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arXiv:1902.04477 · Physical review. B./Physical review. B

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

Bartosz Kuśmierz, Arkadiusz Wójs, G. J. Sreejith

Published Jun 20, 2019• 2 citations
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Abstract

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.

DOI
10.1103/physrevb.99.235141

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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 \ensuremathν=1/2\ensuremath{\nu}=1/2\ensuremathν=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 \ensuremathν=3/5\ensuremath{\nu}=3/5\ensuremathν=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.
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.
The Read-Rezayi wave function is one of the candidates for the fractional quantum Hall effect at filling fraction $\ensuremath{\nu}=2+⅗$, and thereby also its hole conjugate at 2+⅖2+⅖2+⅖. We study a general class of tripartite composite fermion wave functions, which reduce to the Rezayi-Read ground state and quasiholes for appropriate quantum numbers, but also allow a construction of wave functions for quasiparticles and neutral excitations by analogy to the standard composite fermion theory. We present numerical evidence in finite systems that these trial wave functions capture well the low energy physics of a four-body model interaction. We also compare the tripartite composite fermion wave functions with the exact Coulomb eigenstates at 2+⅗2+⅗2+⅗, and find reasonably good agreement. The ground state as well as several excited states of the four-body interaction are seen to evolve adiabatically into the corresponding Coulomb states for N=15N=15N=15 particles. These results support the plausibility of the Read-Rezayi proposal for the 2+⅖2+⅖2+⅖ and 2+⅗2+⅗2+⅗ fractional quantum Hall effect. However, certain other proposals also remain viable, and further study of excitations and edge states will be necessary for a decisive establishment of the physical mechanism of these fractional quantum Hall states.
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.
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.
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.
We investigate the feasibility of many candidate quantum Hall states for two-component bosons in the lowest Landau level. We identify interactions for which spin-singlet incompressible states occur at filling factors $\ensuremath{\nu}=2/3$, 4/54/54/5, and 4/34/34/3, and partially spin-polarized states at filling factors 3/43/43/4 and 3/23/23/2, where ``spin'' serves as a generic label for the two components. We study ground states, excitations, edge states, and entanglement spectrum for systems with up to 16 bosons and construct explicit trial wave functions to clarify the underlying physics. The composite fermion theory very accurately describes the ground states as well as excitations at $\ensuremath{\nu}=2/3$, 4/54/54/5, and 3/43/43/4, although it is less satisfactory for the $\ensuremath{\nu}=3/2$ state. For $\ensuremath{\nu}=4/3$ a ``non-Abelian spin-singlet'' state, which is the exact ground state of a three-body contact interaction, has been proposed to occur even for a two-body contact interaction; our trial wave functions are very accurate for the excitations of the three-body interaction, but they do not describe the excitations of the two-body interaction very well. Instead, we find that the $\ensuremath{\nu}=4/3$ state is more likely to be a spin-singlet state of reverse-flux-attached composite fermions at filling ${\ensuremath{\nu}}^{*}=4$. We also consider incompressible states at integral filling factors $\ensuremath{\nu}=1$ and 2. The incompressible state at $\ensuremath{\nu}=1$ is shown to be well described by the parton-based Jain spin-singlet wave function, and the incompressible state at $\ensuremath{\nu}=2$ as the spin-singlet state of reverse-flux-attached composite fermions at ${\ensuremath{\nu}}^{*}=2$, which provides an example of the bosonic integer topological phase.