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Journal of Physics Condensed Matter

Fractional quantum Hall states on a square lattice

George S. Kliros, N. d’Ambrumenil

Published Jun 10, 1991• 5 citations
Publisher

Abstract

The authors calculate the ground state energies and wavefunctions for small systems of fermions on a lattice using singly and doubly periodic boundary conditions. They find minima in the ground state energy when the magnetic flux per plaquette corresponds to filling fractions with odd denominators. They interpret these minima in terms of fractional quantum Hall states. The pair correlation function at nearest and next-nearest neighbours is calculated as a function of the repulsive interaction strength between particles.

DOI
10.1088/0953-8984/3/23/012

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The authors calculate the ground state energies and wavefunctions for small systems of fermions on a lattice using singly and doubly periodic boundary conditions. They find minima in the ground state energy when the magnetic flux per plaquette corresponds to filling fractions with odd denominators. They interpret these minima in terms of fractional quantum Hall states. The pair correlation function at nearest and next-nearest neighbours is calculated as a function of the repulsive interaction strength between particles.
We calculate the ground state energies and wavefunctions for small systems of fermions on a lattice using singly and doubly periodic boundary conditions. We find minima in the ground state energy when the magnetic flux per plaquette corresponds to filling fractions with odd denominators. We interpret these minima in terms of fractional quantum Hall states. The pair correlation function at nearest and next-nearest neighbours is calculated as a function of the repulsive interaction strength between particles.
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.
We study lattice models of charged particles in uniform magnetic fields. We show how longer range hopping can be engineered to produce a massively degenerate manifold of single-particle ground states with wave functions identical to those making up the lowest Landau level of continuum electrons in a magnetic field. We find that in the presence of local interactions, and at the appropriate filling factors, Laughlin's fractional quantum Hall wave function is an exact many-body ground state of our lattice model. The hopping matrix elements in our model fall off as a Gaussian, and when the flux per plaquette is small compared to the fundamental flux quantum one only needs to include nearest and next-nearest neighbor hoppings. We suggest how to realize this model using atoms in optical lattices, and describe observable consequences of the resulting fractional quantum Hall physics.
The authors study the dispersion of collective excitations for fractional quantum Hall states using Feynman-Bijl ansatz wavefunctions. They take the ground-state pair correlation function from direct diagonalisations of the Hamiltonian for a small number of electrons confined to the surface of a sphere, instead of from the simulation of large systems described by a Laughlin-like wavefunction. The results seem remarkably well converged considering the small systems used. They also treat systems with partially filled N=1 Landau levels.
Intuitive arguments and numerical simulations have long been used to argue for the incompressibility of fractionally quantized Hall states at the most prominent odd-denominator Landau-level filling fractions $\ensuremath{\nu}$ = ppp/(ppp (mmm - 1) + 1), where ppp is an integer and mmm an odd integer. Here, the authors strengthen the case by deriving these states more rigorously via adiabatic localization of magnetic flux onto the particles. They arrive at wave functions suggested by Jain's composite-fermion picture.