Environmental Engineering Reference
In-Depth Information
The above susceptibilities do not correspond directly to physical observ-
ables but, for instance,
χ
xx
(
q
,ω
)(where
S
+
and
S
−
are both replaced
by
S
x
) does. It is straightforward to see (by symmetry or by direct
verification) that
χ
++
(
q
,ω
)=
χ
−−
(
q
,ω
)
≡
0, and hence
4
χ
+
−
(
q
,ω
)+
χ
−
+
(
q
,ω
)
.
The presence of two-site correlations influences the thermal average
χ
xx
(
q
,ω
)=
χ
yy
(
q
,ω
)=
1
S
z
. A determination of the correction to the MF result (3
.
4
.
5
b
)for
S
z
, leading to a
self-consistent RPA
result for the transverse suscepti-
bility, requires a relation between
S
z
and the susceptibility functions
deduced above. The spin commutator-relation, [
S
i
,S
i
]=2
S
z
δ
ii
,
turns out to be satisfied identically, and thus leads to no additional
conditions. Instead we consider the
Wortis expansion
1
2
S
S
i
1
8
S
2
(
S
S
i
S
i
−
2
)
(
S
i
)
2
(
S
i
)
2
=
S
−
−···
(3
.
4
.
12)
1
−
for which the matrix elements between the
p
lowest single-spin (or MF)
levels are correct, where
p
2
S
+ 1 is the number of terms in the expan-
sion. Using (3.4.11), we find from the fluctuation-dissipation theorem
(3.2.18):
≤
N
q
=
1
S
i
S
i
S
−
+
(
q
,t
=0)
∞
N
q
1
π
1
=
1
e
−βhω
χ
−
+
(
q
,ω
)
d
(
hω
)=2
S
z
Φ
,
1
−
−∞
(3
.
4
.
13
a
)
with
N
q
1
e
βE
q
−
Φ=
1
n
q
n
q
=
1
,
(3
.
4
.
13
b
)
;
1
where
n
q
is the population factor for bosons of energy
E
q
.If
S
=
2
,
then
S
z
is determined by the two first terms of (3.4.12), and
S
z
S
z
=
S
−
/S,
Φ
or
2
−
S
z
=
S
2
/
(
S
+Φ)
Φ+2Φ
2
−···
(
S
i
)
2
(
S
i
)
2
In general one may use a 'Hartree-Fock decoupling',
S
i
S
i
)
2
, of the higher-order terms in (3.4.13) in order to show that
2(
N
q
1
S
z
Φ+(2
S
+1)Φ
2
S
+1
=
S
−
−···
S
−
n
q
,
(3
.
4
.
14)
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