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Physical review. B, Condensed matter

Higher-order fractional quantum Hall effect

W. P. Su

Published Aug 15, 1985• 20 citations
Publisher

Abstract

The Hamiltonian describing interacting two-dimensional electrons in a high magnetic field is diagonalized numerically for a small number of particles to obtain the low-lying excitation spectra. The results include estimates of energy gaps for values of $\ensuremath{\nu}$ (the lowest-Landau-level filling factor) equal to certain multiples of $\frac{1}{5}$, $\frac{1}{7}$, $\frac{1}{9}$, and $\frac{1}{11}$. These $\ensuremath{\nu}'\mathrm{s}$ are characterized by the existence of periodic rigid parent states which generate maximum phase space. The even-denominator cases are markedly different.

DOI
10.1103/physrevb.32.2617

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The Hamiltonian describing interacting two-dimensional electrons in a high magnetic field is diagonalized numerically for a small number of particles to obtain the low-lying excitation spectra. The results include estimates of energy gaps for values of \ensuremathν\ensuremath{\nu}\ensuremathν (the lowest-Landau-level filling factor) equal to certain multiples of 15\frac{1}{5}51​, 17\frac{1}{7}71​, 19\frac{1}{9}91​, and 111\frac{1}{11}111​. These \ensuremathν′s\ensuremath{\nu}'\mathrm{s}\ensuremathν′s are characterized by the existence of periodic rigid parent states which generate maximum phase space. The even-denominator cases are markedly different.
Exact diagonalization of a two-dimensional electron gas in a strong magnetic field in the disk geometry shows that there exists a filling-factor range in the second Landau level where the states significantly differ from those in the lowest Landau level. We show that the difference arises because the interaction between composite fermions is not negligible in higher Landau levels, as indicated by a substantial mixing between composite-fermion Landau-like levels, or $\ensuremath{\Lambda}$ levels. We find that the exact ground state is well reproduced by composite-fermion theory with $\ensuremath{\Lambda}$-level mixing incorporated at the lowest level of approximation. Using the same variational approach in the spherical geometry we estimate the excitation gap at filling 1∕31∕31∕3 in the second Landau level (which corresponds to 7∕37∕37∕3 of experiment).
We compute energy gaps for spin-polarized fractional quantum Hall states in the lowest Landau level (LL) at filling fractions $\ensuremath{\nu}=\frac{1}{3},$ 25,\frac{2}{5},52​, 37,\frac{3}{7},73​, and 49\frac{4}{9}94​ using exact diagonalization of systems with up to 16 particles and extrapolation to the infinite system-size limit. The gaps calculated for a pure Coulomb interaction and ignoring finite width effects, disorder and LL mixing agree well with the predictions of composite fermion theory provided the logarithmic corrections to the effective mass are included. This is in contrast with previous estimates, which, as we show, overestimated the gaps at $\ensuremath{\nu}=2/5$ and 3/7 by around 15%. We also study the reduction of the gaps as a result of the nonzero width of the two-dimensional (2D) layer. We show that these effects are accurately accounted for using either Gaussian or ``$z\ifmmode\times\else\texttimes\fi{}$ Gaussian'' (zG)(z\mathrm{G})(zG) trial wave functions, which we show are significantly better variational wave functions than the Fang-Howard wave function. The Gaussian and zGz\mathrm{G}zG wave functions give Haldane pseudopotential parameters which agree with those of self-consistent local density approximation calculations to better than $\ifmmode\pm\else\textpm\fi{}0.2%.$ For quantum well parameters typical of heterostructure samples, we find gap reductions of around 20%. The experimental gaps, after accounting heuristically for disorder, are still around 40% smaller than the computed gaps. However, for the case of tetracene layers in metal-insulator-semiconductor (MIS) devices we find that the measured activation gaps are close to those we compute. We discuss possible reasons why the difference between computed and measured activation gaps is larger in GaAs heterostructures than MIS devices. Finally, we present calculations using systems with up to 18 electrons of the gap at $\ensuremath{\nu}=\frac{5}{2}$ including width corrections.
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) =
Intuitive arguments and numerical simulations have long been used to argue for the incompressibility of fractionally quantized Hall states at the most prominent odd-denominator Landau-level filling fractions $\ensuremath{\nu}$ = ppp/(ppp (mmm - 1) + 1), where ppp is an integer and mmm an odd integer. Here, the authors strengthen the case by deriving these states more rigorously via adiabatic localization of magnetic flux onto the particles. They arrive at wave functions suggested by Jain's composite-fermion picture.
In the hierarchical theory of the fractional quantum Hall effect, the low--energy behaviour of a daughter state in the next level of the hierarchy is described by an interacting system of quasiparticles of the parent state. Taking the filled lowest Landau level as the parent state, we examine analytically the quantitative consequences of this approach for electrons interacting via a pseudopotential interaction. It is shown that the ground state energy per particle in the daughter state at a filling factor 2/32/32/3 is exactly equal to that of a system of quasiholes in the parent state with half filling, precisely as predicted by the hierarchical approach. This is achieved with {\em only up to two--particle interactions} in the effective Hamiltonian for the quasiholes. Their single particle energy and two--particle interaction are derived. The results are generalized to the other filling factors attainable from the filled Landau level.