210
6 Magnetism and Conduction
number of determinants needed to describe the antiferromagnetic state also grows.
For N = 2, we have Φ AF = (|φ 1 φ 2 |−|φ 1 φ 2 |)/
√
2; for N = 3, the eigenfunction is a
sum of three determinants: Φ AF = (|2φ 1 φ 2 φ 3 |−|φ 1 φ 2 φ 3 |−|φ 1 φ 2 φ 3 |)/
√
6; and for
N = 4, we already need a linear combination of six determinants (see Eq. 1.51). It is
easy to imagine that when we consider a crystal with in principle an infinite number
of magnetic sites, the wave function cannot be written down anymore.
Intuitively one could consider the state with alternating α and β spins, as drawn
in Fig. 6.18, as a good representation of the ground state in an antiferromagnetic
lattice. However, it is quite easy to show that this so-called Neél state is not an
eigenfunction of the Heisenberg Hamiltonian and that its energy expectation value is
only an upper bound to the ground state energy. Using the definition of the Heisenberg
Hamiltonian given in Eq. 6.50 with j = i + 1 and applying periodic boundaries as
mentioned in Sect. 3.3, we calculate the energy expectation value of the Neél state
Φ 0 =| S 1 , −S 2 , S 3 ,...,S i , −S j ,...− S N |. The action of the different products of
spin operators on this function is
ˆ
S
+ (i) ˆ
S
− (j)Φ 0 = 0
ˆ
S
− (i) ˆ
S
+ (j)Φ 0 =|S 1 , −S 2 , S 3 ,...,S i − 1, −S j + 1,...,−S N |
(6.72)
ˆ
S z (i) ˆ
S z (j)Φ 0 = S
2 Φ
This shows that Φ 0 is not an eigenfunction of the Heisenberg Hamiltonian and that
the products of spin-up and spin-down operators give both zero contribution to the
energy expectation value, which becomes
E(Φ 0 ) =
1
2
NzS
2 J
(6.73)
where N is the number of sites and z is the number of nearest neighbours of each
magnetic center. To show that this is not the state with the lowest energy, we now
generate a new spin configuration with the same total M S value by applying the
ˆ
S + (k) ˆ
S − (l) + ˆ
S − (k) ˆ
S + (l) operator to the Neél state. States with different M S values
do not interact with Φ 0 and cannot lower the energy of Φ 0 .
Φ 1 = ( ˆ
S
+ (k) ˆ
S
− (l) + ˆ
S
− (k) ˆ
S
+ (l))|S 1 , −S 2 , S 3 ,...S i , −S j ,...− S N |
=|S 1 , −S 2 ,...,S k − 1, −S l + 1,...,−S N |
(6.74)
Again, we have a state that is not an eigenfunction of the Heisenberg Hamiltonian,
which is easily seen by applying the products of spin-up and spin-down operators.
The energy expectation value is
E(Φ 1 ) = J
1
2
zNS
2 − z + 1
(6.75)
6 Magnetism and Conduction
number of determinants needed to describe the antiferromagnetic state also grows.
For N = 2, we have Φ AF = (|φ 1 φ 2 |−|φ 1 φ 2 |)/
√
2; for N = 3, the eigenfunction is a
sum of three determinants: Φ AF = (|2φ 1 φ 2 φ 3 |−|φ 1 φ 2 φ 3 |−|φ 1 φ 2 φ 3 |)/
√
6; and for
N = 4, we already need a linear combination of six determinants (see Eq. 1.51). It is
easy to imagine that when we consider a crystal with in principle an infinite number
of magnetic sites, the wave function cannot be written down anymore.
Intuitively one could consider the state with alternating α and β spins, as drawn
in Fig. 6.18, as a good representation of the ground state in an antiferromagnetic
lattice. However, it is quite easy to show that this so-called Neél state is not an
eigenfunction of the Heisenberg Hamiltonian and that its energy expectation value is
only an upper bound to the ground state energy. Using the definition of the Heisenberg
Hamiltonian given in Eq. 6.50 with j = i + 1 and applying periodic boundaries as
mentioned in Sect. 3.3, we calculate the energy expectation value of the Neél state
Φ 0 =| S 1 , −S 2 , S 3 ,...,S i , −S j ,...− S N |. The action of the different products of
spin operators on this function is
ˆ
S
+ (i) ˆ
S
− (j)Φ 0 = 0
ˆ
S
− (i) ˆ
S
+ (j)Φ 0 =|S 1 , −S 2 , S 3 ,...,S i − 1, −S j + 1,...,−S N |
(6.72)
ˆ
S z (i) ˆ
S z (j)Φ 0 = S
2 Φ
This shows that Φ 0 is not an eigenfunction of the Heisenberg Hamiltonian and that
the products of spin-up and spin-down operators give both zero contribution to the
energy expectation value, which becomes
E(Φ 0 ) =
1
2
NzS
2 J
(6.73)
where N is the number of sites and z is the number of nearest neighbours of each
magnetic center. To show that this is not the state with the lowest energy, we now
generate a new spin configuration with the same total M S value by applying the
ˆ
S + (k) ˆ
S − (l) + ˆ
S − (k) ˆ
S + (l) operator to the Neél state. States with different M S values
do not interact with Φ 0 and cannot lower the energy of Φ 0 .
Φ 1 = ( ˆ
S
+ (k) ˆ
S
− (l) + ˆ
S
− (k) ˆ
S
+ (l))|S 1 , −S 2 , S 3 ,...S i , −S j ,...− S N |
=|S 1 , −S 2 ,...,S k − 1, −S l + 1,...,−S N |
(6.74)
Again, we have a state that is not an eigenfunction of the Heisenberg Hamiltonian,
which is easily seen by applying the products of spin-up and spin-down operators.
The energy expectation value is
E(Φ 1 ) = J
1
2
zNS
2 − z + 1
(6.75)