Dimensional crossover of the integer quantum Hall plateau transition and disordered topological pumping

We study the quantum Hall plateau transition on rectangular tori. As the aspect ratio of the torus is increased, the two-dimensional critical behavior, characterized by a subthermodynamic number of topological states in a vanishing energy window around a critical energy, changes drastically. In the thin-torus limit, the entire spectrum is Anderson-localized; however, an extensive number of states retain a Chern number $C\neq 0$. We resolve this apparent paradox by mapping the thin-torus quantum Hall system onto a disordered Thouless pump, where the Chern number corresponds to the winding number of an electron's path in real space during a pump cycle. We then characterize quantitatively the crossover between the one- and two-dimensional regimes for large but finite aspect ratio, where the average Thouless conductance also shows anomalous scaling.

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