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arXiv:cond-mat/0505135 · Journal of Physics Condensed Matter

An anyon wavefunction for the fractional quantum Hall effect

Orion Ciftja, G. S. Japaridze, Xiaoqian Wang

Published Apr 29, 2005• 11 citations
arXivHTMLPDF

Abstract

An anyon wavefunction (characterized by the statistical factor n ) projected onto the lowest Landau level is derived for the fractional quantum Hall effect states at filling factor ν = n /(2 pn +1) ( p and n are integers). We study the properties of the anyon wavefunction by using detailed Monte Carlo simulations in disc geometry and show that the anyon ground-state energy is a lower bound to the composite fermion one. Our results suggest that the composite fermions can be viewed as a combination of anyons and a fluid of charge–neutral dipoles.

DOI
10.1088/0953-8984/17/19/011

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An anyon wavefunction (characterized by the statistical factor n ) projected onto the lowest Landau level is derived for the fractional quantum Hall effect states at filling factor ν = n /(2 pn +1) ( p and n are integers). We study the properties of the anyon wavefunction by using detailed Monte Carlo simulations in disc geometry and show that the anyon ground-state energy is a lower bound to the composite fermion one. Our results suggest that the composite fermions can be viewed as a combination of anyons and a fluid of charge–neutral dipoles.
An adiabatic approach put forward by Greiter and Wilczek interpolates between the integer quantum Hall effects of electrons and composite fermions by varying the statistical flux bound to electrons continuously from zero to an even integer number of flux quanta, such that the intermediate states represent anyons in an external magnetic field with the same ``effective'' integer filling factor. We consider such anyons on a torus, and we construct representative wave functions for their ground as well as excited states. These wave functions involve higher Landau levels in general, but can be explicitly projected into the lowest Landau level for many parameters. We calculate the variational energy gap between the first excited state and the ground state, and we find that it remains open as the statistical phase is varied. Finally, we obtain from these wave functions, both analytically and numerically, various topological quantities, such as the ground-state degeneracy, the Chern number, and the Hall viscosity.
Single particle basis functions for composite fermions are obtained from which many-composite fermion states confined to the lowest electronic Landau level can be constructed in the standard manner, i.e. by building Slater determinants. This representation enables a Monte Carlo study of systems containing a large number of composite fermions, yielding new quantitative and qualitative information. The ground state energy and the gaps to charged and neutral excitations are computed for a number of fractional quantum Hall effect (FQHE) states, earlier off-limits to a quantitative investigation. The ground state energies are estimated to be accurate to ~0.1% and the gaps at the level of a few percent. It is also shown that at Landau level fillings smaller than or equal to 1/9 the FQHE is unstable to a spontaneous creation of excitons of composite fermions. In addition, this approach provides new conceptual insight into the structure of the composite fermion wave functions, resolving in the affirmative the question of whether it is possible to motivate the composite fermion theory entirely within the lowest Landau level, without appealing to higher Landau levels.
We study the problem of anyons with statistics \ensuremath{\nu} in a strong magnetic field by means of a similarity transformation. The ground state of this problem exhibits an ``integer'' quantum Hall effect at filling factor 1/\ensuremath{\nu} with quasiparticle excitations of charge q/\ensuremath{\nu} and statistics -1/\ensuremath{\nu}. Certain electron fractional quantum Hall states can be considered as a realization of this, for example, the 2/5, 3/7,. . . hierarchy of the 1/3 state. This may explain the observed quasiparticle-quasihole asymmetry in the fractional quantum hall hierarchy.
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.