Environmental Engineering Reference
In-Depth Information
introduce the thermal expectation-values
in the Hamiltonian
which, after a simple rearrangement of terms, can be written
H
=
S
i
=
S
2
1
i
H
i
−
i
=
j
J
(
ij
)(
S
i
−
S
)
·
(
S
j
−
S
)
,
(3
.
4
.
4
a
)
with
+
2
J
S
i
J
S
z
S
z
2
,
H
i
=
−
(
0
)
(
0
)
(3
.
4
.
4
b
)
S
z
and
z
. In the mean-field approximation, discussed in the pre-
vious chapter, the dynamic correlation between spins on different sites is
neglected. This means that the second term in (3
.
4
.
4
a
) is disregarded,
reducing the original many-spin Hamiltonian to a sum of
N
indepen-
dent single-spin Hamiltonians (3
.
4
.
4
b
). In this approximation,
S
=
S
z
is
determined by the self-consistent equation
+
S
+
S
Me
βMJ
(
0
)
S
z
e
βMJ
(
0
)
S
z
S
z
=
(3
.
4
.
5
a
)
M
=
−S
M
=
−S
(the last term in (3
.
4
.
4
b
) does not influence the thermal average) which,
in the limit of low temperatures, is
S
z
e
−βSJ
(
0
)
.
S
−
(3
.
4
.
5
b
)
In order to incorporate the influence of two-site correlations, to
leading order, we consider the Green function
G
±
(
ii
,t
)=
S
i
(
t
);
S
i
.
(3
.
4
.
6)
Accordingto(3
.
3
.
14
a
), the variation in time of
G
±
(
ii
,t
) depends on
the operator
2
j
(
ij
)
−
+2
S
i
S
j
.
1
[
S
i
2
S
i
S
j
,
H
]=
−
J
The introduction of this commutator in the equation of motion (3
.
3
.
14
a
)
leads to a relation between the original Green function and a new, more
elaborate Green function. Through its equation of motion, this new
function may be expressed in terms of yet another. The power of the
exchange coupling in the Green functions which are generated in this
way is raised by one in each step, and this procedure leads to an infi-
nite hierarchy of coupled functions. An approximate solution may be
obtained by utilizing the condition that the expectation value of
S
i
is
close to its saturation value at low temperatures. Thus, in this limit,
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