SPIN WAVES IN THE FERROMAGNETIC HEAVY RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 262

from H to be

hω

( 0 , 0)

J J

a q ; a q − a +

− q ; a q

− A − B

+ 2 J/N
k τ

(5 . 7 . 22)

c k − q − τ ↑ c k ↓ ; a q j ( − q − τ

=1 .

)

c k − q − τ ↑

; a q

The equation of motion of the new Green function c k ↓

involves the following commutator:
[ c k − q − τ ↑ ]= ε k ↓ − ε k − q − τ ↑ c k − q − τ ↑ c k ↓ , H c k ↓

+ 2 J/N
q

j ( q ) c k − q − τ + q ↓ c k − q ↑ a q c k − q − τ ↑

c k ↓ −

f k − q − τ ↑ a q
(5 . 7 . 23)
obtained by applying the anticommutator relations (5.7.3) and, in the
second equation, an RPA decoupling of the operator products. It is
not necessary here to differentiate between the new and the old Fermi
operators, as the differences introduce corrections only in the third order
of

ε k ↓ − ε k − q − τ ↑ c k − q − τ ↑ c k ↓ + 2 J/N j ( q + ) f k ↓ −

τ

. Introducing this RPA result in the equation of motion for the
Green function

| j ( q ) |

c k − q − τ ↑

c k ↓ ; a q ,weobtain

hω

+ ε k − q − τ ↑

c k − q − τ ↑ ; a q − ε k ↓ c k ↓

2 J/N j ( q +

) f k ↓ − f k − q − τ ↑

a q ; a q − τ =0 ,

(5 . 7 . 24)

which, in combination with (5.7.22), leads to
hω

( q ,ω ) J J ( 0 , 0)+ J J

a q ; a q − a +

− q ; a q

− A − B =1 , (5 . 7 . 25)

where

N
τ

2

f k ↓ − f k − q − τ ↑

2

J

( q ,ω ) = lim
ε→ 0 +

| j ( q + τ ) | .

hω + ihε − ε k ↓ + ε k − q − τ ↑

k

(5 . 7 . 26 a )
This result may be expressed in terms of the susceptibility of the con-
duction electrons. Introducing the spin susceptibility per ion, which is
the usual magnetic susceptibility times (2 µ B ) − 2 V/N ,sothat

d r 1 d r 2

1
N

χ + − s + ( r 1 ); s − ( r 2 ) e −i q · ( r 1 − r 2 )

c . el . ( q ,ω )=

−

N

1

c k − q ↑ ; c k + q ↓ = −

k k

c k ↓ c k ↑

(5 . 7 . 26 b )

N
k

f k ↓ − f k − q ↑

1

= lim
ε→ 0 +

,

hω + ihε − ε k ↓ + ε k − q ↑