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The high frequency of self-energy from Moments

As derived in PRB 73, 035120 (2006) in appendix, the Green's function moments are,

G=M0iω+M1(iω)2+M2(iω)3+

and can be computed from

Mk=(1)k+1[dkdτkG(0+)dkdτkG(0)]

This follows from integration by parts of G(iω)=β0G(τ)eiωτdτ.

We thus have

M0=G(O)G(0+)=δαβM1=[H,ψα]ψβψβ[H,ψα]={[H,ψα],ψβ}M2=[H,[H,ψα]]ψβ+ψβ[H,[H,ψα]]={[H,ψα],[ψβ,H]}M3={[H,[H,ψα]],[ψβ,H]}

where we took into account that H commutes with eβH and hence H can be moved from the first to the last place in the brackets.

First we compute

[H,ψα]=εαiψiUαijkψiψjψkVαkck[ψα,H]=ψiεiαUαijkψkψjψickVkα

where H is the general Anderson impurity Hamiltonian.

M1 is thus

M1αβ=εαβ+(UαijβUαiβj)nij

M2 becomes

M2αβ=εαiεiβ+VαkVkβ+(UαijkUαikj)εkβψiψj+εαk(UβjikUβjki)ψiψj+(UαijkUαikj)(UβlpkUβlkp)ψpψiψjψl+UαkjlUβkipψpψiψjψl+(UαijkUαikj)Uβljkψiψl

To calculate the moments of the self-energy, we need to use

G=1iωεΣSiω+Δ=1iω+M1(iω)2+M2(iω)3+=1iω+ε+Σ(iω)2+S+(ε+Σ)2+(Δ(iω))(iω)3+

We thus see

Σ=M1εS=M2(M1)2V2

Finally, we obtain

Σαβ()=(UαijβUαiβj)nij

and

Sαβ=(UαijkUαikj)(UβlpkUβlkp)(ψpψiψjψlψpψlψiψj)+UαkjlUβkipψpψiψjψl+(UαijkUαikj)Uβljkψiψl
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