2 Concepts in Magnetism
47
where the minus signs appear because of the exchange symmetry. The eigenvalues
are 0, U and (U/2) ±
(U/2) 2 + 2t 2 , so in the limit that t/U 1 the last pair
of eigenvalues are U + 2t
2
/U + O(t
4
/U
3
) and −2t
2
/U + O(t
4
/U
3
). Thus, the
ground state has energy −2t
2
/U . If we try the same problem again with two electrons
with the same spin then they cannot sit on the same site because of the Pauli exclusion
principle. Thus the only state possible is | ↑, ↑↑ and this has energy E = 0. Thus,
there is an energy saving in having the two electrons with opposite spin because you
can go lower than E = 0 and have E = −2t
2
/U + O(t
4
/U
3
). This means that the
exchange interaction has a magnitude J ≈ 2t
2
/U . The moral of the story is that by
having the possibility to mix in the higher energy states in which two spins sit on the
same site (costing U ), it is possible to lower the overall energy. The antiferromagnetic
arrangement allows this process to happen; the ferromagnetic arrangement forbids it.
Superexchange can be considered in more detail [10] and can in certain circumstances
be ferromagnetic. The size and sign of the superexchange interaction is codified in
the Goodenough–Kanamori–Anderson rules [11–14].
2.3 Consequences of the Heisenberg Exchange Interaction
We have seen that at the heart of the exchange interaction is a term ˆ
S a · ˆ
S b , a simple
scalar product between two spin operators. If that scalar product is expanded, we
have
ˆ
S a · ˆ
S b = ˆ
S
x
a
ˆ
S
x
b + ˆ
S
y
a
ˆ
S
y
b + ˆ
S
z
a
ˆ
S
z
b = ˆ
S
z
a
ˆ
S
z
b +
1
2
( ˆ
S
+
a
ˆ
S
−
b + ˆ
S
−
a
ˆ
S
+
b ) ,
(2.22)
where the raising and lowering operators ˆ
S
+ and ˆ
S
− are defined by
ˆ
S
+
= ˆ
S
x
+ i ˆ
S
y
ˆ
S
−
= ˆ
S
x
− i ˆ
S
y
.
(2.23)
Although the term ˆ
S
z
a
ˆ
S
z
b in (2.22) seems to be simple enough to handle, the term
1
2
( ˆ
S
+
a
ˆ
S
−
b + ˆ
S
−
a
ˆ
S
+
b ) will give rise to flip-flop processes in which simultaneously an
up-spin labelled a is lowered and a down-spin labelled b is raised, or vice versa. This
part of the interaction has profound effects.
2.3.1 Two Interacting Spin1
2 Particles
In this section, we will consider two spin1
2
particles coupled by a scalar interaction
described by a Hamiltonian ˆ
H = A ˆ
S a · ˆ
S b , where ˆ
S a and ˆ
S b are the operators for the
spins for the two particles. We can also write the total spin operator ˆ
S
tot = ˆ
S a + ˆ
S b
so that
47
where the minus signs appear because of the exchange symmetry. The eigenvalues
are 0, U and (U/2) ±
(U/2) 2 + 2t 2 , so in the limit that t/U 1 the last pair
of eigenvalues are U + 2t
2
/U + O(t
4
/U
3
) and −2t
2
/U + O(t
4
/U
3
). Thus, the
ground state has energy −2t
2
/U . If we try the same problem again with two electrons
with the same spin then they cannot sit on the same site because of the Pauli exclusion
principle. Thus the only state possible is | ↑, ↑↑ and this has energy E = 0. Thus,
there is an energy saving in having the two electrons with opposite spin because you
can go lower than E = 0 and have E = −2t
2
/U + O(t
4
/U
3
). This means that the
exchange interaction has a magnitude J ≈ 2t
2
/U . The moral of the story is that by
having the possibility to mix in the higher energy states in which two spins sit on the
same site (costing U ), it is possible to lower the overall energy. The antiferromagnetic
arrangement allows this process to happen; the ferromagnetic arrangement forbids it.
Superexchange can be considered in more detail [10] and can in certain circumstances
be ferromagnetic. The size and sign of the superexchange interaction is codified in
the Goodenough–Kanamori–Anderson rules [11–14].
2.3 Consequences of the Heisenberg Exchange Interaction
We have seen that at the heart of the exchange interaction is a term ˆ
S a · ˆ
S b , a simple
scalar product between two spin operators. If that scalar product is expanded, we
have
ˆ
S a · ˆ
S b = ˆ
S
x
a
ˆ
S
x
b + ˆ
S
y
a
ˆ
S
y
b + ˆ
S
z
a
ˆ
S
z
b = ˆ
S
z
a
ˆ
S
z
b +
1
2
( ˆ
S
+
a
ˆ
S
−
b + ˆ
S
−
a
ˆ
S
+
b ) ,
(2.22)
where the raising and lowering operators ˆ
S
+ and ˆ
S
− are defined by
ˆ
S
+
= ˆ
S
x
+ i ˆ
S
y
ˆ
S
−
= ˆ
S
x
− i ˆ
S
y
.
(2.23)
Although the term ˆ
S
z
a
ˆ
S
z
b in (2.22) seems to be simple enough to handle, the term
1
2
( ˆ
S
+
a
ˆ
S
−
b + ˆ
S
−
a
ˆ
S
+
b ) will give rise to flip-flop processes in which simultaneously an
up-spin labelled a is lowered and a down-spin labelled b is raised, or vice versa. This
part of the interaction has profound effects.
2.3.1 Two Interacting Spin1
2 Particles
In this section, we will consider two spin1
2
particles coupled by a scalar interaction
described by a Hamiltonian ˆ
H = A ˆ
S a · ˆ
S b , where ˆ
S a and ˆ
S b are the operators for the
spins for the two particles. We can also write the total spin operator ˆ
S
tot = ˆ
S a + ˆ
S b
so that