SPIN WAVES IN THE FERROMAGNETIC HEAVY RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 258

with
I ( n k ,n k )= N d r 1 d r 2 ψ n k ( r 1 ) φ 4 f ( r 2 )

e 2
| r 1 − r 2 |

ψ n k ( r 2 ) φ 4 f ( r 1 ) ,
(5 . 7 . 5)
where N is the number of rare earth ions. If there are several 4 f electrons
per ion, I ( n k ,n k ) should again be averaged over their wavefunctions.
The Hamiltonian H sf , describing the exchange interaction between the

conduction electrons and the 4 f electrons, is then found to be

N
i

1

1) I ( n k ,n k ) e −i ( k − k ) · R i H sf = − ( g −

nn kk

× ( c n k ↑ c n k ↑ J i ,

(5 . 7 . 6)

c n k ↓ c n k ↓ ) J iz + c n k ↑ c n k ↓ J i + c n k ↓ c n k ↑ −

in second quantization.
In the ordered ferromagnetic phase, we may use the MF approxi-
mation, in which case

1) I ( n k ,n k )( c n k ↑ c n k ↓

H sf (MF) = − ( g − c n k ↑ − c n k ↓ ) J z .

nn k

(5 . 7 . 7)
This Hamiltonian gives rise to both diagonal and off-diagonal contribu-
tions to the energies of the conduction electrons. The diagonal energies
are

ε n k ↑ = ε n k − J z ( g − 1) I ( n k ,n k )

(5 . 7 . 8)

ε n k ↓ = ε n k + J z ( g − 1) I ( n k ,n k ) .

Second-order perturbation theory then gives the energies of the band
electrons as

1) 2
n = n

|I ( n k ,n k )

2

|

2 ( g

ε n k σ = ε n k σ + J z − . (5 . 7 . 9)

ε n k − ε n k

This dependence of the energies of the perturbed band-electrons on their
state of polarization implies that the electron gas itself develops a non-
zero magnetization. In order to calculate this moment, we first note that
(5.7.9) corresponds to a replacement of H s + H sf (MF) by an effective

Hamiltonian for the band electron,
H s =

c n k σ

n k σ

ε n k σ c n k σ , (5 . 7 . 10)

where the new Fermi operators are determined in terms of the old by

c n k ↑ +
n

U k ( n, n ) c n k ↑ = c n k ↑

(5 . 7 . 11 a )

U k ( n, n )

c n k ↓ = c n k ↓ − c n k ↓ ,

n