Holonomy and Wilson loops
Parallel transport around a closed path can rotate a quantum state or a degenerate subspace.
$$W(C)=\mathcal P\exp\!\left(i\oint_C\mathcal A\cdot dk\right)$$
What this means
Parallel transport around a closed path can rotate a quantum state or a degenerate subspace.
Deep dive: mathematical and physical mechanism
For a single band the Wilson loop reduces to an Abelian phase. For an $N$-band subspace, $\mathcal A_{mn}=i\langle u_m|\nabla u_n\rangle$ is matrix valued and path ordering is essential. Wilson-loop eigenphases reveal Wannier-center flow and topological structure.
How it appears in quantum phenomena
Wilson-loop spectra help diagnose time-reversal topological insulators.
Assumptions, caveats and limitations
Non-Abelian holonomy requires a properly isolated multiband subspace.
Connect the ideas
For context, see the Geometric Quantum Mechanics topic guide and the reading references.