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arXiv:cond-mat/9210004 · Europhysics Letters (EPL)

A New Proposal for a Quasi-Electron Trial Wave Function for the FQHE on a Disk

Marcus Kasner, W. Apel

Published Dec 15, 1992• 4 citations
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Abstract

In this letter, we propose a new quasi-electron trial wave function for N interacting electrons in two dimensions moving in a strong magnetic field in a disk geometry. Requiring that the trial wave function exhibits the correct filling factor of a quasi-electron wave function, 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.

DOI
10.1209/0295-5075/20/8/009

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In this letter, we propose a new quasi-electron trial wave function for N interacting electrons in two dimensions moving in a strong magnetic field in a disk geometry. Requiring that the trial wave function exhibits the correct filling factor of a quasi-electron wave function, 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.
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) =
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 various proposals for FQHE quasi-electron 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 13\frac{1}{3}31​ for up to ten electrons in the disk geometry. Jain's trial wave function displays the lowest energy expectation value.
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 analyticity of the lowest Landau level wave functions and the relation between the filling factor and the total angular momentum severely limits the possible forms of trial wave functions of a disk of electrons subject to a strong perpendicular magnetic field. For the number N of electrons up to 12 we have tested these disk trial wave functions for the half-filled Landau level using Monte Carlo and exact diagonalization methods. The agreement between the results for the occupation numbers and ground state energies obtained from these two methods is excellent. We have also compared the profile of the occupation number near the edge with that obtained from a field-theoretical method. The results give qualitatively identical edge profiles. Experimental consequences are briefly discussed.