Hyperspherical theory of the quantum Hall effect: the role of exceptional degeneracy

TL;DR

This paper presents a hyperspherical coordinate approach to understanding the quantum Hall effect, revealing how FQH states emerge from degeneracy patterns and are characterized by a collective quantum number.

Contribution

It introduces a novel hyperspherical framework that explains the emergence of FQH states and their energy gaps through degeneracy patterns and approximate separability.

Findings

FQH states emerge from degeneracy patterns of free-particle eigenfunctions

Coulomb interactions split degenerate states into observable quantized energy levels

Grand angular momentum correlates with experimentally observed FQH states

Abstract

By separating the Schr\"odinger equation for $N$ noninteracting spin-polarized fermions in two-dimensional 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 Schr\"odinger 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.