LINEAR RESPONSE THEORY - Rare Earth Magnetism: Structures and Excitations - page 137

χ BA ( z ) has no poles in the upper half-plane. If this were not the case,
the response

B ( t )

to a small disturbance would diverge exponentially
as a function of time.
The absence of poles in χ BA ( z ), when z 2 is positive, leads to a rela-
tion between the real and imaginary part of χ BA ( ω ), called the Kramers-
Kronig dispersion relation .If χ BA ( z ) has no poles within the contour
C , then it may be expressed in terms of the Cauchy integral along C by

the identity

χ BA ( z )
z −

1
2 πi

dz . χ BA ( z )=

z

C

The contour is chosen to be the half-circle, in the upper half-plane,

centred at the origin and bounded below by the line parallel to the z 1 -
axis through z 2 = ,and z is a point lying within this contour. Since
φ BA ( t ) is a bounded function in the domain > 0, then χ BA ( z )must
go to zero as

C

, whenever z 2 > 0. This implies that the part
of the contour integral along the half-circle must vanish when its radius
goes to infinity, and hence

| z |→∞

∞ + i

χ BA ( ω + i )
ω + i −

1
2 πi

d ( ω + i ) . χ BA ( z ) = lim

→ 0 +

z

−∞ + i

Introducing z = ω + i and applying 'Dirac's formula':

1 1

ω − ω + iπδ ( ω − ω ) ,

lim
→ 0 + i = P ω − ω −

0 + , we finally obtain the Kramers-Kronig rela-

in taking the limit →

tion ( P denotes the principal part of the integral):

∞

χ BA ( ω )
ω −

χ BA ( ω )= 1 dω ,

iπ P

(3 . 1 . 10)

ω

−∞

which relates the real and imaginary components of χ ( ω ).

3.2 Response functions
In this section, we shall deduce an expression for the response function
φ BA ( t ), in terms of the operators B and A and the unperturbed Hamil-
tonian H 0 . In the preceding section, we assumed implicitly the use of

the Schrodinger picture. If instead we adopt the Heisenberg picture,
the wave functions are independent of time, while the operators become
time-dependent. In the Heisenberg picture, the operators are

B ( t )= e iHt/h Be −iHt/h ,

(3 . 2 . 1)