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arXiv:cond-mat/0407448 · Physical Review B

Three-body correlations and finite-size effects in Moore-Read states on a sphere

Arkadiusz Wójs, John J. Quinn

Published Jan 19, 2005• 18 citations
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Abstract

Two- and three-body correlations in partially filled degenerate fermion shells are studied numerically for various interactions between the particles. Three distinct correlation regimes are defined, depending on the superharmonic, subharmonic, or harmonic forms of Haldane pair pseudopotential at short range. The harmonic form applies to electrons in the first excited Landau level $({\mathrm{LL}}_{1})$. Their correlations near half filling are confirmed to have a simple three-body form characteristic of the Moore-Read (MR) Pfaffian state, consisting of the maximum avoidance of the triplet state with the smallest relative angular momentum. To study MR correlations quantitatively, three-body amplitudes and pseudopotentials are introduced and calculated. Exact sum rules derived for the amplitudes imply that MR correlations depend on the actual (pair) interaction largely through the short-range behavior of the corresponding three-body interaction. In the calculated spectra of a model three-body repulsion, higher-energy bands are identified in addition to (previously known) MR ground state and its elementary charged (quasiparticle) and neutral (pair-breaking) excitations. The pair-breaker dispersion curve is determined, and the quasiparticles are correctly described by a generalization of the composite fermion model appropriate for Halperin $p$-type electron pairing with Laughlin correlations between the pairs. The known problem of the exact few-electron ground states having small overlaps with MR trial states on a sphere is also resolved by showing that the short-range three-body pseudopotential coefficients are more sensitive to the surface curvature than the two-body ones (leading to a slow convergence of the overlaps as a function of the electron number). Hence, the MR state and its excitations are believed to be a more accurate description of the experimental $\ensuremath{\nu}=\frac{5}{2}$ quantum Hall states than could be inferred from previous, small-size calculations.

DOI
10.1103/physrevb.71.045324

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Moore-Read states on a sphere: Three-body correlations and finite-size effects

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Energy spectra of a model short‐range three‐body repulsion are calculated for a half‐filled Landau level. The Moore‐Read ground state and its quasielectron (QE), quasihole (QH), magnetoroton (QE+QH), and pair‐breaking excitations are all identified. Two‐ and three‐body correlations of these states are analyzed. The QE/QH excitations are described by a composite fermion model for Laughlin‐correlated electron pairs. Comparison with the results obtained for the Coulomb interaction suggests that finite‐size effects are important in numerical diagonalization for the v = 5/2 quantum Hall state.

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a a U ni versit y of T ennessee , Kno xvi l le, T ennessee37996, U SA b Wro cl aw Uni versi ty of T echno lo gy, 50-370 Wro cl aw, Po l and Corr el at ions in partiall y Ùlled electron and comp osite fermion Landa u levels are studied numerically .I nsight into the nature of the correlations is obtained by using mo del pair pseudop oten tials.Energy spectra of a mo del short-range three-b ody repulsion are calculated.Mo ore{ Read ground state at the half -Ùllin g and its quasielec tron , quasihol e, magnetoroton, and pair-brea ki ng excitation s are all identiÙed.T he quasielectron /qu asihol e excitations are describ ed by a comp osite fermion mo del for Laughlin-correla ted electron pairs.C omparison of energy spectra and wavef unction overlaps obtained for di ˜erent pseudop otentials suggests that Ùnite-size e˜ects can b e imp ortant in numerical diagonal i zati on studies on a sphere.PAC S numb ers: 71.

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The fractional quantum Hall effect (FQHE) observed at half filling of the second Landau level is believed to be caused by a pairing of composite fermions captured by the Moore-Read Pfaffian wave function. The generating Hamiltonian for the Moore-Read Pfaffian is a purely three-body model that breaks particle-hole symmetry and lacks other properties, such as dominate two-body repulsive interactions, expected from a physical model of the FQHE. We use exact diagonalization to study the low-energy states of a more physical two-body generator model derived from the three-body model. We find that the two-body model exhibits the essential features expected from the Moore-Read Pfaffian: pairing, non-Abelian anyon excitations, and a neutral fermion mode. The model also satisfies constraints expected for a physical model of the FQHE at half-filling because it is short range, spatially decaying, particle-hole symmetric, and supports a roton mode with a robust spectral gap in the thermodynamic limit. Hence, this two-body model offers a bridge between artificial three-body generator models for paired states and the physical Coulomb interaction and can be used to further explore properties of non-Abelian physics in the FQHE.

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Two-and three-body correlation functions (number of pairs or triplets vs. relative angular momentum) of electrons or Laughlin quasielectrons (i.e., composite fermions in their first excited Landau level) are studied numerically in several fractional quantum Hall liquids.It is shown directly that the νe = 4/11 liquid (corresponding to a ν = 1/3 filling of composite fermions in their first excited Landau level) is a paired state of quasielectrons, hence interpreted as a condensate of "second-generation" quasiholes of Moore-Read ν = 1/2 state of composite fermions.

