Landau Level Mixing and the Fractional Quantum Hall Effect

TL;DR

This paper derives effective Hamiltonians for the fractional quantum Hall effect in Landau levels n=0 and n=1, incorporating Landau level mixing effects perturbatively up to second order, with a focus on two- and three-body interactions.

Contribution

It provides a new derivation of Landau level mixing effects on the Hamiltonian, including previously omitted virtual processes, using a first quantization approach for n=0.

Findings

Effective Hamiltonians include two- and three-body interactions.

Landau level mixing effects are captured perturbatively up to second order.

The derivation clarifies virtual processes omitted in earlier models.

Abstract

We derive effective Hamiltonians for the fractional quantum Hall effect in n=0 and n=1 Landau levels that account perturbatively for Landau level mixing by electron-electron interactions. To second order in the ratio of electron-electron interaction to cyclotron energy, Landau level mixing is accounted for by constructing effective interaction Hamiltonians that include two-body and three-body contributions characterized by Haldane pseudopotentials. Our study builds upon previous treatments, using as a stepping stone the observation that the effective Hamiltonian is fully determined by the few-body problem with N=2 and N=3 electrons in the partially filled Landau level. For the n=0 case we use a first quantization approach to provide a compact and transparent derivation of the effective Hamiltonian which captures a class of virtual processes omitted in earlier derivations of Landau-level-mixing corrected Haldane pseudopotentials.