Wannier states and bulk-boundary correspondence in fragile topological insulators

Bloch bands with fragile topology present an obstruction to constructing exponentially localized Wannier states, much like strong topological insulators (TIs). However, unlike strong TIs, they lack the gapless edge states characteristic of those phases. This combination of features has made fragile topological phases particularly challenging to understand. In this talk, we employ the recently developed Spatial Localizer to resolve the bulk Wannier structure of fragile topological bands. We further exploit the algebraic structure of the topological classification of topological phases under crystalline symmetries to unveil their bulk–boundary correspondence. We demonstrate they have a higher-order topological nature, with fractionally quantized corner charges in two dimensions. To illustrate our results, we discuss two models: a four-band model on a square lattice and a faithful tight-binding model of the flat bands of twisted bilayer graphene.