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

Physical Review Letters

Anyons, boundary constraint, and hierarchy in fractional quantum Hall effect

Song He, X. C. Xie, Fu‐Chun Zhang

Published Jun 8, 1992• 62 citations
Publisher

Abstract

Calculations on finite-size systems are carried out to examine the hierarchy theory of the fractional quantum Hall effect. The concept of quasiparticle and quasihole is justified to remain valid at high densities. Numerical results reveal a hard-core boundary constraint on the anyonic quasiparticles so that their wave function must vanish as two of them get close in space. The dynamics of this contraint originates from a strong short-range repulsion between the quasiparticles. The observed quasiparticle-quasihole asymmetry in the hierarchy is understood through the hard-core anyon theory.

DOI
10.1103/physrevlett.68.3460

Related papers

Dynamics of quasiholes and quasiparticles at the edges of small lattices

Xikun Li, Błażej Jaworowski, Masudul Haque +1 more

We study quench dynamics of bosonic fractional quantum Hall systems in small lattices with cylindrical boundary conditions and low particle density. The states studied have quasiholes or quasiparticles relative to the bosonic Laughlin state at half-filling. Pinning potentials are placed at edge sites (or sites close to the edges) to trap quasiholes and qausiparticles. The potentials are then turned off, and because the edges of fractional quantum Hall systems host chiral edge modes, we expect chiral dynamics of the quasiholes and quasiparticles. We numerically show that chiral motion of the density distribution is observed and robust for the case with positive potentials (quasiholes), but that there is no noticeable chiral motion for negative potentials (quasiparticles). The comparison of the numerical ground states with model lattice Laughlin wave functions suggests that both positive and negative potentials do create and pin anyons that are not necessarily well separated on small lattices. Initializing the dynamics with the model state also shows the lack of chiral dynamics of quasiparticles. Our results suggest that, in small lattices with low particle density, quasiparticles are strongly adversely affected in dynamical processes, whereas quasiholes are dynamically robust. Published by the American Physical Society 2024

arXiv:2305.18364•Physical Review A•Feb 7, 2024•1 citation

THE HIERARCHY THEORY OF THE FRACTIONAL QUANTUM HALL EFFECT

Jian Cheng Yang, W. P. Su

A recently developed hierarchy theory of the fractional quantum Hall effect (FQHE) is reviewed in this article. We use a many quasiparticle (QP) wave function to obtain the effective partition function of a FQHE system in the path-integral representation, which has been shown to be formally equivalent to an anyon system confined to the lowest Landau level. However the two systems are different in the dimension of the Hilbert space, which implies that the standard hierarchy theory is invalid beyond second hierarchical states. The only way to overcome the difficulty is to place anyons in higher Landau levels. By making a mean-field approximation to the statistical gauge field, the QPs are converted to bosons and a constraint boson system is derived. We find a new type of condensate of the bosonic QPs, which results in a new type of incompressible states of the electrons. These states which involve only two hierarchies should have higher stability over others. That explains why some sequences of filling fractions occur prominently in experiment. Finally, we show that there is a one-to-one correspondence between the new hierarchy theory and the composite fermion theory proposed by Jain.

Modern Physics Letters B•Jan 20, 1993•4 citations

Quantum Hall effect of ideal anyons

Michael Ma, Fu‐Chun Zhang

We study the problem of anyons with statistics \ensuremath{\nu} in a strong magnetic field by means of a similarity transformation. The ground state of this problem exhibits an ``integer'' quantum Hall effect at filling factor 1/\ensuremath{\nu} with quasiparticle excitations of charge q/\ensuremath{\nu} and statistics -1/\ensuremath{\nu}. Certain electron fractional quantum Hall states can be considered as a realization of this, for example, the 2/5, 3/7,. . . hierarchy of the 1/3 state. This may explain the observed quasiparticle-quasihole asymmetry in the fractional quantum hall hierarchy.

Physical Review Letters•Apr 1, 1991•19 citations

Higher stability filling fractions in the hierarchy theory of the fractional quantum Hall effect

Jian Cheng Yang, W. P. Su

The effective partition function of a fractional quantum Hall system, which is described by the Laughlin quasiparticle wave function, is obtained in the path-integral representation. It shows that the quasiparticles cannot be treated as ordinary anyons in view of the freeze of the kinetic energy and the ``bosonic aspect'' of the Hilbert space. By using the constrained anyon systems, we find that one type of filling fractions is more stable than other fractions predicted by the standard hierarchy theory. This work also sheds considerable light on the possible equivalence between the hierarchy theory and the composite-fermion approach proposed by Jain recently.

