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

H.Y. Shik et al. / Nuclear Physics B 666 [FS] (2003) 337–360 355
Fig. 11. The three-dimensional model constructed by conditions 2J 2 = J 3 + J 4 and ˜ J 7 = 2J 11 .
the Hamiltonian. One can also combine those independent set of conditions together to
construct a new three-dimensional model. Of course, the more couplings one introduces,
the more relations one needs and more complicated models will be obtained. To end this
section, let us just mention three more of them:
3D model IV: 2J 8 = J˜
3 = J˜
4 , shown in Fig. 9;
3D model V: 2J 9 = J˜
5 , shown in Fig. 10;
3D model VI: 2J 11 = J˜
7 , shown in Fig. 11.
Let us specify lattice symmetry for models considered. Because these models are built
on the double layer, each unit cell is a tetragonal contains two spins. For models represented
by Figs. 4 and 6, the space symmetry group is C 4 if J ‖ 3 = J ‖ 4 .IfweletJ 3 = J 4 , then they
have all of the space symmetries for the tetragonal lattice: C 4 ,S 4 ,C 4h ,C 4v ,D 4 ,D 4h ,and
D 2d . For models represented by Figs. 5 and 7–11, the space symmetry group is C 4 for any
parameter combinations. If we have J 3 = J 4 , then they also possess all space symmetries
of the tetragonal lattice: C 4 ,S 4 ,C 4h ,C 4v ,D 4 ,D 4h ,andD 2d .
6. Summary
In summary, we obtained bond operator representation for the spin-1 dimers and applied
the bond operator method for any spin S to construct various two and three-dimensional
quantum spin models where the completely dimerized state, whose wave function is a
direct product of all spin dimers, is an eigenstate. We presented three two-dimensional