LINEAR RESPONSE THEORY - Rare Earth Magnetism: Structures and Excitations - page 145

which, according to the fluctuation-dissipation theorem (3.2.18), should
be

∞ ∞

1
2 π

1
π

1

e −βhω χ BA ( ω ) d ( hω ) .
(3 . 3 . 7 b )
Introducing (3.3.5), the integration is straightforward, except in a nar-
row interval around ω =0,andweobtain

S BA ( t =0)= S BA ( ω ) dω =

1 −

−∞ −∞

γ

E α = E α

χ BA ( ω )
πβω

B A

α >< α | S BA ( t =0)= <α | | | α>n α + lim

γ→ 0 +

dω

−γ

αα

e −βhω with βhω in the limit ω

after replacing 1 0. A comparison of

this expression for S BA ( t =0)with(3 . 3 . 7 a ) shows that the last integral
has a definite value:

− →

γ

E α = E α

χ BA ( ω )
πβω

B A B A

α >< α | lim

γ→ 0 +

dω = <α | | | α>n α − .

−γ

αα

(3 . 3 . 8)
The use of the Kramers-Kronig relation (3.1.10), in the form of (3 . 2 . 11 d ),
for calculating χ BA (0) then gives rise to the extra contribution

γ

χ BA ( ω )
ω

1
π

χ BA ( el ) = lim
γ→ 0 +

dω (3 . 3 . 9)

−γ

to the reactive susceptibility at zero frequency, as anticipated in (3 . 3 . 6 b ).
The zero-frequency result, χ BA (0) = χ BA (0), as given by (3.3.6), is the
same as the conventional isothermal susceptibility (2.1.18) for the mag-
netic moments, where the elastic and inelastic contributions are respec-
tively the Curie and the Van Vleck terms. This elastic contribution is
discussed in more detail by, for instance, Suzuki (1971).
The results (3.3.4-6) show that, if the eigenstates of the Hamil-
tonian are discrete and the matrix-elements of the operators B and A
between these states are well-defined, the poles of χ BA ( z ) all lie on the
real axis. This has the consequence that the absorptive part χ BA ( ω )
(3.3.5) becomes a sum of δ -functions, which are only non-zero when hω
is equal to the excitation energies E α −

E α . Insuchasystem,nospon-
taneous transitions occur. In a real macroscopic system, the distribution
of states is continuous, and only the ground state may be considered as a
well-defined discrete state. At non-zero temperatures, the parameters of
the system are subject to fluctuations in space and time. The introduc-
tion of a non-zero probability for a spontaneous transition between the
'levels' α and α can be included in a phenomenological way by replac-
ing the energy difference E α − E α in (3.3.4) by ( E α − E α ) − i Γ α α ( ω ),