Energy Spectra for Fractional Quantum Hall States

TL;DR

This paper calculates the energy spectra of fractional quantum Hall states with various filling factors, revealing how Coulomb interactions and electron pair transitions influence the stability and confinement of Hall resistance at specific fractions.

Contribution

It provides a detailed calculation of energy spectra for FQHS with q<21, incorporating Coulomb energies and pair transition effects, which was not previously done.

Findings

Energy depends linearly on 1/nu for Coulomb energy.

Electron pair transitions are constrained by momentum conservation.

Energy spectrum explains Hall resistance confinement at fractional fillings.

Abstract

Fractional quantum Hall states (FQHS) with the filling factor nu = p/q of q < 21 are examined and their energies are calculated. The classical Coulomb energy is evaluated among many electrons; that energy is linearly dependent on 1/nu. The residual binding energies are also evaluated. The electron pair in nearest Landau-orbitals is more affected via Coulomb transition than an electron pair in non-nearest orbitals. Each nearest electron pair can transfer to some empty orbital pair, but it cannot transfer to the other empty orbital pair because of conservation of momentum. Counting the numbers of the allowed and forbidden transitions, the binding energies are evaluated for filling factors of 126 fraction numbers. Gathering the classical Coulomb energy and the pair transition energy, we obtain the spectrum of energy versus nu. This energy spectrum elucidates the precise confinement of Hall resistance at specific fractional filling factors.