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Static and dynamical properties of systems are ruled by the symmetry. So making clear the symmetry is important.

A spin-1 one dimensional bilinear-biquadratic model has been an interesting subject from the theoretical point of view. This model with only one parameter has a rich ground state phase diagram. Recently, a possibility of experimental realization of this model in optical lattice systems is proposed [1]. Kolezhuk discussed the ground state of this model with single ion anisotropy (D) using the nonlinear sigma model in some dimensional systems [2]. We have determined this ground state phase diagram by numerical calculations using the levelspectroscopy method and twisted boundary condition [3]. Then we find a phase boundary between the XY2 phase and the Neel phase at θ=-π/2. In D = 0 case, this model with θ=-π/2 has SU(3) symmetry [4]. However, in finite D case, this does not have this SU(3) and standard spin SU(2) symmetry.

We show that the one-dimensional spin-1 biquadratic (BQ) model with single ion anisotropy (θ=-π/2, D:finite) has an additional SU(2) symmetry for periodic, open and twisted boundary conditions. We can understand degeneracies in our previous numerical results using this symmetry.

[1] A. Imambekov, M. Lukin, and E. Demler, Phys. Rev. A vol.68, 063602 (2003).
[2] A. Kolezhuk, Phys. Rev. B vol.78, 144428 (2008).
[3] K. Takahashi, K. Hijii, and K. Nomura, in preparation.
[4] I. Affleck, Nucl. Phys. B vol.265, 409 (1986).