Calculating Electron-Electron Self Energy in Fermi Liquid Systems

In summary, the conversation is about calculating the electron-electron self energy of a Fermi Liquid. The person has read through "Green's Functions for Solid State Physicists" by Doniach and Sondheimer, but it doesn't cover the necessary material. They are now looking at "Quantum Field Theoretical Methods in Statistical Physics" by Abrikosov, Gor'kov and Dzyaloshinskii, but are having trouble following the derivation. They are wondering if there are any other references that may be helpful. Suggestions are made for Mahan's "Many-Particle Physics" and David Pines' "The Many-Body Problem".
  • #1
Mute
Homework Helper
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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 [itex]\Gamma_{\alpha \beta \gamma \delta}(x_1x_2,x_3x_4)[/itex], and represented by squares in the Feynman diagrams.

This book does given a calculation of [itex]Im~\Sigma(\varepsilon)[/itex], 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

[tex]Im~\Sigma_R(\varepsilon) = -A(\pi^2T^2 + \varepsilon^2)[/tex],

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
 
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  • #2
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
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.

Thanks for your suggestions!
 

1. What is electron-electron self energy in Fermi liquid systems?

Electron-electron self energy is a measure of the interaction between electrons in a Fermi liquid system. It takes into account the repulsion between electrons due to their negative charges, and is an important factor in understanding the behavior of electrons in a condensed matter system.

2. How is electron-electron self energy calculated?

The electron-electron self energy can be calculated using various theoretical methods, such as the Hartree-Fock approximation or the random phase approximation. These methods take into account the Coulomb interaction between electrons and the Pauli exclusion principle.

3. What is the significance of electron-electron self energy in Fermi liquid systems?

The electron-electron self energy plays a crucial role in determining the electronic properties of Fermi liquids, such as their electrical conductivity and specific heat. It also affects the behavior of electrons at low temperatures, known as many-body effects.

4. How does the electron-electron self energy change with temperature?

At low temperatures, the electron-electron self energy is a dominant factor in determining the behavior of electrons in a Fermi liquid system. As the temperature increases, thermal fluctuations become more significant and can lead to a decrease in the strength of the self energy.

5. Can the electron-electron self energy be experimentally measured?

Yes, there are experimental techniques such as photoemission spectroscopy and tunneling spectroscopy that can indirectly measure the electron-electron self energy in a Fermi liquid system. These methods rely on the detection of changes in the electronic properties of the system due to the presence of the self energy.

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