# Self-energy of a Fermi liquid

A weakly correlated many-electron system can be viewed in a first approximation as a Fermi liquid, meaning that it behaves similarly to a non-interacting electron gas with renormalized parameters.

In this respect, one can calculate the electronic self-energy $\Sigma_{\text{el}}(k, \omega)$ which real and imaginary parts give information about the low-lying excitations of the system (the so-called quasiparticles).

Given an arbitrary interaction between electrons, do we have a general criterion on what the electronic self-energy should be, or how it should behave, for the system to be accurately described by the Fermi liquid theory ? I've heard a lot of awnsers to this question such has "the self-energy must be a smooth varying function" or "one must be able to expand it in powers of $k$ and $\omega$" but I am somewhat not satisfied with these awnsers, and was wondering if there existed a more precise criterion.

Any concrete example with an existing model would be greatly appreciated.

The criterion for the Fermi liquid theory to be justified is that the imaginary part of the self-energy must be vanishingly small around the Fermi surface (both in the energy scale and in the momentum deviation). $$-2\Im\Sigma_\text{el}(k,\omega)\to 0\text{ as }k\to k_F\text{ and }\omega\to 0.$$
The broadening of the quasi-particle peak in the spectral function is determined by the imaginary part of the self-energy. So if the imaginary part is small, the quasi-particle peak is sharp, which means the quasi-particle is well-defined and has a relatively long life-time. Therefore we can focus on the effective theory of quasi-particles around the Fermi surface, which is the essential idea of the Landau Fermi liquid theory. The criterion that $-2\Im\Sigma_\text{el}$ is expected to approach zero progressively as we reduce the energy scale also implies that it can be expand as power series $-2\Im\Sigma_\text{el}(\omega)\sim\omega^2+\cdots$ at low frequency (first order in $\omega$ vanishes because it is an even function of $\omega$). Under Kramers-Kronig relations, the imaginary part determines the real part, so it can be inferred that the real part is also smooth around zero frequency. So the aspects you have mentioned are consistent with the statement that the self-energy correction is progressively small around the Fermi surface, such that the quasi-particles are well-defined.