LINEAR RESPONSE THEORY - Rare Earth Magnetism: Structures and Excitations - page 154

We note that there are corrections to U , given by (3.4.15), of second
order in Φ. The low-temperature properties, as determined by (3.4.14),
(3.4.16), and (3.4.17), agree with the systematic expansion performed by
Dyson (1956), including the leading-order dynamical correction of fourth
power in T (in the cubic case), except for a minor kinematic correction
which is negligible for S ≥ 1.

3.5 The random-phase approximation
Earlier in this chapter, we have demonstrated that many experimentally
observable properties of solids can be expressed in terms of two-particle
correlation functions. Hence it is of great importance to be able to cal-
culate these, or the related Green functions, for realistic systems. We
shall therefore consider the determination of the generalized susceptibil-
ity for rare earth magnets, using the random-phase approximation which
was introduced in the last section, and conclude the chapter by apply-
ing this theory to the simple Heisenberg model, in which the single-ion
anisotropy is neglected.

3.5.1 The generalized susceptibility in the RPA
The starting point for the calculation of the generalized susceptibility
is the (effective) Hamiltonian for the angular momenta which, as usual,
we write as a sum of single- and two-ion terms:

=

i = j J

− 2 H

i H J ( J i )

( ij ) J i · J j . (3 . 5 . 1)

For our present purposes, it is only necessary to specify the two-ion
part and, for simplicity, we consider only the Heisenberg interaction. As
in Section 2.2, we introduce the thermal expectation values

J i in the

Hamiltonian, which may then be written

=

i = j J

− 2 H

i H MF ( i )

( ij )( J i − J i ) · ( J j − J j ) , (3 . 5 . 2)

where

− J i − 2 J i ·

H MF ( i )= H J ( J i ) J ( ij ) J j . (3 . 5 . 3)

j

From the mean-field Hamiltonians as

before. The Hamiltonian (3.5.3) also determines the dynamic suscepti-
bility of the i th ion, in the form of a Cartesian tensor χ o

H MF ( i ), we may calculate J i

i ( ω ), according
to eqns (3.3.4-6), with A and B set equal to the angular-momentum
components J iα . We wish to calculate the linear response J i ( t ) of