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

## TL;DR

This paper investigates how the quantum Hall plateau transition behavior changes from two-dimensional to one-dimensional regimes as the aspect ratio of the system increases, revealing a crossover characterized by localization and topological properties.

## Contribution

It maps the thin-torus quantum Hall system onto a disordered Thouless pump, providing a novel understanding of the dimensional crossover and topological features.

## Key findings

- In the thin-torus limit, the spectrum is Anderson-localized with many states having non-zero Chern number.
- The system exhibits a crossover from 2D to 1D behavior with anomalous scaling of Thouless conductance.
- The mapping to a disordered Thouless pump explains the persistence of topological states in the localized regime.

## Abstract

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.