the edge states of these systems are gapless and their excitations are equipped with the Kac-Moody algebra. The connection to the conformal field theories can be established using this point. The investigation of this correspondence is one of the aims of the present thesis. The connection to the logarithmic conformal field theories (e.g. for the Haldane-Rezaei wave function) is also reported in some works which will be addressed in this thesis.

•Jan 1, 2016•0 citations

$W_{\infty}$ Algebras and Incompressibility in the Quantum Hall Effect

Dimitra Karabali A

We discuss how a large class of incompressible quantum Hall states can be characterized as highest weight states of different representations of the \Winf algebra. Second quantized expressions of the \Winf generators are explicitly derived in the cases of multilayer Hall states, the states proposed by Jain to explain the hierarchical filling fractions and the ones related by particle-hole conjugation.

arXiv:hep-th/9411082•Nov 11, 1994•0 citations

A CLASSIFICATION OF QUANTUM HALL FLUIDS

Jürg Martin Fröhlich, Urban M. Studer, E. Thiran

In this paper, the key ideas of characterizing universality classes of dissipation-free (incompressible) quantum Hall fluids by mathematical objects called quantum Hall lattices are reviewed. Many general theorems about the classification of quantum Hall lattices are stated and their physical implications are discussed. Physically relevant subclasses of quantum Hall lattices are defined and completely classified. The results are carefully compared with experimental data and also with other theoretical schemes (the hierarchy schemes). Several proposals for new experiments are made which could help to settle interesting issues in the theory of the (fractional) quantum Hall effect and thus would lead to a deeper understanding of this remarkable effect.

arXiv:cond-mat/9503113•Mar 20, 1995•3 citations

Beyond paired quantum Hall states: Parafermions and incompressible states in the first excited Landau level

N. Read, Edward H. Rezayi

The Pfaffian quantum Hall states, which can be viewed as involving pairing either of spin-polarized electrons or of composite fermions, are generalized by finding the exact ground states of certain Hamiltonians with $k+1$-body interactions, for all integers $k>~1.$ The remarkably simple wave functions of these states involve clusters of k particles, and are related to correlators of parafermion currents in two-dimensional conformal field theory. The $k=2$ case is the Pfaffian. For $k>~2,$ the quasiparticle excitations of these systems are expected to possess non-Abelian statistics, like those of the Pfaffian. For $k=3,$ these ground states have large overlaps with the ground states of the (two-body) Coulomb-interaction Hamiltonian for electrons in the first excited Landau level at total filling factors $\ensuremath{\nu}=2+3/5,2+2/5.$

arXiv:cond-mat/9809384•Physical review. B, Condensed matter•Mar 15, 1999•943 citations
We show how two-dimensional incompressible quantum fluids and their excitations can be viewed as W1+\ensuremath∞{\mathit{W}}_{1+\mathrm{\ensuremath{\infty}}}W1+\ensuremath∞​ edge conformal field theories, thereby providing an algebraic characterization of incompressibility. The Kac-Radul representation theory of the W1+\ensuremath∞{\mathit{W}}_{1+\mathrm{\ensuremath{\infty}}}W1+\ensuremath∞​ algebra leads then to a purely algebraic complete classification of hierarchical quantum Hall states, which encompasses all measured fractions. Spin-polarized electrons in single-layer devices can only have Abelian anyon excitations.
We present a detailed analysis of the exact numerical spectrum of up to ten interacting electrons in the first Landau level on the disk geometry. We study the edge excitations of the hierarchical plateaus and check the predictions of two relevant conformal field theories: the multicomponent Abelian theory and the ${W}_{1+\ensuremath{\infty}}$ minimal theory of the incompressible fluids. We introduce two criteria for identifying the edge excitations within the low-lying states: the plot of their density profiles and the study of their overlaps with the Jain wave functions in a meaningful basis. We find that the exact bulk and edge excitations are very well reproduced by the Jain states; these, in turn, can be described by the multicomponent Abelian conformal theory. Most notably, we observe that the edge excitations form subfamilies of the low-lying states with a definite pattern, which is explained by the ${W}_{1+\ensuremath{\infty}}$ minimal conformal theory. Actually, the two conformal theories are related by a projection mechanism whose effects are observed in the spectrum. Therefore, the edge excitations of the hierarchical Hall states are consistently described by the ${W}_{1+\ensuremath{\infty}}$ minimal theory, within the finite-size limitations.
Abstract We present a detailed microscopic study of edge excitations for n filled Landau levels. We show that the higher-level wavefunctions possess a non-trivial radial dependence that should be integrated over for properly defining the edge conformal field theory. This analysis let’s us clarify the role of the electron orbital spin s in the edge theory and to discuss its universality, thus providing a further instance of the bulk-boundary correspondence. We find that the values s i for each level, parameterize a Casimir effect or chemical potential shift that could be experimentally observed. These results are generalized to fractional and hierarchical fillings by exploiting the W-infinity symmetry of incompressible Hall fluids.
The quantum Hall effect is one of the most challenging problems in condensed matter physics in which many rare physical phenomena like fractional charges and generally the fractional statistics are involved. In the integer quantum Hall effect the role of the disorder is more than others. The important point in describing these systems is that their bulk states should be gapful and in this sense the system should be incompressible. For describing the electron condensates in an arbitrary filling fraction with the odd denominator one needs the topological U(1) Chern-Simons effective field theory. Our motivation for using these field theories is the emergence of quantized conductivity. This effective field theory also has the ability of constructing the hierarchical fractional condensates with various filling factors. The excitations of these theories have fractional statistics which are investigated in this review. The non-Abelian Chern-Simons for quantum Hall systems will be also investigated. The interesting point is that the edge states of these systems are gapless and their excitations are equipped with the Kac-Moody algebra. The connection to the conformal field theories can be established using this point. The investigation of this correspondence is one of the aims of the present thesis. The connection to the logarithmic conformal field theories (e.g. for the Haldane-Rezaei wave function) is also reported in some works which will be addressed in this thesis.
W∞W_{\infty}W∞​ Algebras and Incompressibility in the Quantum Hall Effect
In this paper, the key ideas of characterizing universality classes of dissipation-free (incompressible) quantum Hall fluids by mathematical objects called quantum Hall lattices are reviewed. Many general theorems about the classification of quantum Hall lattices are stated and their physical implications are discussed. Physically relevant subclasses of quantum Hall lattices are defined and completely classified. The results are carefully compared with experimental data and also with other theoretical schemes (the hierarchy schemes). Several proposals for new experiments are made which could help to settle interesting issues in the theory of the (fractional) quantum Hall effect and thus would lead to a deeper understanding of this remarkable effect.
The Pfaffian quantum Hall states, which can be viewed as involving pairing either of spin-polarized electrons or of composite fermions, are generalized by finding the exact ground states of certain Hamiltonians with k+1k+1k+1-body interactions, for all integers k> 1.k>~1.k> 1. The remarkably simple wave functions of these states involve clusters of k particles, and are related to correlators of parafermion currents in two-dimensional conformal field theory. The k=2k=2k=2 case is the Pfaffian. For k> 2,k>~2,k> 2, the quasiparticle excitations of these systems are expected to possess non-Abelian statistics, like those of the Pfaffian. For k=3,k=3,k=3, these ground states have large overlaps with the ground states of the (two-body) Coulomb-interaction Hamiltonian for electrons in the first excited Landau level at total filling factors $\ensuremath{\nu}=2+3/5,2+2/5.$