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arXiv:cond-mat/0006278

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

S. A. Mikhaǐlov

Published Jun 19, 2000• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.cond-mat/0006278

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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.
We consider a two-dimensional strongly correlated electronic system in a strong perpendicular magnetic field at half-filling of the lowest Landau level (LLL). We seek to build a wave function that, by construction, lies entirely in the Hilbert space of the LLL. Quite generally, a wave function of this nature can be built as a linear combination of all possible Slater determinants formed by using the complete set of single-electron states that belong to the LLL. However, due to the vast number of Slater determinant states required to form such basis functions, the expansion is impractical for any but the smallest systems. Thus, in practice, the expansion must be truncated to a small number of Slater determinants. Among many possible LLL Slater determinant states, we note a particular special class of such wave functions in which electrons occupy either only even, or only odd angular momentum states. We focus on such a class of wave functions and obtain analytic expressions for various quantities of interest. Results seem to suggest that these special wave functions, while interesting and physically appealing, are unlikely to be a very good approximation for the exact ground state at half-filling factor. The overall quality of the description can be improved by including other additional LLL Slater determinant states. It is during this process that we identify another special family of suitable LLL Slater determinant states to be used in an enlarged expansion.
The ground state at 4/11 filling factor is very well understood, Mukherjee et al. [Phys. Rev. Lett. 112, 016801 (2014)] in terms of the 1/3-filled second effective Landau level of the composite fermions whose correlations resemble that of electrons in the ground state of a two-body Haldane pseudo-potential of relative angular momentum 3, V3{V}_{3}V3​. We here propose a closed-form ground-state wave function for V3{V}_{3}V3​ at 1/3 filling factor. We successfully compare it with the exact wave function for systems with a few electrons by calculating their mutual overlap, pair-correlation function, and entanglement spectra. By numerical exact diagonalization for a few electron systems, we find a window of nonzero V3{V}_{3}V3​ is essential together with V1{V}_{1}V1​ for being 4/11 state incompressible. The constructed wave function for the 4/11 state using this proposed wave function has satisfactorily high overlap with previously studied composite-fermion-diagonalized ground-state wave function.
We perform finite-cluster exact diagonalization studies for electrons in the lowest Landau level for filling factors from 1/3 to 1/11. In contrast to previous studies, we focus on the ground-state eigenfunction. This wave function changes continuously for filling factors from 1/3 to 1/9. At a filling factor of 1/11, there is a dramatic qualitative change in the ground-state wave function. Associated with this change is a peak in the structure factor at a finite wave vector as well as excitation energies that are consistent with gapless excitations at 1/11 filling. These results are consistent with recent experimental findings.
We present an exact solution for the interacting electron gas in the half-filled lowest Landau level on a thin torus. The low energy sector consists of non-interacting, one-dimensional, neutral fermions (dipoles). The ground state, which is homogeneous, is the Fermi sea obtained by filling the negative energy states and the excited states are the gapless neutral excitations out of this one-dimensional sea. We identify this ground state as a version of the Rezayi-Read state, and find that it develops continuously, as the circumference grows, into the Rezayi-Read state that is believed to describe the observed metallic phase in the two-dimensional system. This suggests a Luttinger liquid description of the half-filled Landau level.