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where
β
0
H
1
(
τ
1
)
dτ
1
+
U
(
β,
0) = 1
−
···
β
0
···
β
1)
n
n
!
+
(
−
···
T
τ
H
1
(
τ
1
)
···H
1
(
τ
n
)
dτ
1
···
dτ
n
+
···
(7
.
2
.
2
b
)
0
which is suitable for a diagrammatic representation in which the denom-
inator in (7
.
2
.
2
a
) just eliminates all 'un-linked' diagrams. Furthermore,
it can be shown that the retarded Green function is the analytic contin-
uation of the
τ
-ordered function to the real axis in the complex
ω
-plane,
or
G
τ
BA
(
iω
n
→
χ
BA
(
ω
)=
−
lim
→
0
+
ω
+
i
)
,
(7
.
2
.
3)
and we shall therefore use the frequency arguments
iω
n
and
ω
to distin-
guish between respectively the
τ
-ordered and the retarded Green func-
tion.
Considering the simplest case of the Ising model, we wish to calcu-
late the Fourier transform of
G
(
ij, τ
)=
−
T
τ
J
iα
(
τ
)
J
jα
.Wetake
H
0
to
be the single-ion crystal-field Hamiltonian, and the perturbation
H
1
is
then the two-ion part. With this partition, the ensemble average
0
of
a product of operators belonging to different sites is just the product of
the averages of the operators, i.e.
=
j
.This
concentrates attention on the Green function for a single site
G
(
ii, iω
n
),
for which the perturbation expansion leads to a series corresponding to
that considered in the CPA calculation, eqn (5.6.9). The only differences
are that
K
(
i, ω
) is replaced by the
αα
-component
K
(
iω
n
) and, more sig-
nificantly, that the products (
c
i
χ
o
(
ω
))
p
=
c
i
(
χ
o
(
ω
))
p
are replaced by
the 2
p
th order
cumulant
averages or
semi-invariants
J
iα
J
jα
0
=
J
iα
0
J
jα
0
if
i
β
β
2
p
2
p
dτ
2
p
T
τ
J
iα
(
τ
l
)
0
exp
ihω
nl
τ
l
,
(7
.
2
.
4)
1
β
p
(2
p
)
=
S
dτ
1
···
0
0
l
=1
l
=1
with the conditions
l
ω
nl
=0and
ω
n
1
=
ω
n
. The lowest-order semi-
invariant is
g
(
iω
n
)=2
n
01
M
α
∆
/
∆
2
(
ihω
n
)
2
,whichisthe
(2)
=
S
−
−
Fourier transform of
g
(
ω
)=
χ
o
(
ω
). The calculation of the fourth- and higher-order cumulants is
more involved. It is accomplished basically by utilizing the invariance
of the trace (i.e. of the ensemble average) to a cyclic permutation of
the operators, as is discussed, for instance, by Yang and Wang (1974)
and Care and Tucker (1977). If the operators are proportional to Bose
operators this results in
Wick's theorem
, which here implies that
T
τ
J
iα
(
τ
)
J
iα
0
,and
−
g
(
iω
n
→
ω
)=
−
(2
p
)
Bose
=
S
S
(2)
p
. The determination of the cumulant averages is facilitated by
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