# Good Wannier bases in Hilbert modules associated to topological insulators

## TL;DR

This paper investigates the existence of smooth, well-localized Wannier functions in spectral subspaces of operators relevant to topological insulators, linking their existence to $K$-theoretic invariants of associated Hilbert modules.

## Contribution

It establishes a general criterion connecting Wannier basis existence to the freeness of Hilbert modules over group $C^*$-algebras, highlighting the role of $K$-theory in topological insulators.

## Key findings

- Existence of Wannier bases is equivalent to Hilbert module freeness.
- $K$-theoretic invariants classify topological phases.
- Provides criteria for Wannier basis construction in complex systems.

## Abstract

For a large class of physically relevant operators on a manifold with discrete group action, we prove general results on the (non-)existence of a basis of smooth well-localised Wannier functions for their spectral subspaces. This turns out to be equivalent to the freeness of a certain Hilbert module over the group $C^*$-algebra canonically associated to the spectral subspace. This brings into play $K$-theoretic methods and justifies their importance as invariants of topological insulators in physics.