The formulate to calculate berry phase for Bloch wave functions is $$ \gamma = i \sum_{n\in occ}\int_{\mathcal{C}} dk \langle \psi_k^n|\partial_k|\psi_k^n\rangle, $$ where $|\psi_k^n\rangle$ is a Bloch wave function of $n$-th band, $\mathcal{C}$ is a closed loop in momentum space for integration and the summation is performed over all occupied bands.
Let us consider a one-dimensional system with only one occupied band $|\psi_k\rangle$ and $\mathcal{C}$ can only be the whole Brillouin Zone. We can Fourier transform $|\psi_k\rangle$ to have a Wannier function,
$$ |W(R)\rangle = \frac{1}{\sqrt{N}}\sum_{k} e^{-iR\cdot k}|\psi_k\rangle. $$
Therefore, the occupied band can be written as $$ |\psi_k\rangle = \frac{1}{\sqrt{N}}\sum_{R} e^{iR\cdot k}|W(R)\rangle. $$ Insert the above relation to the formulate of berry phase, we have $$ \gamma = -\frac{2 \pi}{Na}\sum_{R}R, $$ where $a$ is lattice constant of unit cell in one dimension.
It is confusing since the value of berry phase is very trivial and not related to the exact form of the Bloch wave function. However, I think the value should depend on the state of the system such as topological state or trivial state of SSH model.
So what is wrong or is there missing in the derivation above ?
