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

Modern Physics Letters B

THE HIERARCHY THEORY OF THE FRACTIONAL QUANTUM HALL EFFECT

Jian Cheng Yang, W. P. Su

Published Jan 20, 1993• 4 citations
Publisher

Abstract

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.

DOI
10.1142/s0217984993000096

Related papers

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

COLLECTIVE FIELD THEORY APPLIED TO THE FRACTIONAL QUANTUM HALL EFFECT

BUNJI SAKITA, Dong-Ning Sheng, Zhao-Bin Su

We present an application of collective field theory to the fractional quantum Hall effect (FQHE). We first express the condition, that the electrons are all in the lowest Landau level, as a constraint equation for the state functional. We then derive the fractional filling factor from this equation together with the no-free-vortex assumption. A hierarchy of filling factors is derived by using the particle-vortex dual transformations. In the final section we discuss an attempt at a dynamical theory of FQHE, which would justify the no-free-vortex assumption. A derivation of Laughlin’s wave function with and without quasi-hole excitations is also given.

International Journal of Modern Physics B•Jan 1, 1991•0 citations

Quantum Hall Wave Functions on the Torus

Mikael Fremling

The fractional quantum Hall effect (FQHE), now entering it's fourth decade, continues to draw attention from the condensed matter community. New experiments in recent years are raising hopes that it will be possible to observe quasi-particles with non-abelian anyonic statistics. These particles could form the building blocks of a quantum computer. The quantum Hall states have topologically protected energy gaps to the low-lying set of excitations. This topological order is not a locally measurable quantity but rather a non-local object, and it is one of the keys to it's stability. From an early stage understanding of the FQHE has been facilitate by constructing trial wave functions. The topological classification of these wave functions have given further insight to the nature of the FQHE. An early, and successful, wave function construction for filling fractions ν=p/(2p+1) was that of composite fermions on planar and spherical geometries. Recently, new developments using conformal field theory have made it possible to also construct the full Haldane-Halperin hierarchy wave functions on planar and spherical geometries. In this thesis we extend this construction to a toroidal geometry, i.e. a flat surface with periodic boundary conditions. One of the defining features of topological states of matter in two dimensions is that the ground state is not unique on surfaces with non trivial topology, such as a torus. The archetypical example is the fractional quantum Hall effect, where a state at filling fraction ν=p/q, has at least a q-fold degeneracy on a torus. This has been shown explicitly for a few cases, such as the Laughlin states and the the Moore-Read states, by explicit construction of candidate electron wave functions with good overlap with numerically found states. In this thesis, we construct explicit torus wave functions for a large class of experimentally important quantum liquids, namely the chiral hierarchy states in the lowest Landau level. These states, which includes the prominently observed positive Jain sequence at filling fractions ν=p/(2p+1), are characterized by having boundary modes with only one chirality. Our construction relies heavily on previous work that expressed the hierarchy wave functions on a plane or a sphere in terms of correlation functions in a conformal field theory. This construction can be taken over to the torus when care is taken to ensure correct behaviour under the modular transformations that leave the geometry of the torus unchanged. Our construction solves the long standing problem of engineering torus wave functions for multi-component many-body states. Since the resulting expressions are rather complicated, we have carefully compared the simplest example, that of ν=2/5, with numerically found wave functions. We have found an extremely good overlap for arbitrary values of the modular parameter τ, that describes the geometry of the torus. Having explicit torus wave functions allows us to use the methods developed by Read and Read \\& Rezayi to numerically compute the quantum Hall viscosity. Hall viscosity is conjectured to be a topologically protected macroscopic transport coefficient characterizing the quantum Hall state. It is related to the shift of the same QH-fluid when it is put on a sphere. The good agreement with the theoretical prediction for the 2/5 state strongly suggests that our wave functions encodes all relevant topologically information. We also consider the Hall viscosity in the limit of a very thin torus. There we find that the viscosity changes as we approach the thin torus limit. Because of this we study the Laughlin state in that limit and see how the change in viscosity arises from a change in the Hamiltonian hopping elements. Finally we conclude that there are both qualitative and quantitative difference between the thin and the square torus. Thus, one has to be careful when interpreting results in the thin torus limit.

•Jan 1, 2015•1 citation

Test of the hierarchical theory for the FQHE

Marcus Kasner

In the hierarchical theory of the fractional quantum Hall effect, the low--energy behaviour of a daughter state in the next level of the hierarchy is described by an interacting system of quasiparticles of the parent state. Taking the filled lowest Landau level as the parent state, we examine analytically the quantitative consequences of this approach for electrons interacting via a pseudopotential interaction. It is shown that the ground state energy per particle in the daughter state at a filling factor $2/3$ is exactly equal to that of a system of quasiholes in the parent state with half filling, precisely as predicted by the hierarchical approach. This is achieved with {\em only up to two--particle interactions} in the effective Hamiltonian for the quasiholes. Their single particle energy and two--particle interaction are derived. The results are generalized to the other filling factors attainable from the filled Landau level.

arXiv:cond-mat/9401002•Jan 3, 1994•0 citations

HIERARCHICAL WAVE FUNCTIONS AND FRACTIONAL STATISTICS IN FRACTIONAL QUANTUM HALL EFFECT ON THE TORUS

DINGPING LI

One kind of hierarchical wave functions of Fractional Quantum Hall Effect (FQHE) on the torus are constructed. The multi-component nature of anyon wave functions and the degeneracy of FQHE on the torus are very clearly reflected in this kind of wave functions. We also calculate the braid statistics of the quasiparticles in FQHE on the torus and show they fit the picture of anyons interacting with magnetic field on the torus obtained from braid group analysis.

