In a symmetric 1D system, the intrinsic topological orders are characterized by the total Zak phase ($\gamma$) summed over all occupied states$$\tag{1}\gamma=\sum_{n}^{}\gamma_n$$where $n$ is the band index. For that, one needs to find the Zak phase for each of the occupied states by integrating the Berry connection across the 1D Brillouin zone:$$\gamma_n=i\int_{BZ}^{} <u_{nk}|\frac{\partial u_{nk}}{\partial k}> dk\tag{2}$$ where $u_{nk}$ is the periodic part of the Bloch states of band $n$. If the system under consideration holds spatial symmetries such as inversion and/or mirror symmetry, then the total Zak phase could be quantized to $0$ or $\pi$ (mod 2$\pi$).

Analytically, the total Zak phase $\gamma$ can be calculated through another topological invariant ($\mathbb{Z_2}$), which is related to the wave function parities at time-reversal invariant points of the BZ. How am I supposed to obtain the parity eigenvalues $\xi(i)$ for the occupied bands?



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