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and a more appropriate form turns out to be
G
(
ω
)=
G
(0)
i
Γ
hω
+
i
Γ
=
G
(0)
Γ
2
Γ
2
(
hω
)
2
+2
ihω
Γ
Γ
2
+(
hω
)
2
2
2
−
.
(7
.
2
.
11
a
)
The real and imaginary parts of this retarded Green function are con-
nected by the Kramers-Kronig relation, and the expansion in powers of
Γ agrees with (7.2.10), when
Γ =
2
K
(0)
/β.
3
β
1
βK
(0)
2
−
6
G
(0) =
−
and
(7
.
2
.
11
b
)
(2
β/
3
N
)
q
J
2
(
q
), and hence
Γ is independent of
T
in this limit. The most important reason for
choosing the Green function given by (7
.
2
.
11
a
) is that it satisfies the
sum rule:
In the high-temperature limit,
K
(0)
∞
β
n
d
(
hω
)Im
G
(
ω
)
coth (
βhω/
2)
1
3
1
π
−
G
(
iω
n
)=
−
0
α
=
x,y,z
=
J
(
J
+1)=2
,
(7
.
2
.
12)
to the degree of accuracy with which G(0) is determined (this is the same
sum rule considered in (4.2.7)). The original expansion series satisfies
this sum rule, to first order in 1
/Z
, but this property is not easily con-
served if a Lorentzian is chosen. The problem with the Lorentzian (with
approximately the same Γ as above) is that it decreases only slowly with
ω
, and the tails lead to a divergence of the integral in (7.2.12), unless a
high-frequency cut-off is introduced. In this system, there is no natural
frequency-scale setting such a cut-off, and the only reasonable way of
determining it is through the sum-rule itself, which is rather unsatisfac-
tory.
In addition to the equations of motion and the Feynman-Dyson
linked-cluster-expansion method discussed here, there are other many-
body perturbation techniques which may be useful for analysing this
kind of system. The most important supplementary theories are those
based on the Mori technique (Mori 1965; Huber 1978; Ohnari 1980),
or similar projection-operator methods (Becker
et al.
1977; Micnas and
Kishore 1981). However, no matter which theory is used, it cannot
circumvent the essential complication of crystal-field systems; the more
single-ion levels which are important, the greater is the complexity of the
dynamical behaviour. This principle is illustrated by the fact that the
methods discussed above have not yet been extended to systems with
more than two levels, singlet or degenerate, per site.
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