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Acta Physica Polonica A

Ground States of Quantum Hall Three-Body ``Short-Range'' Repulsion and Mean Field Approximation: Correlation Functions and Overlaps

Bartosz Kuśmierz, G. J. Sreejith, Arkadiusz Wójs

Published Jan 1, 2019• 0 citations
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Abstract

We analyze properties of trial wave functions in fractional quantum Hall effect, generated by the "short-range", three-body repulsion and its mean field approximation.Ground states of electron repulsion Hamiltonians at filing factor ν = 1/2 are evaluated as a description of physically observed state ν = 5/2.We analyze overlaps between the Moore-Read state and its mean field approximation for different number of particles and compare both states with ground state of the Coulomb interaction in the first excited Landau level.Our study also includes examination of electron-electron correlation functions and electron densities of excited states for both interactions.

DOI
10.12693/aphyspola.135.82

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

Arkadiusz Wójs

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.

AIP conference proceedings•Jan 1, 2005•0 citations
We analyze properties of trial wave functions in fractional quantum Hall effect, generated by the "short-range", three-body repulsion and its mean field approximation.Ground states of electron repulsion Hamiltonians at filing factor ν = 1/2 are evaluated as a description of physically observed state ν = 5/2.We analyze overlaps between the Moore-Read state and its mean field approximation for different number of particles and compare both states with ground state of the Coulomb interaction in the first excited Landau level.Our study also includes examination of electron-electron correlation functions and electron densities of excited states for both interactions.
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.
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 $\ensuremath{\nu}=\frac{5}{2}$ quantum Hall states than could be inferred from previous, small-size calculations.
We investigate the Read-Rezayi parafermion state of correlated electrons at the fractional Landau level filling ν = 3/5.It is a Jack polynomial generated by contact four-body repulsion.We show by exact diagonalization that it is also emerges from a suitable short-range two-body interaction.We find that it closely matches Coulomb ground state in the second Landau level of non-relativistic fermions, and thus possibly describes the ν = 13/5 (and, by conjugation, ν = 12/5) fractional quantum Hall effect in GaAs.
The fractional quantum Hall effect was experimentally discovered in 1982. It was observed that the Hall conductivity σyx of a two-dimensional electron system is quantized, σyx=e2/3h, in the vicinity of the Landau level filling factor ν=1/3. In 1983, Laughlin proposed a trial many-body wave function, which he claimed described a “new state of matter”—a homogeneous incompressible liquid with fractionally charged quasiparticles. Here, I develop an exact diagonalization theory that allows one to calculate the energy and other physical properties of the ground and excited states of a system of N two-dimensional Coulomb interacting electrons in a strong magnetic field. I analyze the energies, electron densities, and other physical properties of the systems with N≤7 electrons continuously as a function of magnetic field in the range 1/4≲ν<1. The results show that both the ground and excited states of the system resemble a sliding Wigner crystal whose parameters are influenced by the magnetic field. Energy gaps in the many-particle spectra appear and disappear as the magnetic field changes. I also calculate the physical properties of the ν=1/3 Laughlin state for N≤8 and compare the results with the exact ones. This comparison, as well as an analysis of some other statements published in the literature, show that the Laughlin state and its fractionally charged excitations do not describe the physical reality, neither at small N nor in the thermodynamic limit. The results obtained shed new light on the nature of the ground and excited states in the fractional quantum Hall effect.
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.