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Physics Letters A

On minimal energy Hartree-Fock states for the 2DEG at fractional filling factors

Alejandro Cabo Montes de

Published Feb 1, 1996• 4 citations
Publisher
DOI
10.1016/0375-9601(95)00961-2

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FRACTIONAL QUANTUM HALL EFFECT: CONSTRUCTION OF THE HARTREE–FOCK STATE BY USING TRANSLATIONAL COVARIANCE

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The formalism introduced in a previous paper is used for discussing the Coulomb interaction of many electrons moving in two space-dimensions in the presence of a strong magnetic field. The matrix element of the Coulomb interaction is evaluated in the new basis, whose states are invariant under discrete translations (up to a gauge transformation). This paper is devoted to the case of low filling factor, thus we limit ourselves to the lowest Landau level and to spins all oriented along the magnetic field. For the case of filling factor νf = 1/u we give an Ansatz on the state of many electrons which provides a good approximated solution of the Hartree–Fock equation. For general filling factor νf = u′/u a trial state is given which converges very rapidly to a solution of the self-consistent equation. We generalize the Hartree–Fock equation by considering some correlation: all quantum states are allowed for the u′ electrons with the same translation quantum numbers. Numerical results are given for the mean energy and the energy bands, for some values of the filling factor (νf = 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5). Our results agree numerically with the Charge Density Wave approach. The boundary conditions are shown to be very important: only large systems (degeneracy of Landau level over 200) are not affected by the boundaries. Therefore results obtained on small scale systems are somewhat unreliable. The relevance of the results for the Fractional Quantum Hall Effect is briefly discussed.

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On the ground state of a completely filled lowest Landau level in two dimensions

S. A. Mikhaǐlov

There exists a widely believed opinion, that the many-body ground state of a two-dimensional electron system at a completely filled lowest Landau level (the filling factor $ν=1$) is described by the so-called Hartree-Fock wave function, and that this solution is the unique, exact eigenstate of the system at $ν=1$. I show that this opinion is erroneous, construct an infinite number of other variational many-body wave functions, and discuss the properties of a few states which have the energy substantially lower than the energy of the Hartree-Fock state.

arXiv:cond-mat/0006278•Jun 19, 2000•0 citations
The formalism introduced in a previous paper is used for discussing the Coulomb interaction of many electrons moving in two space-dimensions in the presence of a strong magnetic field. The matrix element of the Coulomb interaction is evaluated in the new basis, whose states are invariant under discrete translations (up to a gauge transformation). This paper is devoted to the case of low filling factor, thus we limit ourselves to the lowest Landau level and to spins all oriented along the magnetic field. For the case of filling factor νf = 1/u we give an Ansatz on the state of many electrons which provides a good approximated solution of the Hartree–Fock equation. For general filling factor νf = u′/u a trial state is given which converges very rapidly to a solution of the self-consistent equation. We generalize the Hartree–Fock equation by considering some correlation: all quantum states are allowed for the u′ electrons with the same translation quantum numbers. Numerical results are given for the mean energy and the energy bands, for some values of the filling factor (νf = 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5). Our results agree numerically with the Charge Density Wave approach. The boundary conditions are shown to be very important: only large systems (degeneracy of Landau level over 200) are not affected by the boundaries. Therefore results obtained on small scale systems are somewhat unreliable. The relevance of the results for the Fractional Quantum Hall Effect is briefly discussed.
There exists a widely believed opinion, that the many-body ground state of a two-dimensional electron system at a completely filled lowest Landau level (the filling factor ν=1ν=1ν=1) is described by the so-called Hartree-Fock wave function, and that this solution is the unique, exact eigenstate of the system at ν=1ν=1ν=1. I show that this opinion is erroneous, construct an infinite number of other variational many-body wave functions, and discuss the properties of a few states which have the energy substantially lower than the energy of the Hartree-Fock state.