CRYSTAL-FIELD EXCITATIONS IN THE LIGHT RARE EARTHS - Rare Earth Magnetism: Structures and Excitations - page 326

expressing the angular-momentum components as linear combinations
of the standard-basis operators introduced by eqn (3.5.11). These are
not Bose operators, so the 'contractions' determined by the commutators
of the different operators are not c -numbers, but operators which give
rise to new contractions. In the singlet-singlet Ising model, the result is

G ( ii, iω n )= G ( iω n )= g ( iω n )

( n 0 + n 1 ) g ( iω n ) K ( iω n )+ 1

g ( iω n ) K ( iω n ) u ( n, n ) g ( iω n )

β
n

1
n 01

−

+ ··· (7 . 2 . 5 a )

with
u ( n, n )= g ( iω n )
M α

n 01 β.
(7 . 2 . 5 b )
The sum over the Matsubara frequencies may be transformed into an
integral over real frequencies, but it may be advantageous to keep the
frequency sum in numerical calculations. Before proceeding further, we
must clarify a few points. The first is that

n 0 + n 1 −

g ( iω n )
M α

( ihω n ) 2 +∆ 2
2∆ 2

+ 2 +

H 1 cannot, in general, be

consider as being 'small' compared to H 0 . However, each time a term

involving the two-ion coupling is summed over q , we effectively gain a
factor 1 /Z ,where Z is the co-ordination number. Hence, if we use 1 /Z
as a small expansion parameter, the order of the different contributions
may be classified according to how many q -summations they involve.
In the equation above, K ( iω n ) is derived from one summation over q ,
as in (5.6.17), so the series can be identified as being equivalent to an
expansion in 1 /Z . The second point to realize is that it is of importance
to try to estimate how the expansion series behaves to infinite order. A
truncation of the series after a finite number of terms will produce a re-
sponse function with incorrect analytical properties. If we consider the
corresponding series determining G ( q ,iω n ), it is clear that any changes
in the position of the poles, i.e. energy changes and linewidth phenom-
ena, are reflected throughout the whole series, whereas a (small) scaling
of the amplitude of the response function, which might be determined by
the first few terms, is not particularly interesting. In other words, what
we wish to determine is the first- (or higher-) order correction in 1 /Z to
the denominator of the Green function, i.e. to determine the self-energy
Σ( q ,iω n ), defined by

g ( iω n )

1+ g ( iω n ) J αα ( q )+Σ( q ,iω n ) ,

G ( q ,iω n )= (7 . 2 . 6)

assuming the MF-RPA response function to be the starting point. A
systematic prescription for calculating the Green function to any finite