# Calculating the T^2 dependence of the elec-elec scattering rate in Fermi Liquids?

1. Jun 17, 2007

### Mute

Ahoy,

I'm trying to do a calculation of the electron-electron self energy of a Fermi Liquid, which is supposed to exhibit a dependence on the square of the temperature of the system.

I've read through the first six chapters of "Green's Functions for Solid State Physicists" by Doniach and Sondheimer to get some background on using some QFT methods for calculating T=0 Green's functions and temperature Green's functions, but the book doesn't really cover calculating many properties of systems that aren't in the ground state (T=0), nor does it seem to cover combining time and temperature Green's functions.

Accordingly, I've been looking at "Quantum Field Theoretical Methods in Statistical Physics" by Abrikosov, Gor'kov and Dzyaloshinskii, which does cover this material, but does so in such a way that I haven't yet made the connection between what I've learned from Doniach & Sondheimer - namely, D&S don't seem to use field operators and instead did everything directly in terms of the (time dependent) annihilation and creation operators (in either the Heisenberg or Interaction pictures). D&S also didn't seem to make use of these vertex functions, denoted in Abrikosov by as $\Gamma_{\alpha \beta \gamma \delta}(x_1x_2,x_3x_4)$, and represented by squares in the Feynman diagrams.

This book does given a calculation of $Im~\Sigma(\varepsilon)$, which can be calculated as the sum of the irreducible parts of the Feynman diagrams and is related to the scattering rate of the electron-electron interaction, but the section in which it calculates this is rather jumpy and refers to several previous sections of the text which I haven't read throughly/at all, and given the difference in presentation between this text and D&S, I have been unable to straightforwardly follow the derivation leading to the result

$$Im~\Sigma_R(\varepsilon) = -A(\pi^2T^2 + \varepsilon^2)$$,

which is what I'm after.

If necessary, I'm prepared to try and slog through the necessary sections of Abrikosov in more detail, but I was wondering if anyone knew of any references which introduce how to calculate the time and temperature dependent Green's Functions or go through the self-energy derivation in a style closer to that of the D&S book. (Or just any references for this derivation in general - it's always good to see more than one way to derive something, I figure).

Thanks for any suggestions!

--Mute

Last edited: Jun 17, 2007
2. Jun 18, 2007

### ZapperZ

Staff Emeritus
I don't have the text with me right now, but I am almost certain that Mahan's "Many-Particle Physics" covers exactly what you are looking for. You might also want to look at David Pines "The Many-Body Problem", which deals predominantly with the weak-coupling Fermi Liquid problem.

Zz.

3. Jun 27, 2007

### Mute

Thanks. I have my supervisor's copy of Mahan in my office. It approaches the problem from a Boltzmann Equation perspective, so in terms of deriving the self-energy using diagrammatics, it's not so useful, but it's good to keep in mind.

I briefly skimmed through the Many-Body Problem and didn't see anything of immediate usefulness (though I may have missed it). AGD seems to be not too difficult to follow now that I've gotten into it a bit more, so I'm going with it for the time being.