LINEAR RESPONSE THEORY - Rare Earth Magnetism: Structures and Excitations - page 141

and is related to the response function introduced earlier by

h S BA ( t )

t ) .

K BA ( t )= i − S AB ( − (3 . 2 . 14)

The different response functions obey a number of symmetry rela-
tions, due to the invariance of the trace under a cyclic permutation of
the operators. To derive the first, we recall that the Hermitian conjugate
of an operator is defined by

B B † |

α > ) ∗ = <α |

( <α | | α>.

If we assume that a certain set of state vectors | α> constitutes a diag-

H 0 | α> = E α | α> , then it is straightforward to

onal representation, i.e.

show that

B ( t ) A ∗ = A † ( −t ) B † ,

leading to the symmetry relations

K BA ( t )= K B † A † ( t )

and

χ ∗ BA ( z )= χ B † A † ( z ∗ ) . − (3 . 2 . 15)

Another important relation is derived as follows:

Z Tr e −βH 0 e iH 0 t/h Be −iH 0 t/h A

1

B ( t ) A

=

Z Tr e iH 0 ( t + iβh ) /h Be −iH 0 ( t + iβh ) /h e −βH 0 A

1

=

Z Tr e −βH 0 A B ( t + iβh ) =

1

A B ( t + iβh )

= ,

implying that

S BA ( t )= S AB ( − t − iβh ) . (3 . 2 . 16)

In any realistic system which, rather than being isolated, is in con-
tact with a thermal bath at temperature T , the correlation function
S BA ( t ) vanishes in the limits t

→±∞

, corresponding to the condition

A . If we further assume that S BA ( t )isanan-
alytic function in the interval |t 2 |≤β of the complex t -plane, then the
Fourier transform of (3.2.16) is

B ( t = ±∞ ) A = B

S BA ( ω )= e βhω S AB ( − ω ) , (3 . 2 . 17)

which is usually referred to as being the condition of detailed balance .
Combining this condition with the expressions (3.2.12) and (3.2.14), we