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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 ( n ) K ( n ) G ( 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 ( n )with G ( 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 ( n )
1+ G ( n )
K ( 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( n )=
1
n 01
( n 0 + n 1
g ( n ) K ( n ) u ( n, n ) ,
βg ( n )
n
1
n 01 ) K ( n )+
(7 . 2 . 7 b )
where K ( n ) is determined self-consistently, as in (5.6.17),
K ( 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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