Constructing soluble quantum spin models - Department of Physics ...

H.Y. Shik et al. / Nuclear Physics B 666 [FS] (2003) 337–360 357
|00〉=√
2
3 |p 0〉− 1 √
3
|s〉,
|0↓〉 = 1 √
2
(
|p−1 〉−|t −1 〉 ) ,
|↓↑〉 = 1 √
6
|p 0 〉+ 1 √
2
|t 0 〉+ 1 √
3
|s〉,
|↓0〉= 1 √
2
(
|p−1 〉+|t −1 〉 ) ,
|↓↓〉 = |p −2 〉.
(A.7)
(A.8)
(A.9)
(A.10)
(A.11)
Introducing the matrix representations of Spin-1 operators
S z =
[ ]
1 0 0
0 0 0 , S + = √ 2
0 0 −1
[ ] 0 1 0
0 0 1 , S − = √ 2
0 0 0
[ 0 0
] 0
1 0 0 ,
0 1 0
the bond operator representations of the original spin operator S z and S + are given by
S z = ẽ T · (S z ⊗ I ) · ẽ = e T · M T · (S z ⊗ I ) · M · e
= p † 2 p 2 + 1 2 p† 1 p 1 − 1 2 p† 1 t 1 − 1 √
3
p † 0 t 0 − 1 2 p† −1 p −1 − 1 2 p† −1 t −1 − p † −2 p −2
(A.12)
− 1 2 t† 1 p 1 + 1 2 t† 1 t 1 − 1 √
3
t † 0 p 0 − 1 2 t† −1 p −1 − 1 2 t† −1 t −1 −
where I is an 3 × 3 identity matrix, and
S + = ẽ T · (S + ⊗ I ) · ẽ
√
2 ( t
†
3
0 s + ) s† t 0 ,
(A.13)
= p † 2 p 1 + p † 2 t 1 +
√
3
2 p† 1 p 0 + 1 √
2
p † 1 t 0 +
√
3
2 p† 0 p −1 + 1 √
6
p † 0 t −1 + p † −1 p −2
− 1 √
6
t † 1 p 0 + 1 √
2
t † 1 t 0 − 1 √
2
t † 0 p −1 + 1 √
2
t † 0 t −1 − t † −1 p −2
+ 2 √
3
( t
†
1 s − s† t −1
) .
(A.14)
Similarly,
S ′ z = ẽ T · (I
⊗ S z) · ẽ = e T · M T · (I
⊗ S z) · M · e
= p † 2 p 2 + 1 2 p† 1 p 1 + 1 2 p† 1 t 1 + 1 √
3
p † 0 t 0 − 1 2 p† −1 p −1 + 1 2 p† −1 t −1 − p † −2 p −2
+ 1 2 t† 1 p 1 + 1 2 t† 1 t 1 + √ 1 t † 0 p 0 + 1
3 2 t† −1 p −1 − 1 √
2
2 t† −1 t (
−1 + t
†
0
3
s + ) s† t 0 ,
(A.15)