106
7 Time-Ordered or Goldstone Diagrams
Here, any G
0 -line crossing the cut contributes an orbital energy: with positive
sign for hole lines (occupied orbitals), +ε k , +ε l , . . . ; with negative sign for
particle lines (unoccupied orbitals), −ε a , −ε b , . . . . The energy variable ω
arises, if the auxiliary line (ω-line) between the outer vertices (t → t
) crosses
the cut. Here, the sign σ = 1 applies, if the ω-line is directed downwards, and
σ = −1, if the ω-line is directed upwards. Put σ = 0 if the ω-line does not
cross the cut; note that in that case the infinitesimal iη can be dropped.
(G4) Each hole line contributes a factor (−1). This leads to a total factor of (−1)
L+M ,
where L is the number of closed loops and M is the number of hole lines. The
various i-factors total +1:
−i
from the definition of the GF
(−i)
n
from the nth-order perturbation theory
i
n+1
from the n + 1 cuts
⎫
⎪ ⎬
⎪ ⎭
+ 1
Two important consequences of the Goldstone diagram analysis should be noted:
1. As confirmed by the above rules, the time integrations arising in a linked
Feynman diagram can always be performed and lead to well-defined expressions.
This means that the adiabatic switching functions e
−|t ν | originally accompanying the interaction terms are no longer needed to ensure convergence of the time
integrations. This justifies the a priori limit → 0 supposed in Sect. 5.3.
2. The time-ordered diagrams associated with a given Feynman (or Abrikosov)
diagram can be divided into two classes, I, II, according to the ordering of the
external vertices t and t
. In the diagrams of class I (t > t
), any ω-dependent
denominators are of the type (ω · · · + iη). Obviously, the diagrams of class I are
analytic in the upper complex ω-plane and, thus, contribute exclusively to the
G
+ part of the electron propagator. Conversely, the diagrams of class II (t < t
)
are analytic in the lower complex plane and contribute exclusively to G
− . In this
way, the Goldstone diagrams establish independent diagrammatic PT expansions
of the G
+ and G
− parts.
The second-order Abrikosov diagram (Fig. 6.7) gives rise to 4! = 24 time-ordered
diagrams, as there are altogether four vertices, that is, each two outer and inner ones.
Figure 7.3 shows the 12 diagrams of class I (t > t
); the 12 diagrams of class II
(t
> t) are obtained by turning the class I diagrams upside down.
For a demonstration of the use of the Goldstone rules, we evaluate the second
diagram in Fig. 7.3:
7 Time-Ordered or Goldstone Diagrams
Here, any G
0 -line crossing the cut contributes an orbital energy: with positive
sign for hole lines (occupied orbitals), +ε k , +ε l , . . . ; with negative sign for
particle lines (unoccupied orbitals), −ε a , −ε b , . . . . The energy variable ω
arises, if the auxiliary line (ω-line) between the outer vertices (t → t
) crosses
the cut. Here, the sign σ = 1 applies, if the ω-line is directed downwards, and
σ = −1, if the ω-line is directed upwards. Put σ = 0 if the ω-line does not
cross the cut; note that in that case the infinitesimal iη can be dropped.
(G4) Each hole line contributes a factor (−1). This leads to a total factor of (−1)
L+M ,
where L is the number of closed loops and M is the number of hole lines. The
various i-factors total +1:
−i
from the definition of the GF
(−i)
n
from the nth-order perturbation theory
i
n+1
from the n + 1 cuts
⎫
⎪ ⎬
⎪ ⎭
+ 1
Two important consequences of the Goldstone diagram analysis should be noted:
1. As confirmed by the above rules, the time integrations arising in a linked
Feynman diagram can always be performed and lead to well-defined expressions.
This means that the adiabatic switching functions e
−|t ν | originally accompanying the interaction terms are no longer needed to ensure convergence of the time
integrations. This justifies the a priori limit → 0 supposed in Sect. 5.3.
2. The time-ordered diagrams associated with a given Feynman (or Abrikosov)
diagram can be divided into two classes, I, II, according to the ordering of the
external vertices t and t
. In the diagrams of class I (t > t
), any ω-dependent
denominators are of the type (ω · · · + iη). Obviously, the diagrams of class I are
analytic in the upper complex ω-plane and, thus, contribute exclusively to the
G
+ part of the electron propagator. Conversely, the diagrams of class II (t < t
)
are analytic in the lower complex plane and contribute exclusively to G
− . In this
way, the Goldstone diagrams establish independent diagrammatic PT expansions
of the G
+ and G
− parts.
The second-order Abrikosov diagram (Fig. 6.7) gives rise to 4! = 24 time-ordered
diagrams, as there are altogether four vertices, that is, each two outer and inner ones.
Figure 7.3 shows the 12 diagrams of class I (t > t
); the 12 diagrams of class II
(t
> t) are obtained by turning the class I diagrams upside down.
For a demonstration of the use of the Goldstone rules, we evaluate the second
diagram in Fig. 7.3: