198
7 Quadrupole Contributions from Interface and Bulk
Then the following relation is proved using the unitary character of D (|D| = 1)
and Eq. (7.136),
ε ij k = | U (xyz, ij k) | =
D
T U (x
y
z
, ijk)
=
U (x
y
z
, ijk)
.
Suppose that pqr denote x , y , z while ij k denote x, y, z, the following equation
is derived.
D ip D jq D kr ε pqr = D ip D jq D kr
U (x
y
z
, pqr)
=
U (x
y
z
, ijk)
= | U (xyz, ij k) | = ε ij k .
(7.137)
A.3 Definition of Bulk Polarization
In Chap. 7 the bulk polarization P B
G is defined by Eq. (7.36),
P
B
G,p = i
0
−∞
dz
q,r,s
χ
D1,β
pqrs ((, ω 1 , ω 2 )k
β
T ,s (ω 1 ) + χ
D2,β
pqrs ((, ω 1 , ω 2 )k
β
T ,s (ω 2 )
−χ
Q,β
pqrs ((, ω 1 , ω 2 )
k
β
T ,s (ω 1 ) + k
β
T ,s (ω 2 )
· f
β
p (()f
β
q (ω 1 )f
β
r (ω 2 )L I,q (ω 1 )L I,r (ω 2 )E
α
I,q (ω 1 )E
α
I,r (ω 2 )
· exp
i
k
β
T ,z (ω 1 ) + k
β
T ,z (ω 2 ) − k
β
G,z (()
z
,
(7.36)
which is consistent to Refs. [21] and [17], while a slightly different expression P B
G
is found in Refs. [11] and [5]. In this Appendix, we clarify the difference between
the two expressions.
Some other literature [5, 11] employs a different expression, P B
G
, for the bulk
polarization,
P
B
G,p
= i
0
−∞
dz
q,r,s
χ
D1,β
pqrs ((, ω 1 , ω 2 )k
β
T ,s (ω 1 ) + χ
D2,β
pqrs ((, ω 1 , ω 2 )k
β
T ,s (ω 2 )
−χ
Q,β
pqrs ((, ω 1 , ω 2 )k
β
G,s (()
· f
β
p (()f
β
q (ω 1 )f
β
r (ω 2 )L I,q (ω 1 )L I,r (ω 2 )E
α
I,q (ω 1 )E
α
I,r (ω 2 )
· exp
i
k
β
T ,z (ω 1 ) + k
β
T ,z (ω 2 ) − k
β
G,z (()
z
.
(7.138)