LINEAR RESPONSE THEORY - Rare Earth Magnetism: Structures and Excitations - page 144

or equivalently

χ AB ( − ω ) = lim

→ 0 +

χ AB ( − ω + i )

(3 . 3 . 4 b )

A B

α >< α |

<α | | | α>

= lim
→ 0 +

( n α − n α ) .

E α − E α + hω − ih

αα

An interchange of α and α shows this expression to be the same as
(3 . 3 . 4 a ), with replaced by

. The application of Dirac's formula then
yields the absorptive part of the susceptibility (3 . 2 . 11 b )as
χ BA ( ω )= π
αα

−

A | α> ( n α −n α ) δ hω − ( E α −E α ) B | α >< α |

<α|

(3 . 3 . 5)
(equal to K BA ( ω ) / 2 i in accordance with (3.2.12)), whereas the reactive
part (3 . 2 . 11 a )is

E α = E α

B A

α >< α |

<α | | | α>

χ BA ( ω )= n α )+ χ BA ( el ) δ ω 0 , ( n α −

E α − E α − hω

αα

(3 . 3 . 6 a )

where

ω + i = 1if

ω =0

i

δ ω 0 ≡ lim

→ 0 +

0if ω =0 ,

and the elastic term χ BA ( el ), which only contributes in the static limit
ω =0,is

E α = E α

χ BA ( el )= β . (3 . 3 . 6 b )

B A B A

α >< α |

<α | | | α>n α −

αα

We remark that χ BA ( ω )and χ BA ( ω ) are often referred to respectively as
the real and the imaginary part of χ BA ( ω ). This terminology is not valid
in general, but only if the matrix-element products are real, as they are
if, for instance, B = A † . The presence of the elastic term in the reactive
response requires some additional consideration. There are no elastic
contributions to K BA ( t ), nor hence to χ BA ( ω ), because n α − 0

if E α = E α . Nevertheless, the appearance of an extra contribution at
ω = 0, not obtainable directly from K BA ( t ), is possible because the
energy denominator in (3.3.4) vanishes in the limit

n α ≡

0, when
E α = E α . In order to derive this contribution, we consider the equal-
time correlation function

| ω + i |→

( B

B

)( A

A

S BA ( t =0)= − − )

=
αα

(3 . 3 . 7 a )

B A B A

α >< α |

<α | | | α>n α −