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

H.Y. Shik et al. / Nuclear Physics B 666 [FS] (2003) 337–360 343
h 4 =
K−1,L−1
∑
k,l=1,α≠β
1
4 t† αk,l t (
βk,l tαk+1,l t † βk+1,l − t† αk+1,l t βk+1,l
+ t αk,l+1 t † βk,l+1 − t† αk,l+1 t βk,l+1)
.
(29)
In Eq. (26), h 1 is diagonal in boson number operators. All the remaining four-operator
terms (h 2 ,h 3 and h 4 ) give zero when they act on the completely dimerized state ψ D except
the terms t † αk,l s k,lt † αk+1,l s k+1,l and t † αk,l s k,lt † αk,l+1 s k,l+1 in h 2 .Sincet † αi s it † αj s j will destroy
a singlet and create a triplet on both sites i and j, a new state is created, i.e.,
t † αi s it † αj s j ψ D =[1, 1 ′ ]···(i, i ′ )(j, j ′ ) ···[KL,K ′ L ′ ].
(30)
So to make the completely dimerized state ψ D be an eigenstate, the coefficient of h 2
must be zero, which leads to the condition 2J 2 = J 3 + J 4 .
If J 2 = J 3 = J 4 , the Hamiltonian becomes
∑K,L
H = 2J 1
k,l=1
K−1,L−1
∑
+ 2J 2
(
− 3 4 s† k,l s k,l + 1 )
4 t† αk,l t αk,l
k,l=1,α≠β
t † αk,l t βk,l(
tαk+1,l t † βk+1,l + t αk,l+1t † βk,l+1
− t † αk+1,l t βk+1,l − t † αk,l+1 t βk,l+1)
.
(31)
By inspecting the above simplified Hamiltonian, we immediately see that the eigenvalue
of the completely dimerized state E D is −2J 1 × 3/4 × N where N = KL is the total
number of singlets on the double layer. Moreover, in the large J 1 limit, the completely
dimerized state is the ground state and the first excited state should be a single magnon
created by breaking a dimer at any site, with eigenvalue E D + 2J 1 . Thus, without solving
the Hamiltonian explicitly, useful information are already revealed by the bond operator
representation of the model.
For general spin-S case, similar approach will lead to the same conclusions. In order not
to bore the readers, the derivations is not shown here. Instead, we present the derivations
for the spin-1 case for a three-dimensional model in Section 5 and leave the derivations for
general spin-S case to Appendix B.
4. Two-dimensional exactly soluble model
In this section, we construct other exactly soluble models in two dimensions by applying
the bond operator method. Obviously, there could be a lot. A simple case would be a natural
continuation of the net spin model where one connects a set of ladders by inter-ladder
couplings J 5 ,J 6 ,J 7 ,andJ 8 as shown in Fig. 2.