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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?

DanielSank
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ARB
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    Let me answer with a question. Is the sum commutative? – Dox Jul 18 '14 at 12:50
  • Hi Dox, that precisely my question. In Relativistic field theory normally the sum is carried first over k and then over temperature, in condensed matter all the papers in superconductivity that I am using carry first the sum over temperature and then over k. In fact I was wondering if there is not something like a canonical limit as in statistical mechanics the order of the limits. – ARB Jul 18 '14 at 17:46
  • I would recommend reading this question and answer. It's very related to the issue of switching order of sums. – DanielSank Oct 12 '15 at 21:12

2 Answers2

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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. $$\sum_{i\omega}\sum_k\frac{1}{((i\omega)^2-k^2)^2}.$$

Everett You
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  • Hi, there are cases where the "whole" summation converges but only for a specific order. The argument of convergence of the whole implies commutation is only valid for finite elements sums, when the summation is over infinite series this argument is not always valid. – ARB Oct 21 '15 at 15:01
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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.

pathintegral
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  • Hi, thanks for the reference. But do you have any idea of which the underlying reason of this difference? As Dox pointed out sums are commutative, but this is only valid when the number of terms in the sum is finite, there are plenty of examples when considering infinite sequence in a sum depending on how the sum is carried on the limit is different. Then once again which is the underlying reason to carry the sum over k before omega or the other way around? – ARB Sep 27 '15 at 17:43
  • OK I cheated a bit. A common way to estimate Matsubara summations in frequency is to replace it by integrals. However, consider a minimal example $T\sum_m^{-\Lambda} \int_{\Lambda}^{\Lambda} 1/(i\omega_m-k)^2 dk$. If one replaces Matsubara summation by integral from the beginning, one has an integral $\int_{-\Lambda}^{\Lambda}\int_{\Lambda}^{\Lambda} 1/(i\omega-k)^2 dk d\omega$. We know this is a dangerous integral due to its double singularity in the IR end. Deferent procedures can give you positive, negative, or zero result. – pathintegral Sep 28 '15 at 02:46
  • To regulate this, one should keep in mind from the very beginning the Matsubara frequency $\omega_m=(2m+1)\pi T$ never becomes zero. Once one does this then there is no dangerous singularity. So the correct procedure is to perform the momentum integral first, and at the very end, approximate the Matsubara sum by an integral. – pathintegral Sep 28 '15 at 02:54
  • But keep in mind the reference and the minimal example I gave you are just some examples I encountered before. There well may be other peculiarities in Green function summations. Just be careful -- whenever there is a strange singularity always try to regularize it:) – pathintegral Sep 28 '15 at 02:57
  • Could we say that the bottom-line is that we know in advance that these integral are FINITE therefore we must always regularize them to keep them finite? – ARB Sep 29 '15 at 18:27
  • In this case, I think so. In condensed matter physics one doesn't really have infinities. But on the other hand, there are anomalies in condensed matter just like high energy physics. – pathintegral Sep 30 '15 at 02:40
  • This sort of answer is waaaay better if you can include the important parts of the link in the answer itself. Links rot etc. etc. – DanielSank Oct 12 '15 at 21:08