Berry phase and topological bands

For an isolated nondegenerate Bloch band, choose normalized cell-periodic states $|u_n(\mathbf k)\rangle$ smoothly in a local region. Multiplying each state by a momentum-dependent phase changes no projector or band energy. The Berry connection does change: with the convention $\mathbf A_n=i\langle u_n|\nabla_k u_n\rangle$, the replacement $u_n\mapsto e^{i\chi}u_n$ sends $\mathbf A_n$ to $\mathbf A_n-\nabla_k\chi$. A connection is therefore not by itself an observable vector field independent of convention.

For a closed path with compatible endpoint conventions, the Berry phase is meaningful modulo $2\pi$. Its exponential is unchanged by allowed single-valued gauge transformations. Curvature, the curl of the connection locally, is gauge invariant for an isolated band. Across a whole Brillouin zone, one may need more than one smooth phase patch. A nonzero Chern number obstructs one global smooth periodic eigenvector; it does not obstruct the physical band projector from being well defined.

Consider a spinless chain with sublattices A and B, real positive intracell hopping $t_1$ and intercell hopping $t_2$, no on-site energy difference, and cell length one. In a fixed periodic orbital basis its Bloch Hamiltonian has off-diagonal entries $q^*(k)$ and $q(k)$, where $q(k)=t_1+t_2e^{ik}$. There is no identity term here. The band energies are plus and minus $|q|$, and the gap closes when $t_1=t_2$ at momentum $\pi$.

As k traverses the Brillouin zone once in the positive direction, q draws a circle of radius $t_2$ centered at $t_1$ in the complex plane. Its winding about the origin is one when $t_2>t_1$ and zero when $t_2<t_1$. Chiral symmetry follows because the Hamiltonian has only the two off-diagonal Pauli components. For the stated orbital and cell convention, the occupied-band Zak phase equals $\pi$ times this winding modulo $2\pi$. Changing the cell convention requires tracking the reference and termination as well.

Within this symmetry class, continuously changing the hopping ratio cannot change the winding without closing the gap. Adding a sublattice energy difference introduces a third Pauli component and breaks chiral symmetry. Then a gapped interpolation between the two dimerizations becomes possible, so the original winding protection no longer applies. A general geometric phase can still exist. Losing quantization does not mean all geometric effects have disappeared; it means the particular invariant needs different assumptions.

Boundary states require a specified termination and a comparison between the bulk and its exterior. An interface between distinct dimerizations in the chiral model can host a localized midgap state, but finite-size overlap and symmetry-breaking perturbations affect its energy. Likewise a Zak phase depends on an origin convention when translated into polarization. Predict observable charges or edge spectra using a consistent ionic reference and boundary model. Topological language is most useful when its mathematical object, protecting conditions, and proposed observable are all named.