LINEAR RESPONSE THEORY - Rare Earth Magnetism: Structures and Excitations - page 156

This self-consistent equation may be solved under various conditions.
For convenience, we shall consider here only the uniform case of a ferro-
or paramagnet, where H MF ( i ) is the same for all the ions, i.e. J i = J

and χ i ( ω )= χ o ( ω ), in which case we get the final result
χ ( q ,ω )= 1

( q ) − 1 χ o ( ω ) . χ o ( ω )

− J (3 . 5 . 8)

Here 1 is the unit matrix, and we have used the Fourier transform (3.4.2)
of J ( ij )

( q )=
j

( ij ) e −i q · ( R i − R j ) .

J J (3 . 5 . 9)

In the RPA, the effects of the surrounding ions are accounted for
by a time-dependent molecular field, which self-consistently enhances
the response of the isolated ions. The above results are derived from a
kind of hybrid MF-RPA theory, as the single-ion susceptibility χ o

i ( ω )is
still determined in terms of the MF expectation values. A self-consistent
RPA theory might be more accurate but, as we shall see, gives rise to fur-
ther problems. At high temperatures (or close to a phase transition), the
description of the dynamical behaviour obtained in the RPA is incom-
plete, because the thermal fluctuations introduce damping effects which
are not included. However, the static properties may still be described
fairly accurately by the above theory, because the MF approximation is
correct to leading order in β =1 /k B T .
The RPA, which determines the excitation spectrum of the many-
body system to leading order in the two-ion interactions, is simple to
derive and is of general utility. Historically, its applicability was ap-
preciated only gradually, in parallel with the experimental study of a
variety of systems, and results corresponding to eqn (3.5.8) were pre-
sented independently several times in the literature in the early 1970s
(Fulde and Perschel 1971, 1972; Haley and Erdos 1972; Purwins et al .
1973; Holden and Buyers 1974). The approach to this problem in the
last three references is very similar, and we will now present it, following
most closely the account given by Bak (1974).
We start by considering the MF Hamiltonian defined by (3.5.3). The
basis in which H MF ( i ) is diagonal is denoted | ν i > ; ν =0 , 1 ,..., 2 J ,

and we assume that H MF ( i ) is the same for all the ions:

H MF ( i ) | ν i > = E ν | ν i >, (3 . 5 . 10)

with E ν independent of the site index i . The eigenvalue equation defines
the standard-basis operators

a νµ ( i )= | ν i >< µ i | , (3 . 5 . 11)