Z factor and imaginary part of self-energy

[chenhanghui]

Dear EPW developers,
I have two simple questions concerning the electron self-energy that arises from electron-phonon interaction.
I have run a test calculation and obtained something like:

ik = 7 coord.: 0.0000000 0.0000000 0.1000000
-------------------------------------------------------------------
E( 1 )= -1.0964 eV Re[Sigma]= -6.130738 meV Im[Sigma]= 0.512875 meV Z= 0.998537 lam= 0.001465
E( 2 )= -0.1835 eV Re[Sigma]= -4.352771 meV Im[Sigma]= 29.298553 meV Z= 1.025979 lam= -0.025321
E( 3 )= 2.0820 eV Re[Sigma]= 8.491139 meV Im[Sigma]= 15.193242 meV Z= 1.021277 lam= -0.020833

1) The first question is: why Im[Sigma] is positive? Since -2Im[Sigma] = electron linewidth, I would native think all the Im[Sigma] should be negative, so that the linewidth is positive. From the reference S. Ponce et al. Computer Physics Communications 209 (2016) 116-133, from Eq. (3) to Eq. (8) , is there a minus sign missing, since 1/(a+i\delta) = P(1/a) - i \pi \delta(a), where P is the Cauchy principal?

2) The Z factor can exceed 1 for some bands, so that the mass enhancement factor lambda_k is negative. Is this reasonable for electron-phonon interaction? In the context of electron-electron interaction, Z is always smaller than 1 (and non-negative) so that the effective mass is always larger than the bare mass.

Thank you very much for your help.

Hanghui Chen
Department of Physics
NYU Shanghai and New York University

[hlee]

Dear Hanghui Chen:

Previously I also noticed your first issue.
I didn't discuss about it with other members of EPW team, but after looking at the code and the papers, I thought that as you mentioned, Eq. (3) in CPC 209, 116 (2016) is not consistent with Eq. (8) in the same paper. I understood that the minus sign should be added in the right-hand side of Eq. (8) as you said or the left-hand side should be changed with the absolute value of imaginary part of the electron self-energy; correspondingly, the labels in the output files of EPW should be changed ( "Im[Sigma]" should be changed with "-Im[Sigma]" or "ABS(Im[Sigma])" ).

Regarding your second question, I didn't notice about it. Although I don't know about the details of your test run, considering the comparatively small deviation, I guess that something wrong happens in your calculation.

Sincerely,

H. Lee

[chenhanghui]

Dear Dr. Lee,
Thank you very much for your reply. I greatly appreciate that.
After checking the EPW source code more carefully, I find that there are two minus signs missing, which in fact cancel each other in the linewidth calculations. The first minus sign missing is in the Im[Sigma] (Eq. 8, CPC 209, 116 (2016)). That leads to a positive Im[Sigma]. However, when EPW calculates the linewidth, it uses \Gamma_k = 2 ImSigma, instead of \Gamma_k = -2 Im Sigma (Eq. (61), CPC 209, 116 (2016)). I think it might be a convention that EPW adopts.
As for the Z factor, I am now running more converged calculations to see whether Z>1 is a numerical artifact. However, any feedback from EPW developers or other experts will be greatly appreciated.
Thank you again for the useful discussion.
Best regards.

Hanghui Chen
Department of Physics
NYU Shanghai and New York University