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

BRAID GROUP REPRESENTATION AND FRACTIONAL QUANTUM HALL EFFECT

Christopher Hian Ann Ting, C. H. LAI

Published Aug 20, 1991• 2 citations
Publisher

Abstract

We discuss Laughlin's ground state and its derivation in the path integral approach. When the external magnetic field is very strong, we regard all electron-electron interactions such as the Coulomb term as weak perturbations. It seems that from the non-trivial topology of the configuration space alone, it is possible to arrive at a Hamiltonian for which Laughlin wavefunctions are exact ground states.

DOI
10.1142/s0217984991001581

Related papers

Spinning Braid Group Representation and the Fractional Quantum Hall Effect

Ting, C, Lai, C H

The path integral approach to representing braid group is generalized for particles with spin. Introducing the notion of {\\em charged} winding number in the super-plane, we represent the braid group generators as homotopically constrained Feynman kernels. In this framework, super Knizhnik-Zamolodchikov operators appear naturally in the Hamiltonian, suggesting the possibility of {\\em spinning nonabelian} anyons. We then apply our formulation to the study of fractional quantum Hall effect (FQHE). A systematic discussion of the ground states and their quasi-hole excitations is given. We obtain Laughlin, Halperin and Moore-Read states as {\\em exact} ground state solutions to the respective Hamiltonians associated to the braid group representations. The energy gap of the quasi-excitation is also obtainable from this approach.

•Jan 1, 1993•2 citations

Fractional Quantum Hall Effect States as Exact Ground States

Ranjan Kumar Ghosh, Sumathi Rao

We construct many particle Hamiltonians for which the Laughlin and Jain wavefunctions are exact ground states. The Hamiltonians involve fermions in a magnetic field and with inter-particle interactions. For the Laughlin wave-functions, the interactions involve two- and three-body correlations similar to the Chern–Simons interactions, whereas for the projected Jain wave-functions, N-body interactions (which cannot be explicitly written down in general) are involved.

arXiv:cond-mat/9703217•International Journal of Modern Physics B•May 10, 1998•4 citations

DERIVATION OF LAUGHLIN'S WAVEFUNCTION IN ANYON SYSTEM

Z. F. Ezawa, Aiichi Iwazaki

In a semiclassical approximation we derive the ground-state wavefunction of the system of anyons in an external magnetic field, where anyons are interacting via the Coulomb interaction. The statistical interactions are treated in perturbation around the boson limit. The wavefunction coincides with the Laughlin wavefunction except for small corrections due to the Coulomb interaction. We also calculate the ground-state energy and the excitation spectrum.

Modern Physics Letters B•May 20, 1992•5 citations

Fractional quantum Hall effect and quantum symmetry

Gerhard Grensing

Laughlin's quantum Hall wave function is obtained as an exact ground state of an N-particle Hamiltonian, in which the electrons themselves are coupled to the Chern-Simons field. The proof requires quantum field-theoretic methods in the Schr\"odinger picture, and clearly exhibits the relation to conformal field theory since the Knizhnik-Zamolodchikov connection shows up and the Laughlin ground state is recognized as a conformal block. A refined version of this approach can be applied for toroidal boundary conditions, i.e., the electrons are confined to the physical interior of the Hall sample. The result turns out to be a nontrivial modification of the p-fold degenerate Haldane-Rezayi wave function. Furthermore, it is shown that the degeneracy of the ground state is accounted for by the quantum group ${U}_{q}{(su}_{2})$ with deformation parameter $q=\mathrm{exp}(2\ensuremath{\pi}i/p)$ and p odd; put differently, the possible quantum numbers p of the quantum Hall effect, determining the filling fractions $\ensuremath{\nu}=1/p,$ can be explained by an underlying quantum symmetry.

Physical review. B, Condensed matter•Feb 15, 2000•1 citation

On the Laughlin function and its perturbations

Nicolas Rougerie

The Laughlin state is an ansatz for the ground state of a system of 2D quantum particles submitted to a strong magnetic field and strong interactions. The two effects conspire to generate strong and very specific correlations between the particles. I present a mathematical approach to the rigidity these correlations display in their response to perturbations. This is an important ingredient in the theory of the fractional quantum Hall effect. The main message is that potentials generated by impurities and residual interactions can be taken into account by generating uncorrelated quasi-holes on top of Laughlin's wave-function. An appendix contains a conjecture (not due to me) that should be regarded as a major open mathematical problem of the field, relating to the spectral gap of a certain zero-range interaction.

