8.3 Solving the Dyson Equation
131
G(ω)
−1
= ω1 − − (ω)
(8.75)
In particular, such a procedure applies to the diagonal approximation in which the
non-diagonal elements of the self-energy part are neglected. As a consequence, there
is an individual equation for each diagonal element of the electron propagator:
G pp (ω) =
ω − p − pp (ω)
−1
(8.76)
Assuming that pp (ω) is of the form
pp (ω) = pp (∞) +
n
|m pn |
2
ω − n
(8.77)
the zero points of G pp (ω) are given by the solutions of the equation
ω − p − pp (∞) =
n
|m pn |
2
ω − n
(8.78)
As depicted in Fig. 8.11, the function on the left-hand side is a straight line with slope
1 crossing the ω-axis at p + pp (∞), while the right-hand side is a sum over simple
poles located at the positions n . Obviously, the straight line cuts the pole function
once between two successive pole positions, so that for each pair of successive poles
there is a single zero point of G pp (ω)
−1 . This amounts to a “graphical solution” of the
one-component Dyson equation in the diagonal approximation. Of course, the zero
points between successive pole positions can also be determined numerically, using,
e.g., a Newton–Raphson-type procedure. The residues (or pole strengths) of the poles
of G pp (ω) are determined by the slopes of the pole function at the intersection points.
Let ω 0 be a pole position resulting from the single-component Dyson equation for
G pp (ω) and P 0 denote the residue to be determined. Expanding G
−1
pp (ω) in a Taylor
series about ω 0 yields
G
−1
pp (ω) =
ω − p − pp (ω)
=
1
P 0
(ω − ω 0 ) + α(ω − ω 0 )
2
+ · · ·
(8.79)
where 1/P 0 is given by
1
P 0
= 1 −
pp (ω 0 )
(8.80)
The slope of the pole function is always negative so that the pole strength,
P 0 = (1 −
pp (ω 0 ))
−1
(8.81)
is a positive number smaller 1, P 0 ≤ 1.