Fiber bundles and quantum eigenstates
A quantum eigenstate can be defined locally while a single smooth global phase convention may fail to exist.
$$|u_n(k)\rangle\sim e^{i\phi(k)}|u_n(k)\rangle$$
What this means
A quantum eigenstate can be defined locally while a single smooth global phase convention may fail to exist.
Deep dive: mathematical and physical mechanism
At each momentum $k$, an isolated nondegenerate band determines a complex one-dimensional eigenspace. These eigenspaces form a complex line bundle over the Brillouin zone. Local gauges on overlapping patches differ by $U(1)$ transition functions. Nontrivial gluing is measured by the first Chern class.
How it appears in quantum phenomena
Bloch bands with nonzero Chern number cannot admit a globally smooth periodic eigenvector gauge.
Assumptions, caveats and limitations
An isolated band is needed; degenerate subspaces require a non-Abelian bundle description.
Connect the ideas
For context, see the Foundations of Topology topic guide and the reading references.