arXiv:1906.11656•Jun 27, 2019•4 citations

Jastrow form of the Ground State Wave Functions for Fractional Quantum Hall States

Sutirtha Mukherjee, Sudhansu S. Mandal

The topological morphology--order of zeros at the positions of electrons with respect to a specific electron--of Laughlin state at filling fractions $1/m$ ($m$ odd) is homogeneous as every electron feels zeros of order $m$ at the positions of other electrons. Although fairly accurate ground state wave functions for most of the other quantum Hall states in the lowest Landau level are quite well-known, it had been an open problem in expressing the ground state wave functions in terms of flux-attachment to particles, {\em a la}, this morphology of Laughlin state. With a very general consideration of flux-particle relations only, in spherical geometry, we here report a novel method for determining morphologies of these states. Based on these, we construct almost exact ground state wave-functions for the Coulomb interaction. Although the form of interaction may change the ground state wave-function, the same morphology constructs the latter irrespective of the nature of the interaction between electrons.

arXiv:1709.02070•Sep 7, 2017•0 citations
We discuss Laughlin's ground state and its derivation in the path integral approach. When the external magnetic field is very strong, we regard all electron-electron interactions such as the Coulomb term as weak perturbations. It seems that from the non-trivial topology of the configuration space alone, it is possible to arrive at a Hamiltonian for which Laughlin wavefunctions are exact ground states.
The path integral approach to representing braid group is generalized for particles with spin. Introducing the notion of {\\em charged} winding number in the super-plane, we represent the braid group generators as homotopically constrained Feynman kernels. In this framework, super Knizhnik-Zamolodchikov operators appear naturally in the Hamiltonian, suggesting the possibility of {\\em spinning nonabelian} anyons. We then apply our formulation to the study of fractional quantum Hall effect (FQHE). A systematic discussion of the ground states and their quasi-hole excitations is given. We obtain Laughlin, Halperin and Moore-Read states as {\\em exact} ground state solutions to the respective Hamiltonians associated to the braid group representations. The energy gap of the quasi-excitation is also obtainable from this approach.
Laughlin's quantum Hall wave function is obtained as an exact ground state of an N-particle Hamiltonian, in which the electrons themselves are coupled to the Chern-Simons field. The proof requires quantum field-theoretic methods in the Schr\"odinger picture, and clearly exhibits the relation to conformal field theory since the Knizhnik-Zamolodchikov connection shows up and the Laughlin ground state is recognized as a conformal block. A refined version of this approach can be applied for toroidal boundary conditions, i.e., the electrons are confined to the physical interior of the Hall sample. The result turns out to be a nontrivial modification of the p-fold degenerate Haldane-Rezayi wave function. Furthermore, it is shown that the degeneracy of the ground state is accounted for by the quantum group Uq(su2){U}_{q}{(su}_{2})Uq​(su2​) with deformation parameter $q=\mathrm{exp}(2\ensuremath{\pi}i/p)$ and p odd; put differently, the possible quantum numbers p of the quantum Hall effect, determining the filling fractions $\ensuremath{\nu}=1/p,$ can be explained by an underlying quantum symmetry.
The Laughlin state is an ansatz for the ground state of a system of 2D quantum particles submitted to a strong magnetic field and strong interactions. The two effects conspire to generate strong and very specific correlations between the particles. I present a mathematical approach to the rigidity these correlations display in their response to perturbations. This is an important ingredient in the theory of the fractional quantum Hall effect. The main message is that potentials generated by impurities and residual interactions can be taken into account by generating uncorrelated quasi-holes on top of Laughlin's wave-function. An appendix contains a conjecture (not due to me) that should be regarded as a major open mathematical problem of the field, relating to the spectral gap of a certain zero-range interaction.
The topological morphology--order of zeros at the positions of electrons with respect to a specific electron--of Laughlin state at filling fractions 1/m1/m1/m (mmm odd) is homogeneous as every electron feels zeros of order mmm at the positions of other electrons. Although fairly accurate ground state wave functions for most of the other quantum Hall states in the lowest Landau level are quite well-known, it had been an open problem in expressing the ground state wave functions in terms of flux-attachment to particles, {\em a la}, this morphology of Laughlin state. With a very general consideration of flux-particle relations only, in spherical geometry, we here report a novel method for determining morphologies of these states. Based on these, we construct almost exact ground state wave-functions for the Coulomb interaction. Although the form of interaction may change the ground state wave-function, the same morphology constructs the latter irrespective of the nature of the interaction between electrons.