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

Ground-state energy and quasiparticle gaps in ν=N/(2N±1) fractional-quantum-Hall-effect states

Dmitri V. Khveshchenko

Published May 5, 2023• 0 citations
Publisher

Abstract

Applying the transformation of fermion operators to the fermion quasiparticles introduced by Halperin, Lee, and Read, we estimate the scaling behavior of the ground-state energy and quasiparticle gaps as a function of filling fraction for a ''principal sequence'' of fractional-quantum-Hall-effect ν=N/(2N±1) states converging towards the gapless state at half filling. The exponent describing the shape of the cusp δE(ν)∼δνη is found to be greater than 1 and to depend nontrivially on the interaction potential. The dependence of quasiparticle gaps agrees with the results of recent measurements by R. R. Du et al.

DOI
10.17615/jbwb-0320

Related papers

Ground-state energy and quasiparticle gaps in ν=N/(2N±1) fractional-quantum-Hall-effect states

Dmitri V. Khveshchenko

Applying the transformation of fermion operators to the fermion quasiparticles introduced by Halperin, Lee, and Read, we estimate the scaling behavior of the ground-state energy and quasiparticle gaps as a function of filling fraction for a ``principal sequence'' of fractional-quantum-Hall-effect \ensuremath{\nu}=N/(2N\ifmmode\pm\else\textpm\fi{}1) states converging towards the gapless state at half filling. The exponent describing the shape of the cusp \ensuremath{\delta}E(\ensuremath{\nu})\ensuremath{\sim}\ensuremath{\Vert}\ensuremath{\delta}\ensuremath{\nu}${\mathrm{\ensuremath{\Vert}}}^{\mathrm{\ensuremath{\eta}}}$ is found to be greater than 1 and to depend nontrivially on the interaction potential. The dependence of quasiparticle gaps agrees with the results of recent measurements by R. R. Du et al.

arXiv:cond-mat/9401066•Physical review. B, Condensed matter•Apr 15, 1994•2 citations

Configuration-interaction calculations on the fractional quantum Hall effect

G. Fano, F. Ortolani, E. Colombo

With the use of Haldane's spherical model, the fractional quantum Hall effect is studied by means of large-scale configuration-interaction calculations. The exact wave function and the ground-state energy are computed numerically up to ten electrons for filling \ensuremath{\nu}=(1/3), and up to seven electrons for filling \ensuremath{\nu}=(1/5). Quasiparticle, quasihole, and exciton energies are computed. The mutual relation between wave functions on the sphere and wave functions on the disk is fully analyzed. A wave function originally proposed by Halperin for the disk geometry is tested on the sphere; the numerical results up to six electrons for \ensuremath{\nu}=(2/5) and \ensuremath{\nu}=(2/9) are favorable.

Physical review. B, Condensed matter•Aug 15, 1986•303 citations

Generalization of Laughlin’s theory for the fractional quantum Hall effect

Sudhansu S. Mandal

Motivated by the structure of the quasiparticle wavefunction in the composite fermion (CF) theory for fractional quantum Hall filling factor [Formula: see text] (m odd), I consider a suitable quasiparticle operator in differential form, as a modified form of Laughlin's quasiparticle operator, that reproduces quasiparticle wave function as predicted in the CF theory, without a priori assumption of the presence of CFs. I further consider the conjugate of this operator as quasihole operator for obtaining a novel quasihole wave function for 1/m state. Each of these wave functions is interpreted as expelling an electron into a different Hilbert subspace from the original Hilbert space of the Laughlin condensate while still maintaining its correlation (although changed) with the electrons in the condensate such that the expelled electron behaves as a CF with respect to the electrons in the condensate. With this interpretation, I show that the ground state wavefunctions for general states at filling fractions [Formula: see text], respectively, can be constructed as coherent superposition of n coupled Laughlin condensates and their 'conjugates', formed at different Hilbert subspaces. The corresponding wave functions, specially surprising for [Formula: see text] sequence of states, are identical with those proposed in the theory of noninteracting CFs. The states which were considered as fractional quantum Hall effect of interacting CFs, can also be treated in the same footing as for the prominent sequences of states describing as the coupled condensates among which one is a non-Laughlin condensate in a different Hilbert subspace. Further, I predict that the half filling of the lowest Landau level is a quantum critical point for phase transition between two topologically distinct phases each corresponding to a family of states: one consists of large number of coupled Laughlin condensates of filling factor 1/3 and the other corresponds to large number of coupled conjugate Laughlin condensates of filling factor 1, which may be distinguished, respectively, by the absence and presence of upstream edge modes.

