By separating the Schrödinger equation for N noninteracting spin-polarized fermions in twodimensional hyperspherical coordinates, we demonstrate that fractional quantum Hall (FQH) states emerge naturally from degeneracy patterns of the antisymmetric free-particle eigenfunctions. In the presence of Coulomb interactions, the FQH states split off from a degenerate manifold and become observable as distinct quantized energy eigenstates with an energy gap. This alternative classification scheme is based on an approximate separability of the interacting N -fermion Schrödinger equation in the hyperradial coordinate, which sheds light on the emergence of Laughlin states as well as other FQH states. An approximate good collective quantum number, the grand angular momentum K from K-harmonic few-body theory, is shown to correlate with known FQH states at many filling factors observed experimentally.

Landau level mixing should influence the quantum Hall effect for all but the strongest applied magnetic fields. However, the effects of Landau level mixing have proven difficult to model theoretically. We propose a simple method for examining the effects of Landau level mixing on electron pairs for intermediate magnetic fields by incorporating multiple Landau levels into the pseudopotentials on the Haldane sphere.

A relative coordinate breathing mode in the quantum Hall system is predicted to exist with different behavior under either Coulomb or dipole-dipole interactions. While Kohn's theorem [1] predicts that any relative coordinate interaction will fail to alter the center of mass energy spectrum, it can affect excitations in the relative coordinates. One such collective excitation, which we call the hyperradial breathing mode, emerges naturally from a few-body, hyperspherical representation of the problem and depends on the inter-particle interactions, the ground state wave function, and the number of particles participating in the excitation. Possible observations of this excitation will be discussed in the context of both cold rotating atomic simulations and conventional quantum Hall experiments.

Abstract.The quantum Hall effect arises from the quantum behavior of twodimensional, strongly-interacting electrons exposed to a strong, perpendicular magnetic field [1,2]. Conventionally treated from a many-body perspective, we instead treat the system from the few-body perspective using collective coordinates and the hyperspherical adiabatic technique developed originally for atomic systems [3]. The grand angular momentum K from K-harmonic few-body theory, is shown to be an approximate good collective quantum number in this system, and is shown to correlate with known fractional quantum Hall (FQH) states at experimentally observed filling factors.

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