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Physical review. B, Condensed matter

Comparison of fractional-quantum-Hall-effect quasielectron trial wave functions on the disk

Marcus Kasner, W. Apel

Published Oct 15, 1993• 17 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.48.11435

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arXiv:1112.4650•Dec 20, 2011•0 citations
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.
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.
Variational Monte Carlo calculations of the quasielectron and quasihole excitation energies in the fractional quantum Hall effect have been carried out at filling fractions $\ensuremath{\nu}=1/3,$ 1/5, and 1/7. For the quasielectron both the trial wave function originally proposed by Laughlin and the composite-fermion wave function proposed by Jain have been used. We find that for long-range Coulomb interactions the results obtained using these two wave functions are essentially the same, though the energy gap obtained using the composite-fermion quasielectron is slightly smaller, and closer to extrapolated exact-diagonalization results.
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) =
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.
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.