Bloch theorem and Brillouin torus
Periodic lattices let us label electronic states by crystal momentum.
$$\psi_{nk}(r)=e^{ik\cdot r}u_{nk}(r)$$
What this means
Periodic lattices let us label electronic states by crystal momentum.
Deep dive: mathematical and physical mechanism
Translation symmetry yields Bloch eigenstates. Momentum is periodic modulo reciprocal lattice vectors, so the Brillouin zone has toroidal identifications. The topology of occupied-band eigenspaces over this base space distinguishes band phases.
How it appears in quantum phenomena
Electronic and photonic crystals support momentum-space band structures.
Assumptions, caveats and limitations
Disorder may destroy crystal momentum while alternative real-space invariants remain possible.
Connect the ideas
For context, see the Topological Band Theory topic guide and the reading references.