Wannier functions of a topologically trivial group of bands are smoothly continuous to an atomic limit, are exponentially localized, and transform under a band representation.
In a topological material, the Wannier functions for the valence (group of) bands either fail to be exponentially localized or break the crystal symmetry. Some examples
- Chern Insulator: a non-vanishing Chern number indicates an obstruction to the formation of exponentially localized Wannier functions.
- Topological Insulators: In the Kane-Mele model of graphene, within the -odd topological phase, exponentially localized Wannier functions (when the valence and conduction bands are taken together, atomic like Wannier functions can be formed) for the valence bands break time-reversal symmetry. Hence a gap must close to transition to the atomic limit.
Fragile Obstructions
In some cases, a set of topologically obstructed bands may be trivialized by adding trivial bands to the manifold. The topology then becomes trivial as measured by the Wilson loop winding. Such a set of bands are said to have a “fragile” obstruction.