Quasi-1D limit of the integer quantum Hall transition as a disordered Thouless pump

We study the quantum Hall plateau transition on rectangular tori. As the torus aspect ratio is increased, the 2D critical behavior (where a subextensive number of topological states exist 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 nonzero Chern number. This apparent paradox is resolved by mapping the thin-torus quantum Hall system onto a disordered Thouless pump. We show that the Chern number maps onto the winding number of an electron’s path in real space during a pump cycle, and that the electrons' paths become random walks of diverging length, giving rise to the proliferation of large and essentially random Chern numbers across the spectrum. Building on this thin-torus limit result, we characterize quantitatively the crossover between the 1D and 2D regimes for large but finite aspect ratio. Possible realizations of this physics in quantum simulation platforms (e.g. cold atoms, microwave cavity arrays) are discussed.

Reference: M. Ippoliti and R. N. Bhatt, preprint at arXiv:1905.13171.