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Mean-field approximations for short-range four-body interactions at ν=35

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Trial wave functions such as the Moore-Read and Read-Rezayi states, which minimize short-range multibody interactions, are candidate states for describing the fractional quantum Hall effects at filling factors $\ensuremath{\nu}=1/2$ and $2/5$ in the second Landau level. These trial wave functions are unique zero-energy states of three-body and four-body interaction Hamiltonians, respectively, but they are not close to the ground states of the Coulomb interaction. Previous studies using extensive parameter scans have found optimal two-body interactions on the sphere that produce states close to these. Here we focus on short-ranged four-body interaction and study two mean-field approximations that reduce the four-body interactions to two-body interactions on the sphere by replacing composite operators with their incompressible ground-state expectation values. We present the results for pseudopotentials of these approximate interactions. A comparison of finite system spectra on the sphere of the four-body and the approximate interactions at filling fraction $\ensuremath{\nu}=3/5$ shows that these approximations produce good effective descriptions of the low-energy structure of the four-body interaction Hamiltonian. The approach also independently reproduces the optimal two-body interaction inferred from parameter scans. We also show that for $n=3$, but not for $n=4$, the mean-field approximations of the $n$-body interaction are equivalent to particle-hole symmetrization of the interaction. Within the system sizes accessible, analysis of the spectrum of the mean-field two-body Hamiltonian on the torus was inconclusive, and indicates a competing anisotropic state in the system.

