Environmental Engineering Reference
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are considered, the replacement of
H
1
in the density matrix
only gives rise to errors of higher-order in 1
/J
. Because
H
by
H
0
+
H
1
is
quadratic in the Bose operators, this replacement results in a decoupling
of the
H
0
+
H
2
-terms (according to
Wick's theorem
) which is equivalent to the
RPA decoupling utilized previously. Hence, when considering thermal
averages, we have to leading order in 1
/J
, for instance,
a
k
a
k
a
q
+
k
−
k
a
k
a
k
a
q
+
k
−
k
+
a
k
a
k
a
q
+
k
−
k
+
a
q
+
k
−
k
a
k
a
k
=
δ
k
,−
q
a
+
a
k
a
k
a
k
a
k
−
q
a
k
a
−
k
+
δ
k
,
q
a
q
+
δ
k
,
k
a
q
,
(5
.
2
.
29)
where the last line follows from the diagonality of
H
1
in reciprocal
space. We note that it is convenient here that the single-ion operators are
expressed as products of Bose operators which are well-ordered. When
this decoupling is introduced in (5.2.28), it reduces to
H
0
+
]=
A
q
(
T
)
a
q
+
B
q
(
T
)
a
+
[
a
q
,
H
−
q
,
(5
.
2
.
30)
where the effective, renormalized parameters are
A
q
(
T
)=
A
+4
JC
1
m
+6
JC
2
b
+
J
{J
(
0
)
−J
(
k
)
}
(1
−
m
)
N
k
1
+
J{J
(
k
)
−J
(
k
−
q
)
}m
k
(5
.
2
.
31
a
)
and
B
q
(
T
)=
B
1+
1
4
J
+2
JC
1
b
+6
JC
2
m
+12
JC
3
b
−
2
J
{J
(
0
)
−J
(
q
)
}
b
2
N
k
N
k
1
1
+
J
{J
(
0
)
−J
(
k
)
}
b
k
+
J
{J
(
k
)
−J
(
k
−
q
)
}
b
k
.
(5
.
2
.
31
b
)
a
k
a
k
m
k
and
b
k
are respectively the correlation functions (1
/J
)
and
a
k
a
+
(1
/J
)
,and
m
and
b
are the corresponding aver-
ages over
k
. Equation (5.2.30) implies that the operator [
a
q
,
−
k
=(1
/J
)
a
k
a
−
k
], in the
equations of motion of any Green function involving
a
q
, can be replaced
by the expression on the right-hand side. The same result is obtained if,
instead,
H
H
1
are replaced by
A
q
(
T
)and
B
q
(
T
) in (5.2.17). Consequently, the system behaves as if the Hamilto-
nian
H
2
is neglected, and
A
q
and
B
in
H
2
is replaced by
H
0
+
H
1
, which is similar to
H
1
except for the introduction of the effective, temperature-dependent pa-
rameters. The RPA decoupling (5.2.29) introduces errors in the Green
functions, but only in the third order of 1
/J
, and as it leads to an effec-
tive Hamiltonian which is quadratic in the Bose operators, it is a valid
procedure. This internal consistency of the theory to second order in
H
0
+
H
1
+
H
0
+
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