8.1 Diagrammatic Approach to the Self-Energy
115
G(ω) =
G
0
(ω)
−1
− (ω)
−1
(8.11)
According to
G(ω) =
1 − G
0
(ω)(ω)
−1 G
0
(ω)
= G
0
(ω) + G
0
(ω)(ω)G
0
(ω) + . . .
(8.12)
the right-hand side can be expanded as a geometrical series in powers of (ω), which
is the energy representation analogue to Eq. (8.7).
The Dyson equation can be viewed as providing a formal definition of the selfenergy part (ω), e.g., in the form
(ω) = G
0
(ω)
−1
− G(ω)
−1
(8.13)
obtained from Eq. (8.10). Its usefulness, however, derives from the fact that there is
a direct diagrammatic approach to (ω). An approximation to (ω) (obtained for
example from a low-order diagrammatic expansion) will lead via the Dyson equation
to an approximation for the electron propagator in the form of an infinite, if only
partial, summation of terms in the original perturbation expansion. This may result
in a viable computational scheme.
1. Diagram Rules for the Self-energy Part
The diagram rules for the self-energy part can readily be obtained by obvious modifications of the original Feynman diagram rules for the electron propagator:
Employ the original Feynman rules (F1)–(F4) or the Abrikosov rules (A1)–(A4) for
the electron propagator G pq (t, t
) with the following modifications:
– consider only diagrams that cannot be separated into two fragments by cutting one
free fermion line;
– remove the two outer free fermion lines from the respective wiggly interaction line
(interaction dots), while keeping their one-particle indices, p, q, in the interaction
line (dot) expressions;
– the number of free fermion lines is 2n − 1 rather than 2n + 1. The overall phase
i
n of rule (F4), however, still applies, because the two i-factors associated with
the two outer free fermion lines persist.
The second-order Feynman diagram for pq (t, t
) shown in Fig. 8.6 derives from
diagram (A) in Fig. 6.5. Its analytical expression reads
(2,A)
pq (t, t
) =
r,u,v
V pruv V uvqr G
0
u (t, t
)G
0
v (t, t
)G
0
r (t
, t)
(8.14)
Note the absence of time integrations, since there are only (two) external vertices in
the second-order diagram. The second-order Abrikosov diagram for the self-energy,
comprising both second-order Feynman diagrams, is shown in Fig. 8.7. The diagram
can easily be translated into the following analytical expression:
115
G(ω) =
G
0
(ω)
−1
− (ω)
−1
(8.11)
According to
G(ω) =
1 − G
0
(ω)(ω)
−1 G
0
(ω)
= G
0
(ω) + G
0
(ω)(ω)G
0
(ω) + . . .
(8.12)
the right-hand side can be expanded as a geometrical series in powers of (ω), which
is the energy representation analogue to Eq. (8.7).
The Dyson equation can be viewed as providing a formal definition of the selfenergy part (ω), e.g., in the form
(ω) = G
0
(ω)
−1
− G(ω)
−1
(8.13)
obtained from Eq. (8.10). Its usefulness, however, derives from the fact that there is
a direct diagrammatic approach to (ω). An approximation to (ω) (obtained for
example from a low-order diagrammatic expansion) will lead via the Dyson equation
to an approximation for the electron propagator in the form of an infinite, if only
partial, summation of terms in the original perturbation expansion. This may result
in a viable computational scheme.
1. Diagram Rules for the Self-energy Part
The diagram rules for the self-energy part can readily be obtained by obvious modifications of the original Feynman diagram rules for the electron propagator:
Employ the original Feynman rules (F1)–(F4) or the Abrikosov rules (A1)–(A4) for
the electron propagator G pq (t, t
) with the following modifications:
– consider only diagrams that cannot be separated into two fragments by cutting one
free fermion line;
– remove the two outer free fermion lines from the respective wiggly interaction line
(interaction dots), while keeping their one-particle indices, p, q, in the interaction
line (dot) expressions;
– the number of free fermion lines is 2n − 1 rather than 2n + 1. The overall phase
i
n of rule (F4), however, still applies, because the two i-factors associated with
the two outer free fermion lines persist.
The second-order Feynman diagram for pq (t, t
) shown in Fig. 8.6 derives from
diagram (A) in Fig. 6.5. Its analytical expression reads
(2,A)
pq (t, t
) =
r,u,v
V pruv V uvqr G
0
u (t, t
)G
0
v (t, t
)G
0
r (t
, t)
(8.14)
Note the absence of time integrations, since there are only (two) external vertices in
the second-order diagram. The second-order Abrikosov diagram for the self-energy,
comprising both second-order Feynman diagrams, is shown in Fig. 8.7. The diagram
can easily be translated into the following analytical expression: