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Physical review. B, Condensed matter

Application of Fermi-hypernetted-chain theory to composite-fermion quantum Hall states

Orion Ciftja, Stefano Fantoni

Published Nov 15, 1997• 26 citations
Publisher

Abstract

The Fermi-hypernetted-chain (FHNC) theory and the effective hypernetted-chain method are applied to study the composite-fermion (CF) states of the fractional quantum Hall effect. Using this theory we compute, in the thermodynamic limit, the correlation energy, radial distribution function, and static structure factor for all unprojected CF wave functions. The unprojected excitation gaps for $\ensuremath{\nu}=1/3,1/5$ were obtained by adopting in the FHNC a scheme previously used to compute nuclear matter excitation spectra. The results obtained so far are consistent with Monte Carlo simulations and small-number exact diagonalizations.

DOI
10.1103/physrevb.56.13290

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The Fermi-hypernetted-chain (FHNC) theory and the effective hypernetted-chain method are applied to study the composite-fermion (CF) states of the fractional quantum Hall effect. Using this theory we compute, in the thermodynamic limit, the correlation energy, radial distribution function, and static structure factor for all unprojected CF wave functions. The unprojected excitation gaps for \ensuremathν=1/3,1/5\ensuremath{\nu}=1/3,1/5\ensuremathν=1/3,1/5 were obtained by adopting in the FHNC a scheme previously used to compute nuclear matter excitation spectra. The results obtained so far are consistent with Monte Carlo simulations and small-number exact diagonalizations.
The Fermi hypernetted-chain theory is applied to study the half-filled state of the fractional quantum Hall effect in the thermodynamic limit. We study in detail the radial distribution function, the correlation energy, and the quasiparticle-quasihole excitation spectrum of an unprojected Fermi wave function of the form ${\ensuremath{\psi}}_{\ensuremath{\nu}=1/2}^{\mathrm{Fermi}}={\ensuremath{\Pi}}_{j<k}^{N}{(z}_{j}\ensuremath{-}{z}_{k}{)}^{2}\mathrm{Det}{{\ensuremath{\varphi}}_{\stackrel{\ensuremath{\rightarrow}}{k}}(\stackrel{\ensuremath{\rightarrow}}{r})},$ a possible candidate to describe the half-filled state. Adopting a technique originating from nuclear physics, we compute the effective mass of the fermion excitations near the Fermi surface for this wave function. We find it to be exactly the bare mass of the electron, in accordance with the mean field approximation of not imposing the lowest Landau level constraint. Similar calculations were performed on other related wave functions, which, based on the composite fermion picture, describe the half-filled state of the electrons as a limit of infinite-filled composite fermion Landau levels.
The hypernetted-chain theory is applied to study the fractional quantum Hall effect with the Laughlin wave functions. A new method is proposed to include the effect of the elementary diagrams, which improves upon the commonly used modified hypernetted-chain approximation. The correlation energy, the pair distribution function, as well as the magnetoroton excitation spectrum have been computed within this method. The results obtained are in very good agreement with the available Monte Carlo estimates. The method is generalizable to treat other wave functions, like those corresponding to the hierarchical states or those of composite fermion type.
We apply Fermi hypernetted-chain theory to study the spin polarization of higher-order fractional quantum Hall (FQH) states at filling factors in between the primary FQH sequences, $\ensuremath{\nu}=p/({q}_{e}p\ifmmode\pm\else\textpm\fi{}1)$, where qe{q}_{e}qe​ is an even integer and ppp is a nonzero integer. The filling factors related to the higher-order FQH states include $\ensuremath{\nu}=3/8$, 4/114/114/11, 5/135/135/13, 5/175/175/17, 4/134/134/13, 6/176/176/17, 7/117/117/11, and so on. We use a model of strongly interacting fermions with different spin degrees of freedom to explain the states beyond primary FQH sequences. We calculate the correlation energy, the radial distribution function, as well as the static structure function associated with the Halperin wave function adopted for the mixture states of fermions with different spins. The results are comparable with those from the residual interaction between composite fermions.
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.
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.
The Fermi hypernetted-chain (FHNC) theory is applied to spin-density-wave states combined with a Jastrow-type correlation operator. The commonly used approximation scheme FHNC/n is found to be unsuitable for dealing with this class of correlated wave functions. A new scheme, named FHNC-n, is proposed, and numerically tested for the one-dimensional (1D) Hubbard model and the Gutzwiller correlation operator. The results obtained are in very good agreement with the available variational Monte Carlo calculation. This scheme can be used to study 2D and 3D Hubbard models and can easily be extended to treat more general correlation operators, which include triplet and quartet correlations.