8.2 Analytical Properties of the Self-Energy
119
8.2 Analytical Properties of the Self-Energy
The self-energy can be written in the general form
pq (ω) = pq (∞) + M pq (ω)
(8.18)
Here, M pq (ω) and pq (∞) are referred to as dynamical (ω-dependent) part and static
(ω-independent) part, respectively. The notation pq (∞) used for the static selfenergy part reflects the fact that M pq (ω) → 0 for ω → ∞. Analogous to Eq. (3.17)
for the electron propagator, there is a spectral representation for M pq (ω),
M pq (ω) = M
+
pq (ω) + M
−
pq (ω) =
ν
m
(ν)
p m
(ν)∗
q
ω − ω ν + iη
+
μ
m
(μ)
p m
(μ)∗
q
ω − ω μ − iη
(8.19)
The M
±
pq (ω) terms differ in the location of the poles, being in the lower and upper
complex ω-plane, respectively. Like in the case of the electron propagator, M
+
pq (ω)
and M
−
pq (ω) are referred to as (N +1)-particle (or affinity) and (N −1)-particle (or
ionisation) part, respectively. Other than in the spectral representation of the electron
propagator, the positions of the poles ω λ and the amplitudes m
(λ)
p cannot directly be
related to physical quantities.
A general proof of Eq. (8.18) and the spectral representation (8.19) has been
presented in Refs. [2, 3]. In principle, the analytical form of the self-energy part may
be inferred from the spectral properties of G(ω) via Eq. (8.13), which can be seen
as a definition of (ω). To get an idea of the algebra here at work, one may consider
a simple function modelled after the diagonal approximation to Eq. (8.13),
s(ω) = ω − − g(ω)
−1
(8.20)
where
g(ω) =
m
k=1
p k
ω − e k
,
p k = 1
(8.21)
serves as a surrogate for the electron propagator. As the reader may verify (see
Exercise 8.3), the function s(ω) can be written in the desired form,
s(ω) = x +
m−1
k=1
q k
ω − ω k
(8.22)
where ω 1 , . . . , ω m−1 are the m − 1 zeros of g(ω), and x is a constant.
Of course, a corresponding analysis for the (ω) and G(ω) matrices is more
demanding. We confine ourselves to the derivation of Eq. (8.18) and a specification
of the constant self-energy part, (∞). To this end, we write the spectral representation (3.17) of the electron propagator in the generalizing form
119
8.2 Analytical Properties of the Self-Energy
The self-energy can be written in the general form
pq (ω) = pq (∞) + M pq (ω)
(8.18)
Here, M pq (ω) and pq (∞) are referred to as dynamical (ω-dependent) part and static
(ω-independent) part, respectively. The notation pq (∞) used for the static selfenergy part reflects the fact that M pq (ω) → 0 for ω → ∞. Analogous to Eq. (3.17)
for the electron propagator, there is a spectral representation for M pq (ω),
M pq (ω) = M
+
pq (ω) + M
−
pq (ω) =
ν
m
(ν)
p m
(ν)∗
q
ω − ω ν + iη
+
μ
m
(μ)
p m
(μ)∗
q
ω − ω μ − iη
(8.19)
The M
±
pq (ω) terms differ in the location of the poles, being in the lower and upper
complex ω-plane, respectively. Like in the case of the electron propagator, M
+
pq (ω)
and M
−
pq (ω) are referred to as (N +1)-particle (or affinity) and (N −1)-particle (or
ionisation) part, respectively. Other than in the spectral representation of the electron
propagator, the positions of the poles ω λ and the amplitudes m
(λ)
p cannot directly be
related to physical quantities.
A general proof of Eq. (8.18) and the spectral representation (8.19) has been
presented in Refs. [2, 3]. In principle, the analytical form of the self-energy part may
be inferred from the spectral properties of G(ω) via Eq. (8.13), which can be seen
as a definition of (ω). To get an idea of the algebra here at work, one may consider
a simple function modelled after the diagonal approximation to Eq. (8.13),
s(ω) = ω − − g(ω)
−1
(8.20)
where
g(ω) =
m
k=1
p k
ω − e k
,
p k = 1
(8.21)
serves as a surrogate for the electron propagator. As the reader may verify (see
Exercise 8.3), the function s(ω) can be written in the desired form,
s(ω) = x +
m−1
k=1
q k
ω − ω k
(8.22)
where ω 1 , . . . , ω m−1 are the m − 1 zeros of g(ω), and x is a constant.
Of course, a corresponding analysis for the (ω) and G(ω) matrices is more
demanding. We confine ourselves to the derivation of Eq. (8.18) and a specification
of the constant self-energy part, (∞). To this end, we write the spectral representation (3.17) of the electron propagator in the generalizing form