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Acta Physica Polonica A

Jack 3/5 State from Two-Body Interaction

Bartosz Kuśmierz, Y.H. Wu, Arkadiusz Wójs

Published Jan 1, 2016• 7 citations
Publisher

Abstract

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.

DOI
10.12693/aphyspola.129.a-73

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arXiv:1301.6565•Physical Review B•Jun 24, 2013•31 citations
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 family of ``Jack states'' related to antisymmetric Jack polynomials are the exact zero-energy ground states of particular model short-range many-body repulsive interactions, defined by a few nonvanishing leading pseudopotentials. Some Jack states are known or anticipated to accurately describe many-electron incompressible ground states emergent from the two-body Coulomb repulsion in the fractional quantum Hall effect. By extensive numerical diagonalization, we demonstrate the emergence of Jack states from suitable pair interactions. We find empirically a simple formula for the optimal two-body pseudopotentials for the series of most prominent Jack states generated by contact many-body repulsion. Furthermore, we seek a realization of arbitrary Jack states in realistic quantum Hall systems with Coulomb interaction, i.e., in the partially filled lowest and excited Landau levels in quasi-two-dimensional layers of conventional semiconductors such as GaAs or in graphene.
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.
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.
Fractional quantum Hall eect is a remarkable behaviour of correlated electrons, observed exclusively in two dimensions, at low temperatures, and in strong magnetic elds.The most prominent fractional quantum Hall state occurs at Landau level lling factor ν = 1/3 and it is described by the famous Laughlin wave function, which (apart from the trivial Gaussian factor) is an example of Jack polynomial.Fermionic Jack polynomials also describe another pair of candidate fractional quantum Hall states: MooreRead and ReadRezayi states, believed to form at the ν = 1/2 and 3/5 llings of the second Landau level, respectively.Bosonic Jacks on the other hand are candidates for certain correlated states of cold atoms.We examine here a continuous family of fermionic Jack polynomials whose special case is the Laughlin state as approximate wave functions for the 1/3 fractional quantum Hall eect.
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.