In this paper, the berry phase is approximated to be $e^{-i\theta} = \prod_{i=1}^{N} \langle\psi_{n,k_i} | \psi_{n,k_{i+1}} \rangle$. The authors claim that "each Bloch wavefunction appears twice in the above product." My problem is with the first and last inner products, $\psi_{n,1}$ and $\psi_{n,N+1}$. Are these wavefunctions equal? If they are the periodic part of the Bloch wavefunctions, like the preceding text in the paper suggest, then they are not the same function: they will be off by $e^{2 \pi i \mathbf{\hat k} \cdot \mathbf{r}}$ in real space, where $\mathbf{\hat k}$ is the unit vector in the $\mathbf{k}$ direction.


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