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arXiv:1802.06666 · Physical review. B./Physical review. B

Emergence of Jack ground states from two-body pseudopotentials in fractional quantum Hall systems

Bartosz Kuśmierz, Arkadiusz Wójs

Published Jun 15, 2018• 22 citations

Abstract

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.

DOI: 10.1103/physrevb.97.245125

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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.

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.

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.

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.

Model wavefunctions are essential for studying fractional quantum Hall (FQH) phases. They reveal the intricate connections between model FQH states, conformal field theory, Jack polynomials, and pseudopotential Hamiltonians. However, most model wavefunctions studied so far are in the continuum. In this work, we construct model wavefunctions and the corresponding exact parent Hamiltonians for the entire

Z_k

Read--Rezayi series on a lattice. Our model wavefunctions obey a generalized clustering rule implemented on the lattice, which we find is encoded in the Macdonald polynomials providing a unified algebraic description of our lattice-specific model states. This ensures that the model states are exact zero-energy ground states of local density interactions and exhibit an infinitely large gap in the particle entanglement spectrum. Numerical studies of the Laughlin and Moore--Read series find the expected zero-mode and one-flux quasihole counting and positive energy gaps at the system sizes studied, and the available finite-size scaling suggests nonzero thermodynamic gaps. Tuning the model interaction range can drive a phase transition out of the FQH phase. Our work not only advances the theoretical understanding of the underlying mathematical structure of lattice model FQH states but also is potentially useful for designing exotic many-body phases in engineered quantum platforms.

A simple one-dimensional model is proposed, in which N spinless repulsively interacting fermions occupy M > N degenerate states. It is argued that the energy spectrum and the wavefunctions of this system strongly resemble the spectrum and wavefunctions of 2D electrons in the lowest Landau level (the problem of the Fractional Quantum Hall Effect). In particular, Laughlin-type wavefunctions describe ground states at filling factors v = N/M = 1/ q , q odd.. Within this model the complimentary wavefunction for v =1−1/ q is found explicitly, and extremely simple ground state wavefunctions for arbitrary odd-denominator filling factors are proposed.