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Physical review. B, Condensed matter

Interacting electrons in two-dimensional Landau levels: Results for small clusters

S. M. Girvin, Terrence Jach

Published Oct 15, 1983• 133 citations
Publisher

Abstract

We study the two-dimensional electron gas in a quantizing magnetic field for the cases of Coulomb and harmonic interactions among the electrons. Numerical solutions for the quantum states of clusters of up to five electrons show that the strength of the excitation gap is a strongly oscillating function of the density not unlike what is observed in the anomalous quantum Hall effect. We present analytic results for the case of harmonic interactions and show that the variational wave function recently proposed by Laughlin for the Coulomb problem is in fact an exact eigenstate of the harmonic problem.

DOI
10.1103/physrevb.28.4506

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We study the two-dimensional electron gas in a quantizing magnetic field for the cases of Coulomb and harmonic interactions among the electrons. Numerical solutions for the quantum states of clusters of up to five electrons show that the strength of the excitation gap is a strongly oscillating function of the density not unlike what is observed in the anomalous quantum Hall effect. We present analytic results for the case of harmonic interactions and show that the variational wave function recently proposed by Laughlin for the Coulomb problem is in fact an exact eigenstate of the harmonic problem.
A method for constructing a nonspurious set of N-particle harmonic wave functions with arbitrary permutational symmetry is employed to study the behavior of two-dimensional few-electron clusters on an infinite flat plane in a strong magnetic field. The Coulomb interaction is exactly diagonalized within the first Landau level and energy cusps are identified at the appropriate filling factors. The exact energies and pair distribution functions are compared with the corresponding results obtained using Laughlin's trial wave functions. Very good agreement is found with Laughlin's m=3 wave function, but some deviation is indicated for m=5.
The incompressible states of interacting two-dimensional electrons in a high magnetic field has been studied by finite-size calculations in the spherical geometry. The excitation spectrum at 13\frac{1}{3}31​ Landau-level filling is described, and the Laughlin-Jastrow character of the Coulomb-interaction ground state is unambiguously confirmed. As the interaction is varied, a transition to a gapless compressible state is observed.
This chapter presents the microscopic theory of a two-dimensional (2D) electron gas based on an ab initio functional approach to the many-body problem developed by V. A. Khodel et al. and V. R. Shaginyan. It discusses the detailed structure of the theory and scheme of the numerical solution of relevant equations. The chapter also discusses the method of evaluation of the single-particle spectrum of a 2D electron gas. It examines the behavior of a 2D electron gas in external magnetic fields. The chapter focuses on the disappearance of magnetic oscillations in electron systems of metal-oxide-semiconductor field-effect transistors due to merging of spin- and valley-split Landau levels in silicon, being the topological rearrangement of the Landau state, analogous to the formation of the fermion condensate. It explores the impact of the violation of the equidistance of electron spectra in strong magnetic fields on the structure of magnetic oscillations near the critical density.
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).
The Schr\"odinger equation for an interacting spinless electron gas in a nonuniform magnetic field admits an exact solution in Jastrow product form when the fluctuations in the magnetic field track the fluctuations in the scalar potential. For tracking realizations in a two-dimensional electron gas, the degeneracy of the lowest Landau level persists and the ``tracking'' solutions span the ground state subspace. In the context of the fractional quantum Hall problem, the Laughlin wave function is shown to be a tracking solution. Tracking solutions for screened Coulomb interactions are also constructed. The resulting wave functions are proposed as variational wave functions with potentially lower energy in the case of non-negligible Landau level mixing than the Laughlin function.