Large-scale geometry obstructs localization

TL;DR

This paper investigates how large-scale geometric properties of spaces influence the delocalization of spectral subspaces in topological insulators, showing that certain spectral states cannot be localized regardless of symmetry or homogeneity.

Contribution

It provides a geometric explanation for delocalization phenomena in topological insulators, extending to disordered systems on general Riemannian manifolds without symmetry assumptions.

Findings

Spectral subspaces of topological insulators are inherently delocalized due to geometric factors.

Delocalization occurs even in disordered systems and without spatial symmetry.

The results apply broadly to Landau levels in quantum Hall systems on arbitrary manifolds.

Abstract

We explain the coarse geometric origin of the fact that certain spectral subspaces of topological insulator Hamiltonians are delocalized, in the sense that they cannot admit an orthonormal basis of localized wavefunctions, with respect to any uniformly discrete set of localization centers. This is a robust result requiring neither spatial homogeneity nor symmetries, and applies to Landau levels of disordered quantum Hall systems on general Riemannian manifolds.