Flux State in von Neumann lattice and Fractional Hall Effect

TL;DR

This paper presents a new formulation of quantum Hall dynamics using von Neumann lattices, introduces a flux condensation mean field theory for the fractional Hall effect, and connects it to Hofstadter's spectrum, with results aligning with experiments.

Contribution

It introduces a novel mean field theory based on flux condensation for the fractional Hall effect using von Neumann lattices, linking it to Hofstadter's spectrum.

Findings

Derived a topological invariant expression for Hall conductance.

Proposed a flux condensation mean field theory for fractional Hall effect.

Computed energy gaps consistent with experimental data.

Abstract

Formulation of quantum Hall dynamics using von Neumann lattice of guiding center coordinates is presented. A topological invariant expression of the Hall conductance is given and a new mean field theory of the fractional Hall effect based on flux condensation is proposed. Because our mean field Hamiltonian has the same form as Hofstadter Hamiltonian, it is possible to understand characteristic features of the fractional Hall effect from Hofstadter's spectrum. Energy gap and other physical quantities are computed and are compared with the experiments. A reasonable agreement is obtained.