I'm currently studying an article by Maslov, in particular the first section about higher corrections to Fermi-liquid behavior of interacting electron systems. Unfortunately, I've hit a snag when trying to understand an argument concerning the (retarded) self-energy ΣR(ε,k).
Maslov states that in a Fermi liquid, the real part and the imaginary part of the self-energy ΣR(ε,k) are given by
ReΣR(ε,k)=−Aε+Bξk+…
−ImΣR(ε,k)=C(ε2+π2T2)+…
(equations 2.4a and 2.4b). These equations seem reasonable: when plugged into the fermion propagator,
GR(ε,k)=1ε+iδ−ξk−ΣR(ε,k)
the real part slightly modifies the dispersion relation ε=ξk slightly and the imaginary part slightly broadens the peak. That's what I'd call a Fermi liquid: the bare electron peaks are smeared out a bit, but everything else stays as usual.
Now, Maslov goes on to derive higher-order corrections to the imaginary part of the self-energy, for instance of the form
ImΣR(ε)=Cε2+D|ε|3+….
First, I do not quite understand how to interpret this expansion.
How am I to understand the expansions in orders of ε? I suppose that ε is small, but in relation to what? The Fermi level seems to be given by ε=0.
Second, he states that this expansion is to be understood "on the mass-shell".
I take it that "on the mass shell" means to set ξk=ε? But what does the expansion mean, then? Maybe I am supposed to expand in orders of (ε−ξk)?
Now the question that is the most important to me. Maslov argues that the real part of the self-energy can be obtained via the Kramers-Kronig relation from the imaginary part of self-energy. My problem is that the corresponding integrals diverge.
How can
ReΣR(ε,k)=P1π∫∞−∞dωImΣR(ω,k)ω−ε
be understood for non-integrable functions like ImΣR(ε,k)=ε2?
It probably has to do with ε being small, but I don't really understand what is going on.
I should probably mention my motivation for these questions: I have calculated the imaginary part of the self-energy for the one-dimensional Luttinger liquid ξk=|k| as
ImΣR(ε,k)=(|ε|−|k|)θ(|ε|−|k|)sgn(ε)
and would like to make the connection to Maslov's interpretation and results. In particular, I want to calculate the imaginary part of the self-energy with the Kramers-Kronig relations.
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