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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)
H.Y. Shik et al. / Nuclear Physics B 666 [FS] (2003) 337–360 341 The triplets |t α 〉 with α =±1and0beingthez-component of the total spin are defined as |t 1 〉≡t † 1 |vac〉= 1 (16) √ 2 ( |0↑〉 − |↑0〉 ) , (17) |t 0 〉≡t † 0 |vac〉= 1 √ 2 ( |↓↑〉 − |↑↓〉 ) , (18) |t −1 〉≡t † −1 |vac〉= 1 √ 2 ( |↓0〉−|0↓〉 ) . The fivefold multiplets pentad |p β 〉 with β =±2, ±1, and 0 are defined as |p 2 〉≡p † 2 |vac〉 = |↑↑〉, (19) (20) |p 1 〉≡p † 1 |vac〉= 1 √ 2 ( |0↑〉 + |↑0〉 ) , (21) |p 0 〉≡p † 0 |vac〉= 1 √ 6 ( |↑↓〉 + 2|00〉 + |↓↑〉 ) , (22) Conversely, |p −1 〉≡p † −1 |vac〉= 1 √ 2 ( |0↓〉 + |↓0〉 ) , (23) |p −2 〉≡p † −2 |vac〉=|↓↓〉. S z = p † 2 p 2 + 1 2 p† 1 p 1 − 1 2 p† −1 p −1 − p † −2 p −2 + 1 2 t† 1 t 1 − 1 2 t† −1 t −1 − 1 2 ( p † 1 t 1 + t † 1 p ) 1 ( 1 − √3 p † 0 t 0 + t † 0 p 0 ) 1( − p † −1 2 t −1 + t † −1 p ) −1 − √ 2 ( t † 0 3 s + ) s† t 0 , S ′ z = p † 2 p 2 + 1 2 p† 1 p 1 − 1 2 p† −1 p −1 − p † −2 p −2 + 1 2 t† 1 t 1 − 1 2 t† −1 t −1 + 1 2 + ( p † 1 t 1 + t † 1 p ) 1 ( 1 + √3 p † 0 t 0 + t † 0 p 0 √ 2 ( t † 3 0 s + s† t 0 ) , S + = p † 2 p 1 + p † −1 p −2 + ) 1( + p † 2 −1 t −1 + t † −1 p ) −1 √ 3 ( p † 1 2 p 0 + p † 0 p ) 1 ( −1 + √2 t † 1 t 0 + t † 0 t ) −1 + p † 2 t 1 − t † −1 p −2 + √ 1 ( p † 1 t 0 − t † 0 p ) 1 ( −1 + √6 p † 0 t −1 − t † 1 p ) 0 2 + 2 √ 3 ( t † 1 s − s† t −1 ) ,
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