I am working on the Ginzburg-Landau model for Charge density waves, and I am carrying out the sum of Green's functions to calculate the terms in the GL model. Is the sum's order over $ \vec{k} $ (or eventually $ \vec{r} $) and $\omega_n$ important? Mathematically the question is the following,
$$ \sum_{\vec{k}} \sum_{\omega_n} \stackrel{?}{=} \sum_{\omega_n} \sum_{\vec{k}} \, . $$
If it is not, when does it happen or under which conditions there is a difference?
If the whole summation converges, then the two summations commute.
For example, the following summation diverges, so the two summations do not commute, $$\sum_{i\omega}\sum_k\frac{1}{(i\omega-k)^2}.$$ However if we consider the following convergent summations, you can change the summation order freely. $$\sum_{i\omega}\sum_k\frac{1}{((i\omega)^2-k^2)^2}.$$
Yes, sometimes opposite limits give opposite signs.
Check Eq. (B5) of
http://journals.aps.org/prb/abstract/10.1103/PhysRevB.90.035149
if you integrate over frequency first -- it gives the wrong result.
