Physical review. B, Condensed matter

Hypernetted-chain approximation and quasiparticle energies in the (1/3 fractional quantized Hall effect

H. A. Fertig, Bertrand I. Halperin

Published Oct 15, 1987• 13 citations

Abstract

We review some details of the hypernetted-chain (HNC) approximation and its application to the Laughlin quasihole and quasielectron trial wave functions. We then develop a generalized HNC approximation for calculating the three-body correlation function ${g}_{112}$ of a classical plasma with two particle species. We apply this approximation to the Laughlin trial wave functions for the case \ensuremath{\nu}=(1/3 and we estimate their energies. Our quasihole energy is in good agreement with past calculations, but our quasielectron energy is considerably higher than the accepted result. However, the approximation has the correct property that near a quasielectron, the electron pair correlation function vanishes as \ensuremath{\Vert}${\mathrm{r}}_{1}$${\mathrm{\ensuremath{-}}\mathrm{r}}_{2}$${\ensuremath{\Vert}}^{2}$ for \ensuremath{\Vert}${\mathrm{r}}_{1}$${\mathrm{\ensuremath{-}}\mathrm{r}}_{2}$\ensuremath{\Vert}\ensuremath{\rightarrow}0, where ${\mathrm{r}}_{1}$ and ${\mathrm{r}}_{2}$ are electron positions. We show that an approximation analogous to Laughlin's pseudopotential approximation does not have this property.

DOI: 10.1103/physrevb.36.6302

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Physical review. B, Condensed matter

Hypernetted-chain approximation and quasiparticle energies in the (1/3 fractional quantized Hall effect

H. A. Fertig, Bertrand I. Halperin

Published Oct 15, 1987• 13 citations

Publisher

Abstract

{g}_{112}

of a classical plasma with two particle species. We apply this approximation to the Laughlin trial wave functions for the case \ensuremath{\nu}=(1/3 and we estimate their energies. Our quasihole energy is in good agreement with past calculations, but our quasielectron energy is considerably higher than the accepted result. However, the approximation has the correct property that near a quasielectron, the electron pair correlation function vanishes as \ensuremath{\Vert}

{\mathrm{r}}_{1}$${\mathrm{\ensuremath{-}}\mathrm{r}}_{2}$${\ensuremath{\Vert}}^{2}

for \ensuremath{\Vert}

{\mathrm{r}}_{1}$${\mathrm{\ensuremath{-}}\mathrm{r}}_{2}

\ensuremath{\Vert}\ensuremath{\rightarrow}0, where

{\mathrm{r}}_{1}

and

{\mathrm{r}}_{2}

are electron positions. We show that an approximation analogous to Laughlin's pseudopotential approximation does not have this property.

DOI: 10.1103/physrevb.36.6302

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A new hypernetted-chain treatment for Laughlin quantum Hall states

Orion Ciftja, Stefano Fantoni

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J Kjäll, Eddy Ardonne, V Dwivedi +2 more

Abstract Matrix product state techniques provide a very efficient way to numerically evaluate certain classes of quantum Hall wave functions that can be written as correlators in two-dimensional conformal field theories. Important examples are the Laughlin and Moore-Read ground states and their quasihole excitations. In this paper, we extend the matrix product state techniques to evaluate quasielectron wave functions, a more complex task because the corresponding conformal field theory operator is not local. We use our method to obtain density profiles for states with multiple quasielectrons and quasiholes, and to calculate the (mutual) statistical phases of the excitations with high precision. The wave functions we study are subject to a known difficulty: the position of a quasielectron depends on the presence of other quasiparticles, even when their separation is large compared to the magnetic length. Quasielectron wave functions constructed using the composite fermion picture, which are topologically equivalent to the quasielectrons we study, have the same problem. This flaw is serious in that it gives wrong results for the statistical phases obtained by braiding distant quasiparticles. We analyze this problem in detail and show that it originates from an incomplete screening of the topological charges, which invalidates the plasma analogy. We demonstrate that this can be remedied in the case when the separation between the quasiparticles is large, which allows us to obtain the correct statistical phases. Finally, we propose that a modification of the Laughlin state, that allows for local quasielectron operators, should have good topological properties for arbitrary configurations of excitations.Abstract Matrix product state techniques provide a very efficient way to numerically evaluate certain classes of quantum Hall wave functions that can be written as correlators in two-dimensional conformal field theories. Important examples are the Laughlin and Moore-Read ground states and their quasihole excitations. In this paper, we extend the matrix product state techniques to evaluate quasielectron wave functions, a more complex task because the corresponding conformal field theory operator is not local. We use our method to obtain density profiles for states with multiple quasielectrons and quasiholes, and to calculate the (mutual) statistical phases of the excitations with high precision. The wave functions we study are subject to a known difficulty: the position of a quasielectron depends on the presence of other quasiparticles, even when their separation is large compared to the magnetic length. Quasielectron wave functions constructed using the composite fermion picture, which are topologically equivalent to the quasielectrons we study, have the same problem. This flaw is serious in that it gives wrong results for the statistical phases obtained by braiding distant quasiparticles. We analyze this problem in detail and show that it originates from an incomplete screening of the topological charges, which invalidates the plasma analogy. We demonstrate that this can be remedied in the case when the separation between the quasiparticles is large, which allows us to obtain the correct statistical phases. Finally, we propose that a modification of the Laughlin state, that allows for local quasielectron operators, should have good topological properties for arbitrary configurations of excitations.

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{g}_{112}

{\mathrm{r}}_{1}$${\mathrm{\ensuremath{-}}\mathrm{r}}_{2}$${\ensuremath{\Vert}}^{2}

for \ensuremath{\Vert}

{\mathrm{r}}_{1}$${\mathrm{\ensuremath{-}}\mathrm{r}}_{2}

\ensuremath{\Vert}\ensuremath{\rightarrow}0, where

{\mathrm{r}}_{1}

and

{\mathrm{r}}_{2}

are electron positions. We show that an approximation analogous to Laughlin's pseudopotential approximation does not have this property.

\pm1/4

of electron charge, and obey Abelian fractional statistics. In the spherical geometry, the total flux quanta

N_ϕ

is shown to be related to the number of electrons

N

N_ϕ = 2N-(5-q)

with

q

being the relative angular momentum between the quasielectrons in each pair which takes values of non-negative even integers. The overlaps are calculated between the proposed trial wave functions, including the ground state, quasiexciton states, and quasihole states, and the exact states of the finite size systems at

N_ϕ = 2N-3

. The near unity overlaps are obtained in the lowest Landau level, while the moderate overlaps are obtained in the second Landau level. The relevance of the wave functions to the yet to be discovered fractional quantum Hall effect in the lowest Landau level, as well as the 5/2 quantum Hall effect is also discussed.

\frac{1}{3},

\frac{2}{3},

corresponding to the electron fillings $\ensuremath{\nu}=\frac{3}{8},$

\frac{4}{11},

\frac{5}{13}.

The equivalent quasihole (QH) states occur at ${\ensuremath{\nu}}_{\mathrm{QH}}=\frac{1}{4},$

\frac{1}{5},

\frac{2}{7},

corresponding to $\ensuremath{\nu}=\frac{3}{10},$

\frac{4}{13},

\frac{5}{17}.

All these six novel FQH states were recently discovered experimentally. Detailed analysis indicates that QE or QH correlations in these states are different from those of well-known FQH electron states (e.g., Laughlin or Moore--Read states), leaving the origin of their incompressibility uncertain. Halperin's idea of Laughlin states of QP pairs is also explored, but is does not seem adequate.