LINEAR RESPONSE THEORY - Rare Earth Magnetism: Structures and Excitations - page 155

the system to a small perturbative field h j ( t )= gµ B H j ( t ) (the Zeeman
term due to a stationary field is taken as included in H J ( J i )). From

(3.5.2), we may extract all terms depending on J i and collect them in
an effective Hamiltonian

H i , which determines the time-dependence of
J i . Transformed to the Heisenberg picture, this Hamiltonian is

)+ h i ( t ) . − J i ( t ) − J i · H i ( t )= H MF ( i, t ) J ( ij )( J j ( t ) − J j

j

(3 . 5 . 4)
We note that a given site i appears twice in the second term of (3.5.2),
and that the additional term

J i · h i has no consequences in the limit
when h i goes to zero. The differences J j ( t ) − J j ( t ) fluctuate in a vir-
tually uncorrelated manner from ion to ion, and their contribution to
the sum in (3.5.4) is therefore small. Thus, to a good approximation,
these fluctuations may be neglected, corresponding to replacing J j ( t )
in (3.5.4) by = i ). This is just the random-phase ap-

proximation (RPA), introduced in the previous section, and so called
on account of the assumption that J j ( t )

J j ( t ) (when j

may be described in
terms of a random phase-factor. It is clearly best justified when the
fluctuations are small, i.e. at low temperatures, and when many sites
contribute to the sum, i.e. in three-dimensional systems with long-range
interactions. The latter condition reflects the fact that an increase in the
number of (nearest) neighbours improves the resemblance of the sum in
(3.5.4) to an ensemble average. If we introduce the RPA in eqn (3.5.4),
the only dynamical variable which remains is J i ( t ), and the Hamiltonian
becomes equivalent to

− J j ( t )

H MF ( i ), except that the probing field h i ( t )isre-
placed by an effective field h eff
i

( t ). With J i ( ω ) defined as the Fourier

transform of J i ( t ) − J i , then, according to eqn (3.1.9),

= χ i ( ω ) h eff

J i ( ω ) ( ω ) ,

i

where the effective field is
h eff
i

( ω )= h i ( ω )+
j

J ( ij ) J j ( ω ) . (3 . 5 . 5)

This may be compared with the response determined by the two-ion
susceptibility functions of the system, defined such that

=
j

J i ( ω ) χ ( ij, ω ) h j ( ω ) . (3 . 5 . 6)

The two ways of writing the response should coincide for all h j ( ω ), which
implies that, within the RPA,
χ ( ij, ω )= χ i ( ω ) δ ij +

( ij ) χ ( j j, ω ) .

j J

(3 . 5 . 7)