Integer quantum Hall transition in a fraction of a Landau level

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 δ functions Nδ is smaller than the number of flux quanta through the system (Nφ), there is a manifold of (Nφ-Nδ) 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) quantitatively the same as that for the lowest Landau level without δ-function impurities, but with a reduced number Nφ′≡Nφ-Nδ 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.