Physical Review Letters•Apr 13, 1992•7 citations

Trapping Abelian anyons in fractional quantum Hall droplets

Zi-Xiang Hu, Xin Wan, Peter Schmitteckert

We study the trapping of Abelian anyons (quasiholes and quasiparticles) by a local potential (e.g., induced by an atomic force microscopy tip) in a microscopic model of fractional quantum Hall liquids with long-range Coulomb interaction and edge confining potential. We find, in particular, that at Laughlin filling fraction $\ensuremath{\nu}=1∕3$, both quasihole and quasiparticle states can emerge as the ground state of the system in the presence of the trapping potential. As expected, we find that the presence of an Abelian quasihole has no effect on the edge spectrum of the quantum liquid, unlike in the non-Abelian case [X. Wan et al., Phys. Rev. Lett. 97, 256804 (2006)]. Although quasiholes and quasiparticles can emerge generically in the system, their stability depends on the strength of the confining potential and the strength and the range of the trapping potential. We discuss the relevance of the calculation to the high-accuracy generation and control of individual anyons in potential experiments, in particular, in the context of topological quantum computing.

arXiv:0712.1461•Physical Review B•Feb 28, 2008•16 citations

Anyonic description of low-energy physics in the fractional quantum Hall effect

Jian Cheng Yang, W. P. Su

The diagonalization of a Hamiltonian in the subspace spanned by many quasiparticle states has been performed. The result, compared to the full Hilbert space diagonalization, shows that the anyonic picture of the hierarchy theory provides a very accurate description of the low-energy physics of the fractional quantum Hall system. The implication of this study on the equivalence between the hierarchy theory and the composite fermion theory is discussed.

Physical Review Letters•Feb 22, 1993•17 citations
Calculations on finite-size systems are carried out to examine the hierarchy theory of the fractional quantum Hall effect. The concept of quasiparticle and quasihole is justified to remain valid at high densities. Numerical results reveal a hard-core boundary constraint on the anyonic quasiparticles so that their wave function must vanish as two of them get close in space. The dynamics of this contraint originates from a strong short-range repulsion between the quasiparticles. The observed quasiparticle-quasihole asymmetry in the hierarchy is understood through the hard-core anyon theory.
We study quench dynamics of bosonic fractional quantum Hall systems in small lattices with cylindrical boundary conditions and low particle density. The states studied have quasiholes or quasiparticles relative to the bosonic Laughlin state at half-filling. Pinning potentials are placed at edge sites (or sites close to the edges) to trap quasiholes and qausiparticles. The potentials are then turned off, and because the edges of fractional quantum Hall systems host chiral edge modes, we expect chiral dynamics of the quasiholes and quasiparticles. We numerically show that chiral motion of the density distribution is observed and robust for the case with positive potentials (quasiholes), but that there is no noticeable chiral motion for negative potentials (quasiparticles). The comparison of the numerical ground states with model lattice Laughlin wave functions suggests that both positive and negative potentials do create and pin anyons that are not necessarily well separated on small lattices. Initializing the dynamics with the model state also shows the lack of chiral dynamics of quasiparticles. Our results suggest that, in small lattices with low particle density, quasiparticles are strongly adversely affected in dynamical processes, whereas quasiholes are dynamically robust. Published by the American Physical Society 2024
A recently developed hierarchy theory of the fractional quantum Hall effect (FQHE) is reviewed in this article. We use a many quasiparticle (QP) wave function to obtain the effective partition function of a FQHE system in the path-integral representation, which has been shown to be formally equivalent to an anyon system confined to the lowest Landau level. However the two systems are different in the dimension of the Hilbert space, which implies that the standard hierarchy theory is invalid beyond second hierarchical states. The only way to overcome the difficulty is to place anyons in higher Landau levels. By making a mean-field approximation to the statistical gauge field, the QPs are converted to bosons and a constraint boson system is derived. We find a new type of condensate of the bosonic QPs, which results in a new type of incompressible states of the electrons. These states which involve only two hierarchies should have higher stability over others. That explains why some sequences of filling fractions occur prominently in experiment. Finally, we show that there is a one-to-one correspondence between the new hierarchy theory and the composite fermion theory proposed by Jain.
We study the problem of anyons with statistics \ensuremath{\nu} in a strong magnetic field by means of a similarity transformation. The ground state of this problem exhibits an ``integer'' quantum Hall effect at filling factor 1/\ensuremath{\nu} with quasiparticle excitations of charge q/\ensuremath{\nu} and statistics -1/\ensuremath{\nu}. Certain electron fractional quantum Hall states can be considered as a realization of this, for example, the 2/5, 3/7,. . . hierarchy of the 1/3 state. This may explain the observed quasiparticle-quasihole asymmetry in the fractional quantum hall hierarchy.
The effective partition function of a fractional quantum Hall system, which is described by the Laughlin quasiparticle wave function, is obtained in the path-integral representation. It shows that the quasiparticles cannot be treated as ordinary anyons in view of the freeze of the kinetic energy and the ``bosonic aspect'' of the Hilbert space. By using the constrained anyon systems, we find that one type of filling fractions is more stable than other fractions predicted by the standard hierarchy theory. This work also sheds considerable light on the possible equivalence between the hierarchy theory and the composite-fermion approach proposed by Jain recently.
We study the trapping of Abelian anyons (quasiholes and quasiparticles) by a local potential (e.g., induced by an atomic force microscopy tip) in a microscopic model of fractional quantum Hall liquids with long-range Coulomb interaction and edge confining potential. We find, in particular, that at Laughlin filling fraction $\ensuremath{\nu}=1∕3$, both quasihole and quasiparticle states can emerge as the ground state of the system in the presence of the trapping potential. As expected, we find that the presence of an Abelian quasihole has no effect on the edge spectrum of the quantum liquid, unlike in the non-Abelian case [X. Wan et al., Phys. Rev. Lett. 97, 256804 (2006)]. Although quasiholes and quasiparticles can emerge generically in the system, their stability depends on the strength of the confining potential and the strength and the range of the trapping potential. We discuss the relevance of the calculation to the high-accuracy generation and control of individual anyons in potential experiments, in particular, in the context of topological quantum computing.