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arXiv:1301.6565 · Physical Review B

Tripartite composite fermion states

G. J. Sreejith, Ying-Hai Wu, Arkadiusz Wójs, Jainendra K. Jain

Published Jun 24, 2013• 31 citations
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Abstract

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 $2+⅖$. 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 $2+⅗$, 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 $N=15$ particles. These results support the plausibility of the Read-Rezayi proposal for the $2+⅖$ and $2+⅗$ 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.

DOI
10.1103/physrevb.87.245125

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The Read-Rezayi wave function is one of the candidates for the fractional quantum Hall effect at filling fraction \ensuremathν=2+⅗\ensuremath{\nu}=2+⅗\ensuremathν=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.
We propose trial wave functions for quasiparticle and exciton excitations of the Moore-Read Pfaffian fractional quantum Hall states, both for bosons and for fermions, and study these numerically. Our construction of trial wave functions employs a picture of the bosonic Moore-Read state as a symmetrized double layer composite fermion state. We obtain the number of independent angular momentum multiplets of quasiparticle and exciton trial states for systems of up to 20 electrons. We find that the counting for quasielectrons at large angular momentum on the sphere matches that expected from the conformal field theory (CFT) that describes the Moore-Read state's boundary theory. In particular, the counting for quasielectrons is the same as for quasiholes, in accordance with the idea that the CFT describing both sides of the Fractional Quantum Hall plateau should be the same. We also show that our trial wave functions have good overlaps with exact wave functions obtained using various interactions, including second Landau level Coulomb interactions and the three-body delta interaction for which the Pfaffian states and their quasiholes are exact ground states. We discuss how these results relate to recent work by Sreejith et al. on a similar set of trial wave functions for excitations over the Pfaffian state as well as to earlier work by Hansson et al., which has produced trial wave functions for quasiparticles based on conformal field theory methods and by Bernevig and Haldane, which produced trial wave functions based on clustering properties and ``squeezing.''
The Read-Rezayi (RR) parafermion states form a series of exotic non-Abelian fractional quantum Hall (FQH) states at filling $\ensuremath{\nu}=k/(k+2)$. Computationally, the wave functions of these states are prohibitively expensive to generate for large systems. We introduce a series of parton states, denoted 2‾k1k+1{\overline{2}}^{k}{1}^{k+1}2k1k+1, and show that they lie in the same universality classes as the particle-hole-conjugate RR (``anti-RR'') states. Our analytical results imply that a $[U{(1)}_{k+1}\ifmmode\times\else\texttimes\fi{}U{(2k)}_{\ensuremath{-}1}]/[SU{(k)}_{\ensuremath{-}2}\ifmmode\times\else\texttimes\fi{}U{(1)}_{\ensuremath{-}1}]$ coset conformal field theory describes the edge excitations of the 2‾k1k+1{\overline{2}}^{k}{1}^{k+1}2k1k+1 state, suggesting nontrivial dualities with respect to previously known descriptions. The parton construction allows wave functions in anti-RR phases to be generated for hundreds of particles. We further propose the parton sequence n‾2‾214\overline{n}{\overline{2}}^{2}{1}^{4}n2214, with n=1,2,3n=1,2,3n=1,2,3, to describe the FQH states observed at $\ensuremath{\nu}=2+1/2$, 2+2/52+2/52+2/5, and 2+3/82+3/82+3/8.
We study a class of ansatz wave functions in which composite fermions form two correlated "partitions." These "bipartite" composite fermion states are demonstrated to be very accurate for electrons in a strong magnetic field interacting via a short-range 3-body interaction potential over a broad range of filling factors. Furthermore, this approach gives accurate approximations for the exact Coulomb ground state at 2+3/5 and 2+4/7 and is thus a promising candidate for the observed fractional quantum Hall states at the hole conjugate fractions at 2+2/5 and 2+3/7.
It is commonly assumed in the studies of the fractional quantum Hall effect that the physics of a fractional quantum Hall state, in particular the character of its excitations, is invariant under a continuous deformation of the Hamiltonian during which the gap does not close. We show in this article that, at least for finite systems, as the interaction is changed from a model three body interaction to Coulomb, the ground state at filling factor $\ensuremath{\nu}=2/5$ evolves continuously from the so-called Gaffnian wave function to the composite fermion wave function, but the quasiholes alter their character in a nonperturbative manner. This is attributed to the fact that the Coulomb interaction opens a gap in the Gaffnian quasihole sector, pushing many of the states to very high energies. Interestingly, the states below the gap are found to have a one-to-one correspondence with the composite fermion theory, suggesting that the Gaffnian model contains composite fermions, and that the Gaffnian quasiholes are unstable to the formation of composite fermions when a two-body interaction term is switched on. General implications of this study are discussed.
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.
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.