TKNN quantization and Hall conductance
Hall conductivity can be locked to integers because it is the integral of band geometry.
$$\sigma_{xy}=\frac{e^2}{h}\sum_{n\in\mathrm{occ}}C_n$$
What this means
Hall conductivity can be locked to integers because it is the integral of band geometry.
Deep dive: mathematical and physical mechanism
Linear-response theory relates the transverse current to Berry curvature of filled bands. Integrating over the Brillouin zone gives $e^2/h$ times the total Chern number. This explains integer quantization in clean gapped noninteracting 2D band models.
How it appears in quantum phenomena
Integer quantum Hall plateaus and Chern insulators.
Assumptions, caveats and limitations
Real Hall plateaus additionally rely on disorder/localization and appropriate transport conditions.
Connect the ideas
For context, see the Topological Band Theory topic guide and the reading references.