Symmetry-protected trivial phases and quantum phase transitions in an anisotropic antiferromagnetic spin-1 biquadratic model

The ground-state phase diagram is obtained for an antiferromagnetic spin-1 anisotropic biquadratic model. With the help of symmetry and duality transformations, three symmetry-protected trivial phases and one dimerized symmetry-breaking phase are found. Local and nonlocal order parameters are identified to characterize these phases. Quantum phase transitions between the symmetry-protected phases belong to the Gaussian universality class with central charge c=1, and quantum phase transitions from the symmetry-protected trivial phases to the dimerized phase belong to the Ising universality class with central charge c=1/2. In addition, the model admits three characteristic lines of factorized ground states, which are located in the symmetry-protected trivial phases instead of a symmetry-breaking phase, in sharp contrast to other known cases.