In Fu and Kane's paper from 2006, the authors define the $\mathbb{Z}_2$ invariant for time-reversal invariant topological insulators as an obstruction to Stoke's theorem, $$ \begin{equation} \nu=\frac{1}{2\pi}\left[\oint_{\partial\text{HBZ}}\mathrm{d}\mathbf{k}\cdot\mathbf{A}(\mathbf{k})-\int_{\text{HBZ}}\mathrm{d}^2\mathbf{k}F(\mathbf{k})\right]\text{(mod 2)}, \end{equation} $$ where $\mathbf{A}(\mathbf{k})=\sum_{n,\sigma}\langle u_{n,\sigma}(\mathbf{k})|\mathrm{i}\nabla|u_{n,\sigma}(\mathbf{k})\rangle$ and $F=\nabla\times\mathbf{A}(\mathbf{k})$ (where $|u_{n,\sigma}(\mathbf{k})\rangle$s are the periodic Bloch functions). The integrals are evaluated over half of the Brillouin zone. This formula is not gauge invariant and the authors impose a time reversal constraint on the periodic Bloch functions, $$ \begin{equation} |u_{n,\uparrow}(-\mathbf{k})\rangle=\Theta|u_{n,\downarrow}(\mathbf{k})\rangle \end{equation} $$ $$ \begin{equation} |u_{n,\downarrow}(-\mathbf{k})\rangle=-\Theta|u_{n,\uparrow}(\mathbf{k})\rangle \end{equation}. $$ However, I do not understand how this fixes the gauge. I can always define the new Bloch states,
$$ \begin{equation} |u_{n,\uparrow}(\mathbf{k})\rangle'=\mathrm{e}^{\mathrm{i}\theta_{\mathbf{k}}}|u_{n,\uparrow}(\mathbf{k})\rangle \end{equation} $$ $$ \begin{equation} |u_{n,\downarrow}(\mathbf{k})\rangle'=\mathrm{e}^{-\mathrm{i}\theta_{-\mathbf{k}}}|u_{n,\downarrow}(\mathbf{k})\rangle \end{equation} $$ which satisfies the gauge constraint for any arbitrary choice of $\theta_{\mathbf{k}}$. The Berry connection gets modified to, $$ \begin{equation} \mathbf{A}(\mathbf{k})\rightarrow\mathbf{A}(\mathbf{k})+\nabla(\theta_{\mathbf{k}}-\theta_{-\mathbf{k}}). \end{equation} $$ If I choose a $\theta$ that has a singularity in only one-half of the Brillouin zone, it can change the value of $\nu$ by 1. My question is, do we need any additional constraints on the Bloch functions to define $\nu$ unambiguously modulo 2? Thank you!
