In condensed matter physics, I heard that if chern number of a band $n$ is non zero, it is impossible to choose a gauge such that $\psi_{nk}$ is smooth in the whole brillouin zone.

However, it is know that $\psi_{nk}$ can be written as:

$$\psi_{nk}(x)=\sum_R e^{ikR}a_R(x)$$

where $a_R(x)$ is wannier function.

It seems that the above equation gives a continuous $\psi_{nk}$ over $k$ and contradicts the fact such gauge does not exist in some cases.

So how to resolve such a contradiction?

This post imported from StackExchange Physics at 2016-03-05 11:44 (UTC), posted by SE-user Chong Wang