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

340 H.Y. Shik et al. / Nuclear Physics B 666 [FS] (2003) 337–360
2. Bond operator
2.1. Spin-1/2 bond operator
In this section, the so called “bond operator” representation is discussed. For a dimer
consisting of two spins S and S ′ , instead of using the spin operator representation for S α
and S α ′ ,whereα = x,y,z, an alternative representation, known as the bond operator could
also be used. Such representation was first introduced by Sachdev and Bhatt in 1990 [1],
in which the original two spin-1/2 operators, S and S ′ , are represented by four bosonic
operators s † ,t x † ,t y † and t z † , which create the singlet state |s〉 and triplet states |t x 〉, |t y 〉,and
|t z 〉 out of the vacuum state |vac〉, respectively,
|s〉≡s † |vac〉= 1 ( )
√ |↑↓〉 − |↓↑〉 , (9)
2
|t x 〉≡t † −1 ( )
x |vac〉= √ |↑↑〉 − |↓↓〉 , (10)
2
|t y 〉≡t y † |vac〉= i ( )
√ |↑↑〉 + |↓↓〉 , (11)
2
(12)
|t z 〉≡t † z |vac〉= 1 √
2
(
|↑↓ + |↓↑〉
)
,
and conversely
S α = 1 s
2( † t α + t α † s − iɛ αβγ t † β t )
γ ,
(13)
S ′ α = 1 2(
−s † t α − t † α s − iɛ αβγ t † β t γ
)
,
where α, β, andγ take the values of x,y,andz. The Levi-Civita symbol ɛ is the totally
antisymmetric tensor, and all repeated indices are summed over. For physical state there is
either a singlet or triplet, so the following constraint should be imposed,
s † s + t † α t α = 1.
(14)
2.2. Spin-1 bond operator
Similar to the spin-1/2 case, we can construct the spin-1 bond operators. We leave their
constructions to Appendix A. The original two spin-1 operators, S and S ′ , are represented
by nine bosonic operators s † , t † 1 , t† 0 , t† −1 , p† 2 , p† 1 , p† 0 , p† −1 and p† −2
which create the
singlet state |s〉, triplet states |t α 〉, and pentad states |p β 〉 out of the vacuum state |vac〉,
respectively.
The singlet |s〉 is defined as
|s〉≡s † |vac〉= 1 √
3
(
|↑↓〉 − |00〉 + |↓↑〉
)
.
(15)