arXiv:1902.04477•Physical review. B./Physical review. B•Jun 20, 2019•2 citations
Two- and three-body correlations in partially filled degenerate fermion shells are studied numerically for various interactions between the particles. Three distinct correlation regimes are defined, depending on the superharmonic, subharmonic, or harmonic forms of Haldane pair pseudopotential at short range. The harmonic form applies to electrons in the first excited Landau level (LL1)({\mathrm{LL}}_{1})(LL1​). Their correlations near half filling are confirmed to have a simple three-body form characteristic of the Moore-Read (MR) Pfaffian state, consisting of the maximum avoidance of the triplet state with the smallest relative angular momentum. To study MR correlations quantitatively, three-body amplitudes and pseudopotentials are introduced and calculated. Exact sum rules derived for the amplitudes imply that MR correlations depend on the actual (pair) interaction largely through the short-range behavior of the corresponding three-body interaction. In the calculated spectra of a model three-body repulsion, higher-energy bands are identified in addition to (previously known) MR ground state and its elementary charged (quasiparticle) and neutral (pair-breaking) excitations. The pair-breaker dispersion curve is determined, and the quasiparticles are correctly described by a generalization of the composite fermion model appropriate for Halperin ppp-type electron pairing with Laughlin correlations between the pairs. The known problem of the exact few-electron ground states having small overlaps with MR trial states on a sphere is also resolved by showing that the short-range three-body pseudopotential coefficients are more sensitive to the surface curvature than the two-body ones (leading to a slow convergence of the overlaps as a function of the electron number). Hence, the MR state and its excitations are believed to be a more accurate description of the experimental \ensuremathν=52\ensuremath{\nu}=\frac{5}{2}\ensuremathν=25​ quantum Hall states than could be inferred from previous, small-size calculations.
Energy spectra of a model short‐range three‐body repulsion are calculated for a half‐filled Landau level. The Moore‐Read ground state and its quasielectron (QE), quasihole (QH), magnetoroton (QE+QH), and pair‐breaking excitations are all identified. Two‐ and three‐body correlations of these states are analyzed. The QE/QH excitations are described by a composite fermion model for Laughlin‐correlated electron pairs. Comparison with the results obtained for the Coulomb interaction suggests that finite‐size effects are important in numerical diagonalization for the v = 5/2 quantum Hall state.
The electron correlations near the half-filling of the lowest and excited Landau levels (LL's) are studied using numerical diagonalization. It is shown that in the low-lying states electrons avoid pair states with relative angular momenta R\mathcal{R}R corresponding to positive anharmonicity of the interaction pseudopotential V(R).V(\mathcal{R}).V(R). In the lowest LL, the superharmonic behavior of V(R)V(\mathcal{R})V(R) causes Laughlin correlations (avoiding pairs with R=1)\mathcal{R}=1)R=1) and the Laughlin-Jain series of incompressible ground states. In the first excited LL, V(R)V(\mathcal{R})V(R) is harmonic at short range and a different series of incompressible states results. Similar correlations occur in the paired Moore-Read $\ensuremath{\nu}=\frac{5}{2}$ state and in the $\ensuremath{\nu}=\frac{7}{3}$ and 83\frac{8}{3}38​ states, all having small total parentage from R=1\mathcal{R}=1R=1 and 3 and large parentage from R=5.\mathcal{R}=5.R=5. The $\ensuremath{\nu}=\frac{7}{3}$ and 83\frac{8}{3}38​ states are different from Laughlin $\ensuremath{\nu}=\frac{1}{3}$ and 23\frac{2}{3}32​ states and, in finite systems, occur at a different LL degeneracy (flux). The series of Laughlin-correlated states of electron pairs at $\ensuremath{\nu}{=2+2/(q}_{2}+2)=\frac{8}{3},$ 52,\frac{5}{2},25​, 125,\frac{12}{5},512​, and 73\frac{7}{3}37​ is proposed, although only in the $\ensuremath{\nu}=\frac{5}{2}$ state pairing has been confirmed numerically. In the second excited LL, V(R)V(\mathcal{R})V(R) is subharmonic at short range and (near the half-filling) the electrons group into spatially separated larger $\ensuremath{\nu}=1$ droplets to minimize the number of strongly repulsive pair states at R=3\mathcal{R}=3R=3 and 5.
a a U ni versit y of T ennessee , Kno xvi l le, T ennessee37996, U SA b Wro cl aw Uni versi ty of T echno lo gy, 50-370 Wro cl aw, Po l and Corr el at ions in partiall y Ùlled electron and comp osite fermion Landa u levels are studied numerically .I nsight into the nature of the correlations is obtained by using mo del pair pseudop oten tials.Energy spectra of a mo del short-range three-b ody repulsion are calculated.Mo ore{ Read ground state at the half -Ùllin g and its quasielec tron , quasihol e, magnetoroton, and pair-brea ki ng excitation s are all identiÙed.T he quasielectron /qu asihol e excitations are describ ed by a comp osite fermion mo del for Laughlin-correla ted electron pairs.C omparison of energy spectra and wavef unction overlaps obtained for di ˜erent pseudop otentials suggests that Ùnite-size e˜ects can b e imp ortant in numerical diagonal i zati on studies on a sphere.PAC S numb ers: 71.
The fractional quantum Hall effect (FQHE) observed at half filling of the second Landau level is believed to be caused by a pairing of composite fermions captured by the Moore-Read Pfaffian wave function. The generating Hamiltonian for the Moore-Read Pfaffian is a purely three-body model that breaks particle-hole symmetry and lacks other properties, such as dominate two-body repulsive interactions, expected from a physical model of the FQHE. We use exact diagonalization to study the low-energy states of a more physical two-body generator model derived from the three-body model. We find that the two-body model exhibits the essential features expected from the Moore-Read Pfaffian: pairing, non-Abelian anyon excitations, and a neutral fermion mode. The model also satisfies constraints expected for a physical model of the FQHE at half-filling because it is short range, spatially decaying, particle-hole symmetric, and supports a roton mode with a robust spectral gap in the thermodynamic limit. Hence, this two-body model offers a bridge between artificial three-body generator models for paired states and the physical Coulomb interaction and can be used to further explore properties of non-Abelian physics in the FQHE.
Two-and three-body correlation functions (number of pairs or triplets vs. relative angular momentum) of electrons or Laughlin quasielectrons (i.e., composite fermions in their first excited Landau level) are studied numerically in several fractional quantum Hall liquids.It is shown directly that the νe = 4/11 liquid (corresponding to a ν = 1/3 filling of composite fermions in their first excited Landau level) is a paired state of quasielectrons, hence interpreted as a condensate of "second-generation" quasiholes of Moore-Read ν = 1/2 state of composite fermions.
Trial wave functions such as the Moore-Read and Read-Rezayi states, which minimize short-range multibody interactions, are candidate states for describing the fractional quantum Hall effects at filling factors $\ensuremath{\nu}=1/2$ and 2/52/52/5 in the second Landau level. These trial wave functions are unique zero-energy states of three-body and four-body interaction Hamiltonians, respectively, but they are not close to the ground states of the Coulomb interaction. Previous studies using extensive parameter scans have found optimal two-body interactions on the sphere that produce states close to these. Here we focus on short-ranged four-body interaction and study two mean-field approximations that reduce the four-body interactions to two-body interactions on the sphere by replacing composite operators with their incompressible ground-state expectation values. We present the results for pseudopotentials of these approximate interactions. A comparison of finite system spectra on the sphere of the four-body and the approximate interactions at filling fraction $\ensuremath{\nu}=3/5$ shows that these approximations produce good effective descriptions of the low-energy structure of the four-body interaction Hamiltonian. The approach also independently reproduces the optimal two-body interaction inferred from parameter scans. We also show that for n=3n=3n=3, but not for n=4n=4n=4, the mean-field approximations of the nnn-body interaction are equivalent to particle-hole symmetrization of the interaction. Within the system sizes accessible, analysis of the spectrum of the mean-field two-body Hamiltonian on the torus was inconclusive, and indicates a competing anisotropic state in the system.