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While computing the Chern number of electronic wave functions \begin{align} \left|\psi\right\rangle = \begin{pmatrix}\cos\left(\frac{\theta}{2}\right) \\ \sin\left(\frac{\theta}{2}\right)e^{i \phi} \end{pmatrix} \end{align} on the Bloch Sphere ($S^2$), it turns out that the wave function has singular dependence (vortex singularity) on $\phi$ at $\theta=\pi$. So, we need to choose a different $U(1)$ gauge for $\left|\psi\right\rangle$, so that the state is well defined at the south pole of the Bloch Sphere. A good choice of gauge is \begin{align} \left|\psi\right\rangle’ = e^{-i\phi}\left|\psi\right\rangle. \end{align} This $\left|\psi\right\rangle’$ is well defined at the south pole. So a single $U(1)$ gauge for the wave function is not going to work, here we needed two gauges which are valid on different patches of the Bloch Sphere. This nature of the gauge is what gives rise to the nonzero Chern number.

Now, in this example, there are two vortex singularities on each pole. I was trying to find any examples where there are more singularities on the state manifold.

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