Real-part of the self-energy from Kramers-Kronig relation

[attacc]

Dear EPW developpers

I have a question on the Electron-Phonon self-energy (for the electrons). With EPW it is very easy to get the imaginary part of the self-energy,
but I image that for the real one the story is more complicated because it requires a sum on many conduction bands.

I would like to know if you ever tried to get the real-part of the self-energy from the imaginary one using the Kramers-Kronig relation?
It works? or it is too unstable?

best
Claudio

[sponce]

Hello Claudio,

Happy to see you have an interest in EPW.

It depends what you want to do with the real part of the self-energy and the accuracy you want.

In EPW, we cannot compute a lot of bands (since we need to Wannierized them). As a result we do not compute the Debye-Waller term (pure real term).

In theory you could get the real part form the KK relation but this requires a freq. integration (therefore lots of bands).
We did try and yes it is quite unstable. We also tried some more clever tricks but still not very satisfactory at the moment.

What we sometimes do, is computing the real part of Fan and imposing Luttinger theorem (nb of particle conservation). This will not give you
access to something like the ZPR but should be relatively accurate at describing the change of bandstructure with k-points (so you can describe the different
renormalization between k-points but not the absolute value (ZPR).

If you want high accuracy ZPR, I would recommend Abinit over Yambo as Abinit uses Sternheimer to avoid bands summation (but you still have to converge on
a crazy number of k/q-points, which is the strength of EPW).

Hope this helps,
Best,
Samuel