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order in 1
/Z
has been given by Stinchcombe (1973), see also Vaks
et al.
(1968). The zero-order result is obtained by the 'boson' approximation
S
(2)
]
p
. Asisapparentfrom(7
.
2
.
5
a
), this corresponds to the
replacement of the second and subsequent terms on the r.h.s. by the in-
finite series generated by
(2
p
)
[
S
g
(
iω
n
)
K
(
iω
n
)
G
(
iω
n
), leading to an equation
for the single-site Green function which is the equivalent to the
Dyson
equation
for bosons (or fermions), or to the CPA equation with
c
=1.
The
q
-dependent Green function may be obtained from the single-site
function by the same procedure as in the CPA case, eqns (5.6.10-17).
In this approximation, the final Green function is that given by the MF-
RPA, corresponding to Σ(
q
,iω
n
) = 0 in (7.2.6). This does not involve
any
q
-summation and may therefore be classified as the (1
/Z
)
0
-order re-
sult. In the
cumulant-expansion
, developed by Stinchcombe (1973) and
others, the difference
−
(2)
)
2
is included, to the next order in 1
/Z
,
as an additional
vertex
appearing in the interaction chain-diagrams of
G
(
q
,iω
n
), independently of the appearance of the
(4)
S
−
(
S
(2)
-vertices. A dif-
ferent approach, which is made possible by the isolation of the single-site
Green function in (7
.
2
.
5
a
), is to generalize this equation once more, so
that it becomes a Dyson equation, by replacing
g
(
iω
n
)with
G
(
iω
n
)in
the second term on the r.h.s. of (7
.
2
.
5
a
), retaining the correct coecient
in this term. The effective-medium equation (5.6.13), with
c
=1,is
valid to first order in 1
/Z
,sothat
S
G
(
iω
n
)
1+
G
(
iω
n
)
K
(
iω
n
)
G
(
q
,iω
n
)=
(7
.
2
.
7
a
)
J
αα
(
q
)
−
and, in combination with the Dyson equation for the single-site Green
function, this leads to a
q
-dependent Green function derived from
Σ(
q
,iω
n
)=Σ(
iω
n
)=
1
n
01
(
n
0
+
n
1
−
g
(
iω
n
)
K
(
iω
n
)
u
(
n, n
)
,
βg
(
iω
n
)
n
1
n
01
)
K
(
iω
n
)+
(7
.
2
.
7
b
)
where
K
(
iω
n
) is determined self-consistently, as in (5.6.17),
K
(
iω
n
)=
q
J
αα
(
q
)
G
(
q
,iω
n
)
G
(
q
,iω
n
)
.
(7
.
2
.
7
c
)
q
The result obtained in this way is close to that derived by Galili and
Zevin (1987) using a more elaborate renormalization procedure, but in
addition to the simplifications attained by utilizing the effective-medium
approximation, the procedure which we have adopted has allowed us to
achieve a fully self-consistent result. We note that, in the application
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