Landau level mixing and the fractional quantum Hall effect

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.