Z3 symmetry-protected topological phases in the SU(3) AKLT model

We study Z(3) symmetry-protected topological (SPT) phases in one-dimensional spin systems with Z(3) x Z(3) symmetry. We construct ground-state wave functions of the matrix product form for nontrivial Z(3) phases and their parent Hamiltonian from a cocycle of the group cohomology H-2(Z(3) x Z(3), U(1)). The Hamiltonian is an SU(3) version of the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, consisting of bilinear and biquadratic terms of su(3) generators in the adjoint representation. A generalization to the SU(N) case, the SU(N) AKLT Hamiltonian, is also presented, which realizes nontrivial Z(N) SPT phases. We use the infinite-size variant of the density matrix renormalization group (iDMRG) method to determine the ground-state phase diagram of the SU(3) bilinear-biquadratic model as a function of the parameter. controlling the ratio of the bilinear and biquadratic coupling constants. The nontrivial Z(3) SPT phase is found for a range of the parameter. including the point of vanishing biquadratic term (theta = 0) as well as the SU(3) AKLT point [theta = arctan(2/9)]. A continuous phase transition to the SU(3) dimer phase takes place at theta approximate to -0.027 pi, with a central charge c approximate to 3.2. For SU(3) symmetric cases, we define string order parameters for the Z3 SPT phases in a similar way to the conventional Haldane phase. We propose simple spin models that effectively realize the SU(3) and SU(4) AKLT models.