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chern number as an obstruction to choose a smooth gauge

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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
asked Mar 5, 2016 in Theoretical Physics by Chong Wang (10 points) [ no revision ]
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
@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
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
@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

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