arXiv:1507.02161•Journal of Physics Condensed Matter•Aug 28, 2018•3 citations

Toward a New Theory of the Fractional Quantum Hall Effect

Sergey A. Mikhailov

The fractional quantum Hall effect was experimentally discovered in 1982. It was observed that the Hall conductivity σyx of a two-dimensional electron system is quantized, σyx=e2/3h, in the vicinity of the Landau level filling factor ν=1/3. In 1983, Laughlin proposed a trial many-body wave function, which he claimed described a “new state of matter”—a homogeneous incompressible liquid with fractionally charged quasiparticles. Here, I develop an exact diagonalization theory that allows one to calculate the energy and other physical properties of the ground and excited states of a system of N two-dimensional Coulomb interacting electrons in a strong magnetic field. I analyze the energies, electron densities, and other physical properties of the systems with N≤7 electrons continuously as a function of magnetic field in the range 1/4≲ν<1. The results show that both the ground and excited states of the system resemble a sliding Wigner crystal whose parameters are influenced by the magnetic field. Energy gaps in the many-particle spectra appear and disappear as the magnetic field changes. I also calculate the physical properties of the ν=1/3 Laughlin state for N≤8 and compare the results with the exact ones. This comparison, as well as an analysis of some other statements published in the literature, show that the Laughlin state and its fractionally charged excitations do not describe the physical reality, neither at small N nor in the thermodynamic limit. The results obtained shed new light on the nature of the ground and excited states in the fractional quantum Hall effect.

arXiv:2206.05152•Nanomaterials•Jan 31, 2024•1 citation

Composite fermions and the energy gap in the fractional quantum Hall effect

N. E. Bonesteel

The energy gaps for the fractional quantum Hall effect at filling fractions 1/3, 1/5, and 1/7 have been calculated by variational Monte Carlo using Jain's composite-fermion wave functions. Results have been obtained before and after projection onto the lowest Landau level. The projected energy gaps agree well with exact-diagonalization results for small systems, lending support to the composite-fermion theory.

arXiv:cond-mat/9406110•Physical review. B, Condensed matter•Apr 15, 1995•18 citations

Droplet wave functions for the fractional quantum Hall effect

Allan H. MacDonald, D. B. Murray

Trial wave functions which, in the thermodynamic limit, describe states relevant to the fractional quantum Hall effect are explicitly constructed for up to eight electrons. Estimates are given for the ground-state energies and pair-correlation functions and for the low-lying excitation energies for states associated with the filling factors \ensuremath{\nu}=(1/3), (2/7), and (2/5). The results help explain why the filling factors \ensuremath{\nu}=${\ensuremath{\nu}}_{m}$=m/(2m+1), m=1,2,..., and \ensuremath{\nu}=1-${\ensuremath{\nu}}_{m}$ show strong quantum Hall effects.

