Lattice Modules
217
the sum of all the n kicks over the whole system is
⎛
⎜
⎜
⎜
⎝
x f
a f
y f
b f
⎞
⎟
⎟
⎟
⎠
=
n−1
m=0
R
(n − m) μ x − φ x
(n − m) μ y − φ y
◦
⎛
⎜
⎜
⎜
⎜
⎝
hds/
√
β x
x (−α x x + a)
T a,k,l,m,n x
k a
l y
m b
n
√
β x hds/β y
x (−α y y + b)
T b,k,l,m,n x
k a
l y
m b
n
⎞
⎟
⎟
⎟
⎟
⎠
◦ R
mμ x + φ x
mμ y + φ y
=
n−1
m=0
R
−2mπ/n − φ x
−2mπ/n − φ y
◦
⎛
⎜
⎜
⎜
⎜
⎝
hds/
√
β x
x (−α x x + a)
T a,k,l,m,n x
k a
l y
m b
n
√
β x hds/β y
x (−α y y + b)
T b,k,l,m,n x
k a
l y
m b
n
⎞
⎟
⎟
⎟
⎟
⎠
◦ R
2mπ/n + φ x
2mπ/n + φ y
.
The last step uses the fact that nμ x = nμ y = 2π. Every term in the above
expression can be written in the form of
n−1
m=0
cos
l
2mπ
n
+ φ j
sin
3−l
2mπ
n
+ φ k
,
where l = 0, 1, 2, 3 and j, k = {x, y}. Using the relations
cos φ =
e
iφ + e
−iφ
2
,
sin φ =
e
iφ
− e
−iφ
2i
,
we have
n−1
m=0
cos
l
2mπ
n
+ φ j
sin
3−l
2mπ
n
+ φ k
=
n−1
m=0
e
i(2mπ/n+φj ) + e
−i(2mπ/n+φj)
2
l
e
i(2mπ/n+φj )
− e
−i(2mπ/n+φj )
2i
3−l
.
Dropping the common parts in each sum, which are functions of φ x and φ y ,
there are only four kinds of sums
n−1
m=0
e
i6mπ/n =
1 − e
in6π/n
1 − e i6π/n = 0,
n−1
m=0
e
−i6mπ/n =
1 − e
−in6π/n
1 − e −i6π/n = 0,
n−1
m=0
e
i2mπ/n =
1 − e
in2π/n
1 − e i2π/n = 0,
n−1
m=0
e
−i2mπ/n =
1 − e
−in2π/n
1 − e −i2π/n = 0,