Zak phase and polarization
The Berry phase across a one-dimensional Brillouin zone is related to electric polarization.
$$\gamma_{\mathrm{Zak}}=i\int_{\mathrm{BZ}}\langle u_k|\partial_k u_k\rangle\,dk$$
What this means
The Berry phase across a one-dimensional Brillouin zone is related to electric polarization.
Deep dive: mathematical and physical mechanism
In one-dimensional band insulators, the Berry phase of occupied states contributes to polarization modulo a polarization quantum. Inversion or chiral symmetry can quantize the Zak phase to $0$ or $\pi$ in suitable gauges and conventions.
How it appears in quantum phenomena
Polarization and boundary charge in one-dimensional insulators.
Assumptions, caveats and limitations
The Zak phase is not universally quantized without symmetry; unit-cell and origin conventions matter.
Connect the ideas
For context, see the Topological Band Theory topic guide and the reading references.