Test of the hierarchical theory for the FQHE

TL;DR

This paper analytically tests the hierarchical theory of the fractional quantum Hall effect by calculating energies of daughter states from parent states, confirming the theory's predictions with simplified interactions.

Contribution

It provides an analytical validation of the hierarchical theory for FQHE, showing that low-energy daughter states match predictions using only two-particle interactions.

Findings

Ground state energy at 2/3 filling matches quasihole system predictions.

Effective Hamiltonian with up to two-particle interactions suffices.

Results extend to other hierarchical filling factors.

Abstract

In the hierarchical theory of the fractional quantum Hall effect, the low--energy behaviour of a daughter state in the next level of the hierarchy is described by an interacting system of quasiparticles of the parent state. Taking the filled lowest Landau level as the parent state, we examine analytically the quantitative consequences of this approach for electrons interacting via a pseudopotential interaction. It is shown that the ground state energy per particle in the daughter state at a filling factor $2/3$ is exactly equal to that of a system of quasiholes in the parent state with half filling, precisely as predicted by the hierarchical approach. This is achieved with {\em only up to two--particle interactions} in the effective Hamiltonian for the quasiholes. Their single particle energy and two--particle interaction are derived. The results are generalized to the other filling factors attainable from the filled Landau level.