Z_3 symmetry-protected topological phases in the SU(3) AKLT model

We study ${\mathbb{Z}}_3$ symmetry-protected topological (SPT) phases in one-dimensional spin systems with $Z_3\times Z_3$ symmetry. We construct ground-state wave functions of the matrix product form for nontrivial ${\mathbb{Z}}_3$ phases and their parent Hamiltonian from a cocycle of the group cohomology $H^2(Z_3\times Z_3,U(1))$. 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 ${\mathbb{Z}}_N$ 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 $\theta$ controlling the ratio of the bilinear and biquadratic coupling constants. The nontrivial ${\mathbb{Z}}_3$ SPT phase is found for a range of the parameter $\theta$ including the point of vanishing biquadratic term ($\theta =0$) as well as the SU(3) AKLT point [$\theta =\arctan (2\mathrm{/}9)$]. A continuous phase transition to the SU(3) dimer phase takes place at $\theta \approx -0.027\pi$, with a central charge $c\approx 3.2$. For SU(3) symmetric cases we define string order parameters for the ${\mathbb{Z}}_3$ 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.