# Connection between Matsubara frequencies and Landau Quasiparticle Interpretation

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In a zero-temperature Fermi liquid, I understand that Landau quasiparticles correspond to poles in the interacting retarded Green's function, with the quasiparticle weight given by the residue of said function:

$$Z=\left(1-\frac{\partial \Re \Sigma}{\partial \omega}\bigg|_{\omega=0}\right)^{-1}$$

where $\Sigma$ is the self-energy. However, this only is for zero temperature. At finite temperature, we get smearing, and Landau quasiparticles gain a lifetime $\tau$, where $1/\tau \sim T^2$. To incorporate finite temperature behavior into the Green's function approach, we have to perform a sum over all Matsubara frequencies when we take the Fourier transform of the finite-temperature Green's function $\mathcal{G}$:

$$\mathcal{G}(\tau)=\frac{1}{\beta}\sum_{\omega_n} e^{-i\omega_n \tau}\mathcal{G}(i\omega_n)$$ where $\omega_n=(2n+1)\pi/\beta$ for fermions and $\tau$ is imaginary time. From the question Matsubara frequencies as poles of distribution fucntion and other sources, I know that the Matsubara frequencies are the poles of the distribution function. Therefore, is there some connection between the quasiparticle concept and the Matsubara frequencies? That is, if quasiparticles are the poles of the interacting Green's function and $\omega_n$ are the poles of the distribution function, is there an explicit connection between the two quantities? Can I use the Matsubara frequencies themselves to describe the requirements for a finite-temperature Fermi liquid? Any explanation or references at the level of Abrikosov, Gorkov, & Dzyaloshinskii would be greatly appreciated.

This post imported from StackExchange Physics at 2019-08-21 22:46 (UTC), posted by SE-user Joshuah Heath

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