A particle's Green's function $G(\omega)$ is analytic everywhere in the complex plane except along the real axis.
Does it mean that a bosonic $G(\omega)$ is possibly nonanalytic at the bosonic Matsubara frequency $i\omega_n=0$?
1 Answer
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Yes, it does. An explicit example: $G(\omega,k)=1/(\omega^2-c^2k^2)$, the Green's function of a phonon, is nonanalytic at the frequency $\omega=0$ or $\mathrm{i}\omega=0$ at the momentum point $k=0$.
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$\begingroup$ would it be accurate to say that any system without mass-gap has a singularity at $\omega=0$? $\endgroup$ Commented Feb 26, 2017 at 17:38
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$\begingroup$ @Everett, does this nonanalyticity have any implication? I believe analyticity is guaranteed at all other Matsubara points, both bosonic and fermionic. $\endgroup$– anonCommented Feb 26, 2017 at 20:43
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$\begingroup$ This non-analyticity simply means that the boson is gapless. $\endgroup$ Commented Feb 27, 2017 at 0:02