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A new Proposal for a Quasielectron Trial Wavefunction for the FQHE on a Disk

Marcus Kasner, W. Apel

Published Jan 1, 1992• 4 citations
OpenAlex

Abstract

Abstract. In this letter, we propose a new quasielectron trial wavefunction for N interacting electrons in two dimensions moving in a strong magnetic field in a disk geometry. Requiring that the trial wavefunction exhibits the correct filling factor of a quasielectron wavefunction, we obtain N + 1 angular momentum eigenfunctions. The expectation values of the energy are calculated and compared with the data of an exact numerical diagonalization. PACS. 71.45N, 73.40L 1 Trial wavefunctions have been used very successfully to describe the lowenergy properties of interacting electrons moving in two dimensions in a strong magnetic field [1]. The prime example is the Laughlin wavefunction [2], which describes the ground state of the system at some of the filling factors of the lowest spin-polarized Landau level (i. e. electronic densities), for which one observes the fractional quantum Hall effect (FQHE) [3]. Deviations from such a state to higher or lower filling factors are then described by quasielectrons or quasiholes, for which trial wavefunctions are also proposed [2]. These are the constituents of the hierarchical picture [4, 5] which is used to describe FQHE states at the remaining rational filling factors observed which are different from 1/m (m odd integer). The quasiparticles have also been proven to be a useful tool in numerical studies of the FQHE system for small particle numbers. Excitation energies have been calculated [5, 6, 7, 8] and even some of the quasiparticle interactions have been obtained [9, 10]. The situation is rather satisfying for the quasiholes, but, in the case of the quasielectrons, questions remain. Some of these we address in the present letter. After introducing our notation, we present a new trial wavefunction for the quasielectron state and then give as an illustration the first numerical results for the expectation-value of the energy. We study the disk geometry in which the single particle basis of the lowest Landau level is given by φm(z) =

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arXiv:cond-mat/9210004•Europhysics Letters (EPL)•Dec 15, 1992•4 citations

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New trial wave functions corresponding to half filling quantum Hall states are proposed. These wave functions are constructed by first pairing up the quasielectrons of the 1/3 Laughlin quantum Hall state, with the same relative angular momentum for each pair, and then making the paired quasielectrons condense into a 1/4 Laughlin state. The quasiparticle excitations of the proposed wave functions carry $\pm1/4$ of electron charge, and obey Abelian fractional statistics. In the spherical geometry, the total flux quanta $N_ϕ$ is shown to be related to the number of electrons $N$ by $N_ϕ = 2N-(5-q)$ with $q$ being the relative angular momentum between the quasielectrons in each pair which takes values of non-negative even integers. The overlaps are calculated between the proposed trial wave functions, including the ground state, quasiexciton states, and quasihole states, and the exact states of the finite size systems at $N_ϕ = 2N-3$. The near unity overlaps are obtained in the lowest Landau level, while the moderate overlaps are obtained in the second Landau level. The relevance of the wave functions to the yet to be discovered fractional quantum Hall effect in the lowest Landau level, as well as the 5/2 quantum Hall effect is also discussed.

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Electrons in a strong magnetic field on a disk

Marcus Kasner, W. Apel

The problem of interacting electrons moving under the influence of a strong magnetic field in two dimensions on a finite disk is reconsidered. First, the results of exact diagonalizations for up to N = 9 electrons for Coulomb as well as for a short–range interaction are used in the search for a peculiar ground state corresponding to filling factor 1/3. Not for the Coulomb, but only for the short–range interaction, can the 1/3– state be safely identified amongst the spectra of various filling factors close to 1/3. Second, the propositions of the concept of quasiparticles, as used in the hierarchical theory, are examined in view of the exact results for the disk geometry. Whereas the theory for the quasiholes is in complete accordance with the spectra, for the quasielectrons, finite size corrections make an analysis difficult. For the quasielectron energy, an extrapolation to N → ∞ is given and compared with the corresponding extrapolations of three different proposals for trial wave functions. While the limiting value for the best trial wave function is very close to the limit of the exact results, the behavior of the finite size corrections of the exact energies and of the trial wave functions, respectively, is qualitatively rather different.

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Electrons in a strong magnetic field on a disk

Marcus Kasner, W. Apel

Abstract The problem of interacting electrons moving under the influence of a strong magnetic field in two dimensions on a finite disk is reconsidered. First, the results of exact diagonalizations for up to N = 9 electrons for Coulomb as well as for a short‐range interaction are used in the search for a peculiar ground state corresponding to filling factor 1/3. Not for the Coulomb, but only for the short‐range interaction, can the 1/3‐state be safely identified amongst the spectra of various filling factors close to 1/3. Second, the propositions of the concept of quasiparticles, as used in the hierarchical theory, are examined in view of the exact results for the disk geometry. Whereas the theory for the quasiholes is in complete accordance with the spectra, for the quasielectrons, finite size corrections make an analysis difficult. For the quasielectron energy, an extrapolation to N→ ∞ is given and compared with the corresponding extrapolations of three different proposals for trial wave functions. While the limiting value for the best trial wave function is very close to the limit of the exact results, the behavior of the finite size corrections of the exact energies and of the trial wave functions, respectively, is qualitatively rather different.

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Marcus Kasner, W. Apel

The various proposals for fractional-quantum-Hall-effect quasielectron trial wave functions are reconsidered. In a short-range model for the electronic interaction, the energy expectation values of four different trial wave functions are calculated at filling factor 1/3 for up to ten electrons in the disk geometry. Jain's trial wave function displays the lowest-energy expectation value.

Physical review. B, Condensed matter•Oct 15, 1993•17 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
Abstract. In this letter, we propose a new quasielectron trial wavefunction for N interacting electrons in two dimensions moving in a strong magnetic field in a disk geometry. Requiring that the trial wavefunction exhibits the correct filling factor of a quasielectron wavefunction, we obtain N + 1 angular momentum eigenfunctions. The expectation values of the energy are calculated and compared with the data of an exact numerical diagonalization. PACS. 71.45N, 73.40L 1 Trial wavefunctions have been used very successfully to describe the lowenergy properties of interacting electrons moving in two dimensions in a strong magnetic field [1]. The prime example is the Laughlin wavefunction [2], which describes the ground state of the system at some of the filling factors of the lowest spin-polarized Landau level (i. e. electronic densities), for which one observes the fractional quantum Hall effect (FQHE) [3]. Deviations from such a state to higher or lower filling factors are then described by quasielectrons or quasiholes, for which trial wavefunctions are also proposed [2]. These are the constituents of the hierarchical picture [4, 5] which is used to describe FQHE states at the remaining rational filling factors observed which are different from 1/m (m odd integer). The quasiparticles have also been proven to be a useful tool in numerical studies of the FQHE system for small particle numbers. Excitation energies have been calculated [5, 6, 7, 8] and even some of the quasiparticle interactions have been obtained [9, 10]. The situation is rather satisfying for the quasiholes, but, in the case of the quasielectrons, questions remain. Some of these we address in the present letter. After introducing our notation, we present a new trial wavefunction for the quasielectron state and then give as an illustration the first numerical results for the expectation-value of the energy. We study the disk geometry in which the single particle basis of the lowest Landau level is given by φm(z) =
New trial wave functions corresponding to half filling quantum Hall states are proposed. These wave functions are constructed by first pairing up the quasielectrons of the 1/3 Laughlin quantum Hall state, with the same relative angular momentum for each pair, and then making the paired quasielectrons condense into a 1/4 Laughlin state. The quasiparticle excitations of the proposed wave functions carry ±1/4\pm1/4±1/4 of electron charge, and obey Abelian fractional statistics. In the spherical geometry, the total flux quanta NϕN_ϕNϕ​ is shown to be related to the number of electrons NNN by Nϕ=2N−(5−q)N_ϕ = 2N-(5-q)Nϕ​=2N−(5−q) with qqq being the relative angular momentum between the quasielectrons in each pair which takes values of non-negative even integers. The overlaps are calculated between the proposed trial wave functions, including the ground state, quasiexciton states, and quasihole states, and the exact states of the finite size systems at Nϕ=2N−3N_ϕ = 2N-3Nϕ​=2N−3. The near unity overlaps are obtained in the lowest Landau level, while the moderate overlaps are obtained in the second Landau level. The relevance of the wave functions to the yet to be discovered fractional quantum Hall effect in the lowest Landau level, as well as the 5/2 quantum Hall effect is also discussed.
The problem of interacting electrons moving under the influence of a strong magnetic field in two dimensions on a finite disk is reconsidered. First, the results of exact diagonalizations for up to N = 9 electrons for Coulomb as well as for a short–range interaction are used in the search for a peculiar ground state corresponding to filling factor 1/3. Not for the Coulomb, but only for the short–range interaction, can the 1/3– state be safely identified amongst the spectra of various filling factors close to 1/3. Second, the propositions of the concept of quasiparticles, as used in the hierarchical theory, are examined in view of the exact results for the disk geometry. Whereas the theory for the quasiholes is in complete accordance with the spectra, for the quasielectrons, finite size corrections make an analysis difficult. For the quasielectron energy, an extrapolation to N → ∞ is given and compared with the corresponding extrapolations of three different proposals for trial wave functions. While the limiting value for the best trial wave function is very close to the limit of the exact results, the behavior of the finite size corrections of the exact energies and of the trial wave functions, respectively, is qualitatively rather different.
Abstract The problem of interacting electrons moving under the influence of a strong magnetic field in two dimensions on a finite disk is reconsidered. First, the results of exact diagonalizations for up to N = 9 electrons for Coulomb as well as for a short‐range interaction are used in the search for a peculiar ground state corresponding to filling factor 1/3. Not for the Coulomb, but only for the short‐range interaction, can the 1/3‐state be safely identified amongst the spectra of various filling factors close to 1/3. Second, the propositions of the concept of quasiparticles, as used in the hierarchical theory, are examined in view of the exact results for the disk geometry. Whereas the theory for the quasiholes is in complete accordance with the spectra, for the quasielectrons, finite size corrections make an analysis difficult. For the quasielectron energy, an extrapolation to N→ ∞ is given and compared with the corresponding extrapolations of three different proposals for trial wave functions. While the limiting value for the best trial wave function is very close to the limit of the exact results, the behavior of the finite size corrections of the exact energies and of the trial wave functions, respectively, is qualitatively rather different.
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.