arXiv:cond-mat/9212033•International Journal of Modern Physics B•Jul 10, 1993•9 citations

Fractional Quantum Hall Effect from Phenomenological Bosonization

Vladimir A. Zyuzin

In this paper we propose a model of the fractional quantum Hall effect within conventional one-dimensional bosonization. It is shown that in this formalism the resulting bosonized fermion operator corresponding to momenta of Landau gauge wave function is effectively two-dimensional. At special filling factors the bulk gets gapped, and the theory is described by a sine-Gordon model. The edges are shown to be gapless, chiral, and carrying a fractional charge. The hierarchy of obtained fractional charges is consistent with existing experiments and theories. It is also possible to draw a connection to composite fermion description and to the Laughlin many-body wave function.

arXiv:1111.6931•Bulletin of the American Physical Society•Nov 29, 2011•0 citations
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.
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 present an application of collective field theory to the fractional quantum Hall effect (FQHE). We first express the condition, that the electrons are all in the lowest Landau level, as a constraint equation for the state functional. We then derive the fractional filling factor from this equation together with the no-free-vortex assumption. A hierarchy of filling factors is derived by using the particle-vortex dual transformations. In the final section we discuss an attempt at a dynamical theory of FQHE, which would justify the no-free-vortex assumption. A derivation of Laughlin’s wave function with and without quasi-hole excitations is also given.
The fractional quantum Hall effect (FQHE), now entering it's fourth decade, continues to draw attention from the condensed matter community. New experiments in recent years are raising hopes that it will be possible to observe quasi-particles with non-abelian anyonic statistics. These particles could form the building blocks of a quantum computer. The quantum Hall states have topologically protected energy gaps to the low-lying set of excitations. This topological order is not a locally measurable quantity but rather a non-local object, and it is one of the keys to it's stability. From an early stage understanding of the FQHE has been facilitate by constructing trial wave functions. The topological classification of these wave functions have given further insight to the nature of the FQHE. An early, and successful, wave function construction for filling fractions ν=p/(2p+1) was that of composite fermions on planar and spherical geometries. Recently, new developments using conformal field theory have made it possible to also construct the full Haldane-Halperin hierarchy wave functions on planar and spherical geometries. In this thesis we extend this construction to a toroidal geometry, i.e. a flat surface with periodic boundary conditions. One of the defining features of topological states of matter in two dimensions is that the ground state is not unique on surfaces with non trivial topology, such as a torus. The archetypical example is the fractional quantum Hall effect, where a state at filling fraction ν=p/q, has at least a q-fold degeneracy on a torus. This has been shown explicitly for a few cases, such as the Laughlin states and the the Moore-Read states, by explicit construction of candidate electron wave functions with good overlap with numerically found states. In this thesis, we construct explicit torus wave functions for a large class of experimentally important quantum liquids, namely the chiral hierarchy states in the lowest Landau level. These states, which includes the prominently observed positive Jain sequence at filling fractions ν=p/(2p+1), are characterized by having boundary modes with only one chirality. Our construction relies heavily on previous work that expressed the hierarchy wave functions on a plane or a sphere in terms of correlation functions in a conformal field theory. This construction can be taken over to the torus when care is taken to ensure correct behaviour under the modular transformations that leave the geometry of the torus unchanged. Our construction solves the long standing problem of engineering torus wave functions for multi-component many-body states. Since the resulting expressions are rather complicated, we have carefully compared the simplest example, that of ν=2/5, with numerically found wave functions. We have found an extremely good overlap for arbitrary values of the modular parameter τ, that describes the geometry of the torus. Having explicit torus wave functions allows us to use the methods developed by Read and Read \\& Rezayi to numerically compute the quantum Hall viscosity. Hall viscosity is conjectured to be a topologically protected macroscopic transport coefficient characterizing the quantum Hall state. It is related to the shift of the same QH-fluid when it is put on a sphere. The good agreement with the theoretical prediction for the 2/5 state strongly suggests that our wave functions encodes all relevant topologically information. We also consider the Hall viscosity in the limit of a very thin torus. There we find that the viscosity changes as we approach the thin torus limit. Because of this we study the Laughlin state in that limit and see how the change in viscosity arises from a change in the Hamiltonian hopping elements. Finally we conclude that there are both qualitative and quantitative difference between the thin and the square torus. Thus, one has to be careful when interpreting results in the thin torus limit.
In the hierarchical theory of the fractional quantum Hall effect, the low--energy behaviour of a daughter state in the next level of the hierarchy is described by an interacting system of quasiparticles of the parent state. Taking the filled lowest Landau level as the parent state, we examine analytically the quantitative consequences of this approach for electrons interacting via a pseudopotential interaction. It is shown that the ground state energy per particle in the daughter state at a filling factor 2/32/32/3 is exactly equal to that of a system of quasiholes in the parent state with half filling, precisely as predicted by the hierarchical approach. This is achieved with {\em only up to two--particle interactions} in the effective Hamiltonian for the quasiholes. Their single particle energy and two--particle interaction are derived. The results are generalized to the other filling factors attainable from the filled Landau level.
In this paper we propose a model of the fractional quantum Hall effect within conventional one-dimensional bosonization. It is shown that in this formalism the resulting bosonized fermion operator corresponding to momenta of Landau gauge wave function is effectively two-dimensional. At special filling factors the bulk gets gapped, and the theory is described by a sine-Gordon model. The edges are shown to be gapless, chiral, and carrying a fractional charge. The hierarchy of obtained fractional charges is consistent with existing experiments and theories. It is also possible to draw a connection to composite fermion description and to the Laughlin many-body wave function.