Integer quantum Hall transition in a $\textit{fraction}$ of a Landau level

Abstract

We investigate the quantum Hall problem in the lowest Landau level in two dimensions, in the presence of an arbitrary number of $δ$-function potentials arranged in different geometric configurations. When the number of delta functions $N_δ$ is smaller than the number of flux quanta through the system ($N_ϕ$), there is a manifold of $(N_ϕ-N_δ)$ degenerate states at the original Landau level energy. We prove that the total Chern number of this set of states is +1 regardless of the number or position of the $δ$ functions. Furthermore, we find numerically that, upon the addition of disorder, this subspace includes a quantum Hall transition which is (in a well-defined sense) $\textit{quantitatively}$ the same as that for the lowest Landau level without $δ$-function impurities, but with a reduced number $N_ϕ' \equiv N_ϕ-N_δ$ of magnetic flux quanta. We discuss the implications of these results for studies of the integer plateau transitions, as well as for the many-body problem in the presence of electron-electron interactions.