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Physical review. B, Condensed matter

Hypernetted-chain treatment and the extended shadow wave functions for fractional quantum Hall hierarchical states

Orion Ciftja, Stefano Fantoni, Klaus A. Gernoth

Published May 15, 1997• 12 citations
Publisher

Abstract

The hypernetted-chain theory is applied to study hierarchical states in the fractional quantum Hall effect. It is noted that a class of wave functions introduced by Girvin [Phys. Rev. B 29, 6012 (1984)] and MacDonald, Aers, and Dharma-wardana [Phys. Rev. B 31, 5529 (1985)], based on charge-conjugation procedures, is of the extended shadow wave-function type. The correlation energy, pair distribution function, and static structure function have been calculated in the thermodynamic limit at various filling factors. The results obtained agree with those of previous calculations performed with a finite number of electrons.

DOI
10.1103/physrevb.55.13739

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The hypernetted-chain theory is applied to study hierarchical states in the fractional quantum Hall effect. It is noted that a class of wave functions introduced by Girvin [Phys. Rev. B 29, 6012 (1984)] and MacDonald, Aers, and Dharma-wardana [Phys. Rev. B 31, 5529 (1985)], based on charge-conjugation procedures, is of the extended shadow wave-function type. The correlation energy, pair distribution function, and static structure function have been calculated in the thermodynamic limit at various filling factors. The results obtained agree with those of previous calculations performed with a finite number of electrons.
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Trial wave functions which, in the thermodynamic limit, describe states relevant to the fractional quantum Hall effect are explicitly constructed for up to eight electrons. Estimates are given for the ground-state energies and pair-correlation functions and for the low-lying excitation energies for states associated with the filling factors \ensuremath{\nu}=(1/3), (2/7), and (2/5). The results help explain why the filling factors \ensuremath{\nu}=${\ensuremath{\nu}}_{m}$=m/(2m+1), m=1,2,..., and \ensuremath{\nu}=1-${\ensuremath{\nu}}_{m}$ show strong quantum Hall effects.
The microscopic approach for studying the half-filled state of the fractional quantum Hall effect is based on the idea of proposing a trial Fermi wave function of the Jastrow-Slater form, which is then fully projected onto the lowest Landau level. A simplified starting point is to drop the projection operator and to consider an unprojected wave function. A recent study claims that such a wave function approximated in a Jastrow form may still constitute a good starting point on the study of the half-filled state. In this paper we formalize the effective hypernetted-chain approximation and apply it to the unprojected Fermi wave function, which describes the even-denominator-filling states. We test the above approximation by using the Fermi hypernetted-chain theory, which constitutes the natural choice for the present case. Our results suggest that the approximation of the Slater determinant of plane waves as a Jastrow wave function may not be a very accurate approximation. We conclude that the lowest Landau-level projection operator cannot be neglected if one wants a better quantitative understanding of the phenomena.