Physical review. B, Condensed matter•Aug 15, 1985•35 citations
Applying the transformation of fermion operators to the fermion quasiparticles introduced by Halperin, Lee, and Read, we estimate the scaling behavior of the ground-state energy and quasiparticle gaps as a function of filling fraction for a ''principal sequence'' of fractional-quantum-Hall-effect ν=N/(2N±1) states converging towards the gapless state at half filling. The exponent describing the shape of the cusp δE(ν)∼δνη is found to be greater than 1 and to depend nontrivially on the interaction potential. The dependence of quasiparticle gaps agrees with the results of recent measurements by R. R. Du et al.
Applying the transformation of fermion operators to the fermion quasiparticles introduced by Halperin, Lee, and Read, we estimate the scaling behavior of the ground-state energy and quasiparticle gaps as a function of filling fraction for a ``principal sequence'' of fractional-quantum-Hall-effect \ensuremath{\nu}=N/(2N\ifmmode\pm\else\textpm\fi{}1) states converging towards the gapless state at half filling. The exponent describing the shape of the cusp \ensuremath{\delta}E(\ensuremath{\nu})\ensuremath{\sim}\ensuremath{\Vert}\ensuremath{\delta}\ensuremath{\nu}${\mathrm{\ensuremath{\Vert}}}^{\mathrm{\ensuremath{\eta}}}$ is found to be greater than 1 and to depend nontrivially on the interaction potential. The dependence of quasiparticle gaps agrees with the results of recent measurements by R. R. Du et al.
With the use of Haldane's spherical model, the fractional quantum Hall effect is studied by means of large-scale configuration-interaction calculations. The exact wave function and the ground-state energy are computed numerically up to ten electrons for filling \ensuremath{\nu}=(1/3), and up to seven electrons for filling \ensuremath{\nu}=(1/5). Quasiparticle, quasihole, and exciton energies are computed. The mutual relation between wave functions on the sphere and wave functions on the disk is fully analyzed. A wave function originally proposed by Halperin for the disk geometry is tested on the sphere; the numerical results up to six electrons for \ensuremath{\nu}=(2/5) and \ensuremath{\nu}=(2/9) are favorable.
Motivated by the structure of the quasiparticle wavefunction in the composite fermion (CF) theory for fractional quantum Hall filling factor [Formula: see text] (m odd), I consider a suitable quasiparticle operator in differential form, as a modified form of Laughlin's quasiparticle operator, that reproduces quasiparticle wave function as predicted in the CF theory, without a priori assumption of the presence of CFs. I further consider the conjugate of this operator as quasihole operator for obtaining a novel quasihole wave function for 1/m state. Each of these wave functions is interpreted as expelling an electron into a different Hilbert subspace from the original Hilbert space of the Laughlin condensate while still maintaining its correlation (although changed) with the electrons in the condensate such that the expelled electron behaves as a CF with respect to the electrons in the condensate. With this interpretation, I show that the ground state wavefunctions for general states at filling fractions [Formula: see text], respectively, can be constructed as coherent superposition of n coupled Laughlin condensates and their 'conjugates', formed at different Hilbert subspaces. The corresponding wave functions, specially surprising for [Formula: see text] sequence of states, are identical with those proposed in the theory of noninteracting CFs. The states which were considered as fractional quantum Hall effect of interacting CFs, can also be treated in the same footing as for the prominent sequences of states describing as the coupled condensates among which one is a non-Laughlin condensate in a different Hilbert subspace. Further, I predict that the half filling of the lowest Landau level is a quantum critical point for phase transition between two topologically distinct phases each corresponding to a family of states: one consists of large number of coupled Laughlin condensates of filling factor 1/3 and the other corresponds to large number of coupled conjugate Laughlin condensates of filling factor 1, which may be distinguished, respectively, by the absence and presence of upstream edge modes.
The fractional quantum Hall effect was experimentally discovered in 1982. It was observed that the Hall conductivity σyx of a two-dimensional electron system is quantized, σyx=e2/3h, in the vicinity of the Landau level filling factor ν=1/3. In 1983, Laughlin proposed a trial many-body wave function, which he claimed described a “new state of matter”—a homogeneous incompressible liquid with fractionally charged quasiparticles. Here, I develop an exact diagonalization theory that allows one to calculate the energy and other physical properties of the ground and excited states of a system of N two-dimensional Coulomb interacting electrons in a strong magnetic field. I analyze the energies, electron densities, and other physical properties of the systems with N≤7 electrons continuously as a function of magnetic field in the range 1/4≲ν<1. The results show that both the ground and excited states of the system resemble a sliding Wigner crystal whose parameters are influenced by the magnetic field. Energy gaps in the many-particle spectra appear and disappear as the magnetic field changes. I also calculate the physical properties of the ν=1/3 Laughlin state for N≤8 and compare the results with the exact ones. This comparison, as well as an analysis of some other statements published in the literature, show that the Laughlin state and its fractionally charged excitations do not describe the physical reality, neither at small N nor in the thermodynamic limit. The results obtained shed new light on the nature of the ground and excited states in the fractional quantum Hall effect.
Trial wave functions which, in the thermodynamic limit, describe states relevant to the fractional quantum Hall effect are explicitly constructed for up to eight electrons. Estimates are given for the ground-state energies and pair-correlation functions and for the low-lying excitation energies for states associated with the filling factors \ensuremath{\nu}=(1/3), (2/7), and (2/5). The results help explain why the filling factors \ensuremath{\nu}=${\ensuremath{\nu}}_{m}$=m/(2m+1), m=1,2,..., and \ensuremath{\nu}=1-${\ensuremath{\nu}}_{m}$ show strong quantum Hall effects.