LINEAR RESPONSE THEORY - Rare Earth Magnetism: Structures and Excitations - page 160

of the Green functions, or a linear combination of them, would lead to
an accurate determination of (the most natural choice would be to

use G ν 0 , 0 ν ( q ,ω ) ). However, a stringent justification of a specific choice
would require an analysis of the errors introduced by the RPA decou-
pling. We conclude that a reliable improvement of the theory can only
be obtained by a more accurate treatment of the higher-order Green
functions than that provided by the RPA. General programs for ac-
complishing this have been developed, but they have only been carried
through in the simplest cases, and we reserve the discussion of these
analyses to subsequent sections, where a number of specific systems are
considered.

a νν

3.5.2 MF-RPA theory of the Heisenberg ferromagnet
We conclude this chapter by applying the RPA to the Heisenberg model,
thereby demonstrating the relation between (3.5.8) and the results pre-
sented in the previous section. In order to do this, we must calculate
χ o ( ω ). The eigenstates of the MF Hamiltonian (3 . 4 . 4 b )are S z = M> , |

with M = − S, − S +1 , ··· ,S , and we neglect the constant contribution

to the eigenvalues

S z S z

E M = − M J ( 0 ) 0 = − M ∆ ith∆= J ( 0 ) 0 ,

denoting the MF expectation-value (3 . 4 . 5 a )of S z by S z

0 . According
to (3 . 3 . 4 a ), we then have (only terms with α = M +1 and α = M
contribute):

( ω )= S− 1

<M +1 | S + | M><M| S − | M +1 >
E M − E M +1 − hω

χ + − ( n M +1 − n M )

M = −S

e β ( M +1)∆

e βM ∆

S− 1

1
Z

S ( S +1) − M ( M +1)
∆ − hω

= −

−S

Z S
−S +1

S ( S +1) − ( M − 1) M e βM ∆

1
∆ − hω

1

=

S ( S +1) − M ( M +1) e βM ∆

S− 1

−

−S

S

2 S z 0

1
∆ − hω

1
Z

2 Me βM ∆

= =

hω , ∆ −

−S

as all the sums may be taken as extending from − S to S . Similarly

χ o

− + ( ω )= χ + − (

ω ), whereas χ ++ ( ω )= χ o

− ( ω ) = 0, from which we

−−