Delocalized spectra of Landau operators on helical surfaces

TL;DR

This paper investigates the spectral properties of Landau operators on helical surfaces with asymptotically constant curvature, revealing the absence of spectral gaps above the lowest Landau level through delocalized index theory.

Contribution

It introduces a novel analysis of Landau operators on non-flat, helical surfaces using Roe algebra techniques, extending spectral theory in geometric quantum physics.

Findings

Landau levels remain isolated in the considered setting.

No spectral gaps exist above the lowest Landau level on helical surfaces.

Delocalized coarse indices can be assigned to Landau levels.

Abstract

On a flat surface, the Landau operator, or quantum Hall Hamiltonian, has spectrum a discrete set of infinitely degenerate Landau levels. We consider surfaces with asymptotically constant curvature away from a possibly non-compact submanifold, the helicoid being our main example. The Landau levels remain isolated, provided the spectrum is considered in an appropriate Hilbert module over the Roe algebra of the surface delocalized away from the submanifold. Delocalized coarse indices may then be assigned to them. As an application, we prove that Landau operators on helical surfaces have no spectral gaps above the lowest Landau level.