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

TL;DR

This paper studies the integer quantum Hall transition within a fraction of a Landau level, revealing that the transition's characteristics are preserved despite the presence of delta-function impurities, with implications for understanding plateau transitions and interactions.

Contribution

It demonstrates that a subset of Landau level states with delta-function impurities retains the same quantum Hall transition properties as the clean system, with a reduced flux quantum count.

Findings

Total Chern number remains +1 regardless of impurity configuration.

Quantum Hall transition persists in the impurity-affected subspace, matching the clean system's transition.

Transition behavior is quantitatively the same with fewer flux quanta, indicating robustness of the transition.

Abstract

We investigate the quantum Hall problem in the lowest Landau level in two dimensions, in the presence of an arbitrary number of $\delta$-function potentials arranged in different geometric configurations. When the number of delta functions $N_\delta$ is smaller than the number of flux quanta through the system ($N_\phi$), there is a manifold of $(N_\phi-N_\delta)$ 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 $\delta$ 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 $\delta$-function impurities, but with a reduced number $N_\phi' \equiv N_\phi-N_\delta$ 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.