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  Question about the retarded Green's function and the analytic continuation of the Matsubara Green's function

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This question is a little bit mathematical. It is about the relation between correlation function in the Matsubara frequency and the retarded correlation function in the real frequency. The following question is based on the contents in the book by Flensberg and Bruus

The retarded correlation function $G^R (\omega)$ can be obtained by taking the analytic contiuation of the correlation function $G(\omega)$ in the Matsubara frequency, i.e. $G^R (\omega) = G (i\omega_n \rightarrow \omega + i\eta)$ with $\omega_n$ being the Matsubara frequency, $\omega$ being the real frequency and $\eta \rightarrow 0$ being a infinitesimal positive number. This theorem can be proved by using the Lehmann representation.

However, I found that, while reading the textbook by Flensberg and Bruus, the example of the density-density correlation function seems to violated the above theorem.

The density-denisty correlation function of non-interacting Fermi gas $\chi$ in the imaginary time domain is given by \begin{equation} \chi (q, \tau )=-\frac{1}{V}\langle T_\tau \rho (q,\tau)\rho(-q,0) \rangle, \end{equation} with $\tau$ being the imaginary time. Here, $V$ is the system volume and $\rho (q) = \sum_{k,\sigma = \pm} c^\dagger_{k\sigma}c_{k+q,\sigma}$ is the particle density in the momentum space. While taking the analytic continuation of the Matsubara density-density correlation function, we obtain \begin{equation} \chi (q, i\omega_n \rightarrow \omega + i\eta )=-\frac{1}{V} \langle \rho _{q=0}\rangle \langle \rho_{q=0}\rangle+ \frac{1}{V} \sum_{k, \sigma} \frac{n_F (\epsilon_k)-n_F (\epsilon_{k+q})}{\omega + \epsilon_k-\epsilon_{k+q}+i\eta}. \end{equation}

On the other hand, the retarded density-density correlation $\chi^R$ is defined as \begin{equation} \chi^R (q, t-t')=-i\theta (t-t')\frac{1}{V} \langle [\rho(q,t), \,\,\rho (-q,t')] \rangle. \end{equation} with $t$ being the real time. By Wick's contraction, $\chi^R$ in the frequency space takes the form \begin{equation} \chi^R (q, \omega )=\frac{1}{V} \sum_{k, \sigma} \frac{n_F (\epsilon_k)-n_F (\epsilon_{k+q})}{\omega + \epsilon_k-\epsilon_{k+q}+i\eta}. \end{equation}.

As compared $\chi (q, i\omega_n \rightarrow \omega + i\eta )$ with $\chi^R (q, \omega )$, there is an extra term $-\frac{1}{V} \langle \rho _{q=0}\rangle \langle \rho_{q=0}\rangle$ in $\chi (q, i\omega_n \rightarrow \omega + i\eta )$.

Does it mean that the theorem relevant to the analytic continuation mentioned above is violated, although the extra term is harmless?

Thanks!

This post imported from StackExchange Physics at 2019-03-15 09:54 (UTC), posted by SE-user Chang
asked Mar 13 in Theoretical Physics by Chang (20 points) [ no revision ]

Is the discussions in any of the below three PDFs of interest?

Analytic Continuation with Padé Decomposition
http://cpl.iphy.ac.cn/EN/article/downloadArticleFile.do?attachType=PDF&id=70857

The Lindhard Function
http://psibox.dk/download/2010PolarizationFreeElectron.pdf

Density and current response functions in strongly disordered electron systems:
Diffusion, electrical conductivity and Einstein relation
https://arxiv.org/pdf/cond-mat/0203414.pdf

Note that all of the above three PDFs talk about Matsubara frequencies.

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