# Is the Berry phase defined in terms of the periodic part of the Bloch wavefunction or the wavefunction itself?

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.