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arXiv:cond-mat/0310015

More about the Laughlin droplet

N. Barberán

Published Oct 1, 2003• 0 citations
arXivHTMLPDF

Abstract

We compare the exact diagonalization ground wave function (calculated wothout any restriction) of a two dimensional droplet submitted to a perpendicular magnetic field with the Laughlin ansatz as the number of electrons increases. The fully polarized case for filling factor 1/3 without lateral confining potential or Zeeman effect is considered. We observe that the overlap worsens as the number of electrons increases.

DOI
10.48550/arxiv.cond-mat/0310015

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We compare the exact diagonalization ground wave function (calculated wothout any restriction) of a two dimensional droplet submitted to a perpendicular magnetic field with the Laughlin ansatz as the number of electrons increases. The fully polarized case for filling factor 1/3 without lateral confining potential or Zeeman effect is considered. We observe that the overlap worsens as the number of electrons increases.
A scaling procedure on the particle number is developed to construct the wave function of the uniform liquid ground state of the two-dimensional electron system in a strong magnetic field. When the filling factor is equal to the inverse of an odd number, this wave function is the correlated basis function introduced by Laughlin. In more general cases, when the filling factor is equal to a rational number, this wave function also can be regarded as a Laughlin wave function on the scale of a certain cluster. By this procedure, we can prove that an energy gap is always present between such a uniform liquid ground state and the excited states when the filling factor is equal to a rational number.
We calculate the ground-state energy and the effective spin Land\'e g factor (g∗{\mathit{g}}^{\mathrm{*}}g∗) of a two-dimensional electron gas in strong magnetic fields using a variational Monte Carlo method and many-body trial wave functions composed of the Jastrow correlation factor and the Laughlin-type functions. We find that g∗{\mathit{g}}^{\mathrm{*}}g∗ increases with decreasing of electron density and also shows increasing behavior as the filling factor \ensuremath{\nu} decreases in the region of 12. At \ensuremath{\nu}=1.0, g∗{\mathit{g}}^{\mathrm{*}}g∗ is estimated to be about 11 for a dimensionless parameter rs{\mathit{r}}_{\mathit{s}}rs​=1.33, as compared to the measured value of 7.3, while considering impurity and phonon scatterings gives a better agreement of g∗{\mathit{g}}^{\mathrm{*}}g∗=9. \textcopyright{} 1996 The American Physical Society.
The analyticity of the lowest Landau level wave functions and the relation between the filling factor and the total angular momentum severely limits the possible forms of trial wave functions of a disk of electrons subject to a strong perpendicular magnetic field. For the number N of electrons up to 12 we have tested these disk trial wave functions for the half-filled Landau level using Monte Carlo and exact diagonalization methods. The agreement between the results for the occupation numbers and ground state energies obtained from these two methods is excellent. We have also compared the profile of the occupation number near the edge with that obtained from a field-theoretical method. The results give qualitatively identical edge profiles. Experimental consequences are briefly discussed.
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.
A droplet of $N\ensuremath{\sim}30$ electrons is created by lateral confinement of a two-dimensional electron gas. The evolution of its chemical potential ${\ensuremath{\mu}}_{N}$ with magnetic field BBB is inferred from resonant tunneling experiments and is used to study the dependence of its magnetization on BBB.The experimental data, quantitatively described by Hartree-Fock theory, suggest a divergent spin susceptibility at Bc{B}_{c}Bc​ implying a ground state of minimum total spin just below Bc{B}_{c}Bc​. In the spin polarized regime we observe a new phase also predicted by Hartree-Fock theory.