chern number as an obstruction to choose a smooth gauge

Question

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

Comments

It is also the obstruction to the existence of localized Wannier functions.

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

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@MengCheng, Really? But even if $a_R(x)$ is not localized, the equation in the original question still seems to provide a smooth gauge.

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

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No, that's wrong. You have to be more careful about the analytical properties. For example, if the bands touch, you can still formally have Wannier functions, but then they have power-law tails.

This post imported from StackExchange Physics at 2016-03-05 11:45 (UTC), posted by SE-user Meng Cheng

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@MengCheng, what happens when Wannier functions are not localized, does that $\sum_R$ blows up?

This post imported from StackExchange Physics at 2016-03-05 11:45 (UTC), posted by SE-user Chong Wang