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I am writing a code for the numerical evaluation of susceptibilities. The formalism is explicitly written on the Matsubara axis (fermionic case) and in the heart of the procedure lie multiple integrations of the form:

$$ M = \frac{1}{\beta}\sum_n \frac{F(i\omega_n)}{i\omega_n - z} $$

where z can be either complex or real constant. Since the $F$ function is known only numerically for a finite number of Matsubara frequencies, I wonder if it's possible to use some kind of analytical expression to deal with the numerical tails that are missing and converge the integration without expanding endlessly the axis.

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Does $F$ have any poles?

If not, the result of the sum would be $M=n_{f}(z) F(z)$, where $n_f$ is the Fermi distribution.

In general, the sum over Matsubara frequencies of a function $G$ is $$ \sum_{n} G(i\omega_n)= -\xi \beta \sum_{x\in\text{poles of G}} Res(n(x) G(x)) $$

with $n$ the Bose-Einstein of Fermi distribution, according to the matsubara frequencies you are summing over. There are a couple of technical requirements (no poles in the imaginary axis, asymptotic behaviour...) but it's a standard formula covered in most condesed matter textbooks.

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  • $\begingroup$ Thank you for the quick reply! The (hybridization) function F is written formally as $F(i\omega_n) = \sum\limits_k \frac{|V_k|^2}{i\omega_n - \epsilon_k}$ and note that I do not have explicit access to $V_k, e_k$. I just have the numerical data ( I could just do an polynomial expansion of course ) $\endgroup$
    – dthed
    Commented Oct 2, 2018 at 21:57
  • $\begingroup$ then you'll have to appy the second formula in my answer. You have an additional pole in $\epsilon_k $ with residue $n(\epsilon_k) V_k / (\epsilon_k - z)$. You'll have of course to sum over $k$ $\endgroup$
    – tbt
    Commented Oct 2, 2018 at 22:05
  • $\begingroup$ As I wrote in my previous comment, I don't have access to the parameters $V_k$ and $\epsilon_k$ and so I cannot explicitly use the otherwise very useful formula that you provided. $\endgroup$
    – dthed
    Commented Oct 2, 2018 at 22:10
  • $\begingroup$ I can't think of anything else that would be useful at the moment $\endgroup$
    – tbt
    Commented Oct 2, 2018 at 22:21
  • $\begingroup$ If you think anything please contact. Thanks for the help! $\endgroup$
    – dthed
    Commented Oct 2, 2018 at 22:24

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