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Acta Physica Polonica A

Jack Polynomials and Fractional Quantum Hall Effect at ν = 1/3

Bartosz Kuśmierz, Arkadiusz Wójs

Published Aug 1, 2016• 3 citations
Publisher

Abstract

We investigate properties of strongly correlated, spinless electrons confined within given Landau level at filling factor ν = 1/3.Our analysis is based on the formalism of the Jack polynomials.Selected Jack polynomial wave functions are compared with ground states of the Coulomb interaction Hamiltonians, in different materials and the Landau levels, obtained by exact diagonalization.We show that certain Jack wave functions can be used as a description of fractional quantum Hall states.

DOI
10.12693/aphyspola.130.607

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We investigate properties of strongly correlated, spinless electrons confined within given Landau level at filling factor ν = 1/3.Our analysis is based on the formalism of the Jack polynomials.Selected Jack polynomial wave functions are compared with ground states of the Coulomb interaction Hamiltonians, in different materials and the Landau levels, obtained by exact diagonalization.We show that certain Jack wave functions can be used as a description of fractional quantum Hall states.
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.
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.
We focus on a certain aspect of trial wave function approach in the fractional quantum Hall effect.We analyze the role of partition orderings and discuss the possible numerical search for the partition determining the subspace of the Hilbert space containing a particular quantum Hall wave function.This research is inspired by analogical properties of certain polynomials which are the object of interest of the symmetric function theory, especially the Jack polynomials (related to the so-called "Jack states").Presented method may be used in the search of candidate trial wave functions.We also justify (in certain cases) diagonalization of the Coulomb repulsion Hamiltonian restricted to certain subspaces.We focus on the states at filling factor ν = 1/3 in the lowest and second Landau level.
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 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.
States of strongly interacting particles are of fundamental interest in physics and can produce exotic emergent phenomena and topological structures. We consider here two-dimensional electrons in a magnetic field, and, departing from the standard practice of restricting to the lowest LL, introduce a model short-range interaction that is infinitely strong compared to the cyclotron energy. We demonstrate that this model lends itself to an exact solution for the ground as well as excited states at arbitrary filling factors ν<1/2p and produces a fractional quantum Hall effect at fractions of the form ν=n/(2pn+1), where n and p are integers. The fractional quantum Hall states of our model share many topological properties with the corresponding Coulomb ground states in the lowest Landau level, such as the edge physics and the fractional charge of the excitations.