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arXiv:cond-mat/0702101

Mapping interacting onto non-interacting quantum Hall systems

Johannes N. Kriel, Frederik G. Scholtz

Published Feb 5, 2007• 0 citations
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Abstract

We construct an explicit duality between the interacting quantum Hall system in the lowest Landau level and a non-interacting Landau problem. This is done by absorbing the interaction into the gauge field in the form of an effective magnetic vector potential. The result is analogous to, and illuminates the microscopic origin of, the well-known composite fermion model, but has several advantageous properties. Using this duality we derive, for an arbitrary short-range interaction, analytic expressions for the ground state energy and the excitation gap as functions of the filling fraction. We find good agreement with existing results.

DOI
10.48550/arxiv.cond-mat/0702101

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We construct an explicit duality between the interacting quantum Hall system in the lowest Landau level and a non-interacting Landau problem. This is done by absorbing the interaction into the gauge field in the form of an effective magnetic vector potential. The result is analogous to, and illuminates the microscopic origin of, the well-known composite fermion model, but has several advantageous properties. Using this duality we derive, for an arbitrary short-range interaction, analytic expressions for the ground state energy and the excitation gap as functions of the filling fraction. We find good agreement with existing results.
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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.
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 show that a modified version of Son's Dirac composite fermion theory proposed by Seiberg et al gives a candidate unified description of the gapped and gapless fractional quantum Hall states within a single Landau level. Our main tool is the successive application of three-dimensional dualities to partially filled Landau levels of composite fermions, which imply that this theory has a complicated landscape of gapped vacua and critical points. This construction is the Lagrangian, or effective field theory, analogue of the flux attachment procedure. The critical points exist at even denominator filling and are well-described by a Fermi surface for a weakly coupled composite fermion coupled to an abelian Chern-Simons theory. The gapped states include odd-denominator filling fraction states with an abelian Chern-Simons description which we show matches the one expected for hierarchy states, as well as non-abelian states at even-denominator filling that arise from pair instabilities of the composite fermion's Fermi surface.
It is argued that a short-range interaction induces a quasi-Landau-level structure in the filling-factor dependence of the ground-state energy of electrons in the lowest Landau level. The discussion is based on exact numerical results obtained from small-system calculations, and the adiabatic principle of Greiter and Wilczek. It is shown that by attaching a single negative flux quantum to each electron adiabatically, the noninteracting electron ground states occupying higher Landau levels are smoothly transformed into the ground states of a hard-core boson gas in the lowest Landau level. This implies a close correspondence between the noninteracting fermions and the hard-core bosons in the lowest Landau level. Numerical results suggest the possibility of a smooth transition from hard-core bosons to fermions with a short-range interaction, by continuously varying the interaction potential.