• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,064 questions , 2,215 unanswered
5,347 answers , 22,728 comments
1,470 users with positive rep
818 active unimported users
More ...

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

+ 4 like - 0 dislike

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?


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

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights