Lindhard RPA Dielectric Function Electron Gas

[DrDu]

The longitudinal dielectric function of a gas of free electrons (+ homogeneous positive background) is often described in the Lindhard- or Random Phase Approximation (RPA).
The dielectric function depends on both frequency omega and wavevector k. However, it is non-analytic at the point omega=0, k=0. Namely its value depends on how the constant ratio of omega/k is chosen in the limit omega to 0. What is the physics behind this behaviour?

 

[Physics Monkey]

The singular structure comes from the existence of gapless excitations at the Fermi surface. Is this answer too brief/trivial for what you were looking for?

 

[DrDu]

Dear Physics Monkey,

Too brief yes, too trivial no. I was thinking the following: when taking the limit omega to 0 before k to 0 (static screening) the electrons have all the time of the world to adjust to the field. In the other limit ( k to 0 before omega to 0) they would have to move with too high velocity over too large a distance.
I also think that the latter limit changes drastically if scattering/band structure is to be included.

 

[Physics Monkey]

This is also the basic picture I have. In the case of k going to zero first, one knows a lot about the response of the free gas because the k = 0 current is basically the momentum. Even if you include electron-electron interactions the k = 0 current has a simple structure dictated by momentum conservation. This is another way to understand how the k = 0 finite omega result is special. Of course, this changes as you say once one introduces band structure or non-translation invariant scattering (like impurities) etc.